continuous phasetransitions 12 - cornell...

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Continuous phasetransitions 12

12.1 Universality 303

12.2 Scale invariance 310

12.3 Examples of critical points316

Fig. 12.1 The Ising model at Tc,the critical temperature separating themagnetized phase T < Tc from thezero-magnetization phase T > Tc. Thewhite and black regions represent pos-itive and negative magnetizations s =±1. Unlike the abrupt transitions stud-ied in Chapter 11, here the magnetiza-tion goes to zero continuously as T →

Tc from below.

Continuous phase transitions are fascinating. As we raise the tempera-ture of a magnet, the magnetization will vanish continuously at a criticaltemperature Tc. At Tc we observe large fluctuations in the magnetiza-tion (Fig. 12.1); instead of picking one of the up-spin, down-spin, orzero-magnetization states, this model magnet at Tc is a kind of fractal1

1The term fractal was coined to de-scribe sets which have characteris-tic dimensions that are not integers;it roughly corresponds to non-integerHausdorff dimensions in mathematics.The term has entered the popular cul-ture, and is associated with strange,rugged sets like those depicted in thefigures here.

blend of all three. This fascinating behavior is not confined to equi-librium thermal phase transitions. Figure 12.2 shows the percolationtransition. An early paper which started the widespread study of thistopic [82] described punching holes at random places in a conductingsheet of paper and measuring the conductance. Their measurement fellto a very small value as the number of holes approached the criticalconcentration, because the conducting paths were few and tortuous justbefore the sheet fell apart. Thus this model too shows a continuous tran-sition: a qualitative change in behavior at a point where the propertiesare singular but continuous.Many physical systems involve events of a wide range of sizes, the

largest of which are often catastrophic. Figure 12.3(a) shows the en-ergy released in earthquakes versus time during 1995. The Earth’s crustresponds to the slow motion of the tectonic plates in continental driftthrough a series of sharp, impulsive earthquakes. The same kind ofcrackling noise arises in many other systems, from crumpled paper [61]to Rice Krispies

TM

[72], to magnets [130]. The number of these impul-sive avalanches for a given size often forms a power law D(s) ∼ s−τ

over many decades of sizes (Fig. 12.3(b)). In the last few decades, it hasbeen recognized that many of these systems can also be studied as criti-cal points—continuous transitions between qualitatively different states.We can understand most of the properties of large avalanches in thesesystems using the same tools developed for studying equilibrium phasetransitions.The renormalization-group and scaling methods we use to study these

critical points are deep and powerful. Much of the history and practice inthe field revolves around complex schemes to implement these methodsfor various specific systems. In this chapter, we will focus on the keyideas most useful in exploring experimental systems and new theoreticalmodels, and will not cover the methods for calculating critical exponents.In Section 12.1 we will examine the striking phenomenon of universal-

ity: two systems, microscopically completely different, can exhibit pre-

Copyright Oxford University Press 2006 v1.0 --

302 Continuous phase transitions

Fig. 12.2 Percolation transition.A percolation model on the computer,where bonds between grid points areremoved rather than circular holes.Let the probability of removing abond be 1−p; then for p near one (noholes) the conductivity is large, butdecreases as p decreases. After enoughholes are punched (at pc = 1/2 forthis model), the biggest cluster justbarely hangs together, with holes onall length scales. At larger proba-bilities of retaining bonds p = 0.51,the largest cluster is intact with onlysmall holes (bottom left); at smallerp = 0.49 the sheet falls into small frag-ments (bottom right; shadings denoteclusters). Percolation has a phasetransition at pc, separating a con-nected phase from a fragmented phase(Exercises 2.20 and 12.25).



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Fig. 12.3 Earthquake sizes.(a) Earthquake energy release in1995 versus time. This time series,when sped up, sounds like cracklingnoise [72]. (b) Histogram of thenumber of earthquakes in 1995 asa function of their size S. Noticethe logarithmic scales; the smallestearthquakes shown are a million timessmaller and a thousand times moreprobable than the largest earthquakes.The fact that this distribution iswell described by a power law is theGutenberg–Richter law ∼ S−2/3.

cisely the same critical behavior near their phase transitions. We willprovide a theoretical rationale for universality in terms of a renormaliza-tion-group flow in a space of all possible systems.

Fig. 12.4 The Burridge–Knopoffmodel of earthquakes, with theearthquake fault modeled by blockspulled from above and sliding with fric-tion on a surface below. It was later re-alized by Carlson and Langer [26] thatthis model evolves into a state with alarge range of earthquake sizes even forregular arrays of identical blocks.

In Section 12.2 we will explore the characteristic self-similar structuresfound at continuous transitions. Self-similarity is the explanation forthe fractal-like structures seen at critical points: a system at its criticalpoint looks the same when rescaled in length (and time). We will showthat power laws and scaling functions are simply explained from theassumption of self-similarity.Finally, in Section 12.3 we will give an overview of the wide variety of

types of systems that are being understood using renormalization-groupand scaling methods.

12.1 Universality

Quantitative theories of physics are possible because macroscale phe-nomena are often independent of microscopic details. We saw in Chap-ter 2 that the diffusion equation was largely independent of the un-derlying random collision processes. Fluid mechanics relies upon theemergence of simple laws—the Navier-Stokes equations—from complexunderlying microscopic interactions; if the macroscopic fluid motions de-pended in great detail on the shapes and interactions of the constituentmolecules, we could not write simple continuum laws. Ordinary quan-tum mechanics relies on the fact that the behavior of electrons, nuclei,and photons are largely independent of the details of how the nucleusis assembled—non-relativistic quantum mechanics is an effective theorywhich emerges out of more complicated unified theories at low energies.High-energy particle theorists developed the original notions of renor-malization in order to understand how these effective theories emerge inrelativistic quantum systems. Lattice quantum chromodynamics (simu-lating the strong interaction which assembles the nucleus) is useful onlybecause a lattice simulation which breaks translational, rotational, andLorentz symmetries can lead on long length scales to a behavior thatnonetheless exhibits these symmetries. In each of these fields of physics,



many different microscopic models lead to the same low-energy, long-wavelength theory.

Fig. 12.5 A medium-sizedavalanche (flipping 282 785 do-mains) in a model of avalanches andhysteresis in magnets [130] (see Exer-cises 8.17, 12.26 and Fig. 12.11). Theshading depicts the time evolution: theavalanche started in the dark regionin the back, and the last spins to flipare in the upper, front region. Thesharp changes in shading are real, andrepresent sub-avalanches separatedby times where the avalanche almoststops (see Fig. 8.19).

The behavior near continuous transitions is unusually independent ofthe microscopic details of the system—so much so that we give a newname to it, universality. Figure 12.6(a) shows that the liquid and gasdensities ρℓ(T ) and ρg(T ) for a variety of atoms and small moleculesappear quite similar when rescaled to the same critical density and tem-perature. This similarity is partly for mundane reasons: the interactionsbetween the molecules is roughly the same in the different systems up tooverall scales of energy and distance. Hence argon and carbon monoxidesatisfy

ρCO(T ) = AρAr(BT ) (12.1)

for some overall changes of scale A, B. However, Fig. 12.6(b) showsa completely different physical system—interacting electronic spins inmanganese fluoride, going through a ferromagnetic transition. The mag-netic and liquid–gas theory curves through the data are the same if weallow ourselves to not only rescale T and the order parameter (ρ and M ,respectively), but also allow ourselves to use a more general coordinatechange

ρAr(T ) = A(M(BT ), T ) (12.2)

which untilts the axis.2 Nature does not anticipate our choice of ρ and

2Here B = TMc /T ℓg

c is as usualthe rescaling of temperature andA(M,T ) = a1M + a2 + a3T =(ρcρ0/M0)M + ρc(1 + s)− (ρcs/T

ℓgc )T

is a simple shear coordinate transfor-mation from (ρ, T ℓg) to (M,TM ). Asit happens, there is another correctionproportional to (Tc − T )1−α, whereα ∼ 0.1 is the specific heat expo-nent. It can also be seen as a kindof tilt, from a pressure-dependent ef-fective Ising-model coupling strength.It is small for the simple molecules inFig. 12.6(a), but significant for liquidmetals [49]. Both the tilt and this 1−αcorrection are subdominant, meaningthat they vanish faster as we approachTc than the order parameter (Tc−T )β .

T for variables. At the liquid–gas critical point the natural measureof density is temperature dependent, and A(M,T ) is the coordinatechange to the natural coordinates. Apart from this choice of variables,this magnet and these liquid–gas transitions all behave the same at theircritical points.This would perhaps not be a surprise if these two phase diagrams

had parabolic tops; the local maximum of an analytic curve generically3

3The term generic is a mathematicalterm which roughly translates as ‘ex-cept for accidents of zero probability’,like finding a function with zero secondderivative at the maximum.

looks parabolic. But the jumps in magnetization and density near Tc

both vary as (Tc − T )β with the same exponent β ≈ 0.325, distinctlydifferent from the square-root singularity β = 1/2 of a generic analyticfunction.Also, there are many other properties (susceptibility, specific heat,

correlation lengths) which have power-law singularities at the criticalpoint, and all of the exponents of these power laws for the liquid–gassystems agree with the corresponding exponents for the magnets. Thisis universality. When two different systems have the same singular prop-erties at their critical points, we say they are in the same universalityclass. Importantly, the theoretical Ising model (despite its drastic sim-plification of the interactions and morphology) is also in the same univer-sality class as these experimental uniaxial ferromagnets and liquid–gassystems—allowing theoretical physics to be directly predictive in realexperiments.To get a more clear feeling about how universality arises, consider

site and bond percolation in Fig. 12.7. Here we see two microscopicallydifferent systems (left) from which basically the same behavior emerges(right) on long length scales. Just as the systems approach the threshold



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Fig. 12.6 Universality. (a) Uni-versality at the liquid–gas criticalpoint. The liquid–gas coexistence lines(ρ(T )/ρc versus T/Tc) for a variety ofatoms and small molecules, near theircritical points (Tc, ρc) [56]. The curve isa fit to the argon data, ρ/ρc = 1+s(1−T/Tc) ± ρ0(1 − T/Tc)β with s = 0.75,ρ0 = 1.75, and β = 1/3 [56]. (b) Uni-versality: ferromagnetic–paramagneticcritical point. Magnetization versustemperature for a uniaxial antiferro-magnet MnF2 [58]. We have shownboth branches ±M(T ) and swappedthe axes so as to make the analogy withthe liquid–gas critical point (above) ap-parent. Notice that both the magnetand the liquid–gas critical point haveorder parameters that vary as (1 −

T/Tc)β with β ≈ 1/3. (The liquid–gas coexistence curves are tilted; thetwo theory curves would align if wedefined an effective magnetization forthe liquid–gas critical point ρeff = ρ −

0.75ρc(1−T/Tc) (thin midline, above).This is not an accident; both are inthe same universality class, along withthe three-dimensional Ising model, withthe current estimate for β = 0.325 ±

0.005 [152, chapter 28].

of falling apart, they become similar to one another! In particular, allsigns of the original lattice structure and microscopic rules have disap-peared.4 4Notice in particular the emergent

symmetries in the problem. The largepercolation clusters at pc are statis-tically both translation invariant androtation invariant, independent of thegrids that underly them. In addition,we will see that there is an emergentscale invariance—a kind of symmetryconnecting different length scales (as wealso saw for random walks, Fig. 2.2).

Thus we observe in these cases that different microscopic systems lookthe same near critical points, if we ignore the microscopic details andconfine our attention to long length scales. To study this systematically,we need a method to take a kind of continuum limit, but in systemswhich remain inhomogeneous and fluctuating even on the largest scales.This systematic method is called the renormalization group.5

The renormalization group starts with a remarkable abstraction: it

5The word renormalization grew out of quantum electrodynamics, where the effective charge on the electron changes size(norm) as a function of length scale. The word group is usually thought to refer to the family of coarse-graining operationsthat underly the method (with the group product being repeated coarse-graining). However, there is no inverse operation tocoarse-graining, so the renormalization group does not satisfy the definition of a mathematical group.



Fig. 12.7 Universality in percola-tion. Universality suggests that the en-tire morphology of the percolation clus-ter at pc should be independent of mi-croscopic details. On the top, we havebond percolation, where the bonds con-necting nodes on a square lattice areoccupied at random with probability p;the top right shows the infinite clusteron a 1024 × 1024 lattice at pc = 0.5.On the bottom, we have site percola-tion on a triangular lattice, where it isthe hexagonal sites that are occupiedwith probability p = pc = 0.5. Eventhough the microscopic lattices and oc-cupation rules are completely different,the resulting clusters look statisticallyidentical. (One should note that thesite percolation cluster is slightly lessdark. Universality holds up to overallscale changes, here up to a change inthe density.)

works in an enormous ‘system space’. Different points in system spacerepresent different materials under different experimental conditions,and different physical models of these materials with different interac-tions and evolution rules. So, for example, in Fig. 12.8 we can considerthe space of all possible models for hysteresis and avalanches in three-dimensional systems. There is a different dimension in this system spacefor each possible parameter in a theoretical model (disorder, coupling,next-neighbor coupling, dipole fields, . . . ) and also for each parameterin an experiment (chemical composition, temperature, annealing time,. . . ). A given experiment or theoretical model will traverse a line insystem space as a parameter is varied; the line at the top of the figuremight represent an avalanche model (Exercise 8.17) as the strength ofthe disorder R is varied.The renormalization group studies the way in which system space

maps into itself under coarse-graining. The coarse-graining operationshrinks the system and removes microscopic degrees of freedom. Ig-noring the microscopic degrees of freedom yields a new physical sys-tem with identical long-wavelength physics, but with different (renor-malized) values of the parameters. As an example, Fig. 12.9 shows a



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Fig. 12.8 The renormalizationgroup defines a mapping from thespace of physical systems into itself us-ing a coarse-graining procedure. Con-sider the system space of all possiblemodels of avalanches in hysteresis [130].Each model can be coarse-grained intoa new model, removing some fractionof the microscopic degrees of freedomand introducing new rules so that theremaining domains still flip at the sameexternal fields. A fixed-point S∗ underthis coarse-graining mapping will beself-similar (Fig. 12.11) because it mapsinto itself under a change in lengthscale. Points like Rc that flow into S∗

will also show the same self-similar be-havior (except on short length scalesthat are coarse-grained away during theflow to S∗). Models at Rc and S∗ sharethe same universality class. Systemsnear to their critical point coarse-grainaway from S∗ along the unstable curveU ; hence they share universal proper-ties too (Fig. 12.13).

real-space renormalization-group ‘majority rule’ coarse-graining proce-dure applied to the Ising model.6 Several detailed mathematical tech- 6We will not discuss the methods used

to generate effective interactions be-tween the coarse-grained spins.

niques have been developed to implement this coarse-graining opera-tion: not only real-space renormalization groups, but momentum-spaceϵ-expansions, Monte Carlo renormalization groups, etc. These imple-mentations are both approximate and technically challenging; we willnot pursue them in this chapter (but see Exercises 12.22 and 12.24).Under coarse-graining, we often find a fixed-point S∗ for this mapping

in system space. All the systems that flow into this fixed point undercoarse-graining will share the same long-wavelength properties, and willhence be in the same universality class.Figure 12.8 depicts the flows in system space. It is a two-dimensional

picture of an infinite-dimensional space. You can think of it as a planarcross-section in system space, which we have chosen to include the linefor our model and the fixed-point S∗; in this interpretation the arrowsand flows denote projections, since the real flows will point somewhatout of the plane. Alternatively, you can think of it as the curved surfaceswept out by our model in system space as it coarse-grains, in whichcase you should ignore the parts of the figure below the curve U .7 7The unstable manifold of the fixed-

point.Figure 12.8 shows the case of a fixed-point S∗ that has one unstabledirection, leading outward along U . Points deviating from S∗ in thatdirection will not flow to it under coarse-graining, but rather will flow



Fig. 12.9 Ising model at Tc:coarse-graining. Coarse-graining ofa snapshot of the two-dimensional Isingmodel at its critical point. Each coarse-graining operation changes the lengthscale by a factor B = 3. Each coarse-grained spin points in the directiongiven by the majority of the nine fine-grained spins it replaces. This type ofcoarse-graining is the basic operation ofthe real-space renormalization group.



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Fig. 12.10 Generic and self-organized criticality. (a) Oftenthere will be fixed-points that attractin all directions. These fixed-points de-scribe phases rather than phase transi-tions. Most phases are rather simple,with fluctuations that die away on longlength scales. When fluctuations re-main important, they will exhibit self-similarity and power laws called genericscale invariance. (b) The critical man-ifold C in this earthquake model sep-arates a phase of stuck faults from aphase of sliding faults, with the transi-tion due to the external stress F acrossthe fault. Only along C does one findself-similar behavior and a broad spec-trum of earthquakes. (c) The veloc-ity of the fault will vary as a powerlaw v ∼ (F − Fc)β near the criticalforce Fc. The motion of the continen-tal plates, however, drives the fault ata constant, very slow velocity vs, auto-matically setting F to Fc and yieldingearthquakes of all sizes; the model ex-hibits self-organized criticality.

away from it. Fixed-points with unstable directions correspond to con-tinuous transitions between qualitatively different states. In the case ofhysteresis and avalanches, there is a phase consisting of models whereall the avalanches remain small, and another phase consisting of modelswhere one large avalanche sweeps through the system, flipping most ofthe domains. The surface C which flows into S∗ represents systems attheir critical points; hence our model exhibits avalanches of all scales atRc where it crosses C.8 8Because S∗ has only one unstable di-

rection, C has one less dimension thansystem space (mathematically we say Chas co-dimension one) and hence candivide system space into two phases.Here C is the stable manifold for S∗.

Cases like the liquid–gas transition with two tuning parameters (Tc, Pc)determining the critical point will have fixed points with two unstabledirections in system space. What happens when we have no unstabledirections? The fixed-point S∗

a in Fig. 12.10 represents an entire regionof system space that shares long-wavelength properties; it represents aphase of the system. Usually phases do not show fluctuations on allscales. Fluctuations arise near transitions because the system does notknow which of the available neighboring phases to prefer. However,there are cases where the fluctuations persist even inside phases, leadingto generic scale invariance. A good example is the case of the ran-dom walk9 where a broad range of microscopic rules lead to the same 9See Section 2.1 and Exercises 12.23

and 12.24.long-wavelength random walks, and fluctuations remain important onall scales without tuning any parameters.Sometimes the external conditions acting on a system naturally drive

it to stay near or at a critical point, allowing one to spontaneously ob-serve fluctuations on all scales. A good example is provided by cer-tain models of earthquake fault dynamics. Fig. 12.10(b) shows the



renormalization-group flows for these earthquake models. The horizon-tal axis represents the external stress on the earthquake fault. For smallexternal stresses, the faults remain stuck, and there are no earthquakes.For strong external stresses, the faults slide with an average velocity v,with some irregularities but no large events. The earthquake fixed-pointS∗eq describes the transition between the stuck and sliding phases, and

shows earthquakes of all scales. The Earth, however, does not applya constant stress to the fault; rather, continental drift applies a con-stant, extremely small velocity vs (of the order of centimeters per year).Fig. 12.10(c) shows the velocity versus external force for this transition,and illustrates how forcing at a small external velocity naturally sets theearthquake model at its critical point—allowing spontaneous generationof critical fluctuations, called self-organized criticality.

12.2 Scale invariance

The other striking feature of continuous phase transitions is the commonoccurrence of self-similarity, or scale invariance. We can see this vividlyin the snapshots of the critical point in the Ising model (Fig. 12.1), perco-lation (Fig. 12.2), and the avalanche in the hysteresis model (Fig. 12.5).Each shows roughness, irregularities, and holes on all scales at the crit-ical point. This roughness and fractal-looking structure stems at rootfrom a hidden symmetry in the problem: these systems are (statistically)invariant under a change in length scale.Consider Figs 2.2 and 12.11, depicting the self-similarity in a ran-

dom walk and a cross-section of the avalanches in the hysteresis model.In each set, the upper-left figure shows a large system, and each suc-ceeding picture zooms in by another factor of two. In the hystere-sis model, all the figures show a large avalanche spanning the system(black), with a variety of smaller avalanches of various sizes, each withthe same kind of irregular boundary (Fig. 12.5). If you blur your eyesa bit, the figures should look roughly alike. This rescaling and eye-blurring process is the renormalization-group coarse-graining transfor-mation. Figure 12.9 shows one tangible rule sometimes used to imple-ment this coarse-graining operation, applied repeatedly to a snapshotof the Ising model at Tc. Again, the correlations and fluctuations lookthe same after coarse-graining; the Ising model at Tc is statistically self-similar.This scale invariance can be thought of as an emergent symmetry

under changes of length scale. In a system invariant under translations,the expectation of any function of two positions x1, x2 can be written interms of the separation between the two points ⟨g(x1, x2)⟩ = G(x2−x1).In just the same way, scale invariance will allow us to write functions ofN variables in terms of scaling functions of N−1 variables—except thatthese scaling functions are typically multiplied by power laws in one ofthe variables.Let us begin with the case of functions of one variable. Consider the



Fig. 12.11 Avalanches: scale in-variance. Magnifications of a cross-section of all the avalanches in a runof our hysteresis model (Exercises 8.17and 12.26) each one the lower right-hand quarter of the previous. Thesystem started with a billion domains(10003). Each avalanche is shown in adifferent shade. Again, the larger scaleslook statistically the same.



avalanche size distribution D(S) for a model, say the real earthquakes inFig. 12.3(a), or our model for hysteresis, at the critical point. Imaginetaking the same system, but increasing the units of length with whichwe measure the system—stepping back, blurring our eyes, and lookingat the system on a coarse-grained level. Imagine that we multiply thespacing between markings on our rulers by a small amount B = 1+ dℓ.After coarsening, any length scales in the problem (like the correlationlength ξ) will be divided by B.

ξ′ = ξ/B = ξ/(1 + dℓ) = ξ − ξdℓ +O(dℓ2),

dξ/dℓ = −ξ,

ξ[ℓ] = ξℓ = ξ0 exp(−ℓ).

(12.3)

(We shall denote Xℓ the value of a quantity X after coarse-graining byexp(ℓ). Thus X0 is the initial condition to our renormalization-groupflow equations, and hence is the ‘true’ value in the simulation or exper-iment.)Thus exp(ℓ) is the net coarse-graining length as a function of the

parameter ℓ. We assume that the system is flowing to a fixed pointunder our renormalization group, so for large avalanches we expect

D′(S) = D(S); ∂D/∂ℓ = 0. (12.4)

How do D and S change under the renormalization group? Theavalanche sizes S after coarse-graining will be smaller by some factor1010If the size of the avalanche were the

cube of its length, then c would equal3 since (1 + dℓ)3 = 1 + 3dℓ + O(dℓ2).Here c is the fractal dimension of theavalanche.

C = 1 + cdℓ. The overall scale of D(S) will change by some factorA = 1 + adℓ. This factor A is partly due to coarse graining; the sameavalanches occur independent of the units of length with which we mea-sure, but the probability density D(S) per unit size per unit volume willchange. It is partly due to the rescaling factor allowed by the renormal-ization group – here D′ must omit the smallest avalanches S < Ssmallest

(now invisible after blurring our eyes), so the overall normalization factordividing D changes as well under rescaling. Thus our renormalization-group equations are

S′ = S/C = S/(1 + cdℓ), dS/dℓ = −cS,

D′(S′) = AD(S) = D(S)(1 + adℓ), dD/dℓ = aD.(12.5)

Here D′(S′) is the distribution measured with the new ruler: a smalleravalanche with a larger probability density. Note that the flow equa-tion 12.5 for D specifies the total derivative dD/dℓ = −aD (see Sec-tion 4.1), while being at a fixed point (eqn 12.4). specifies the partialderivative ∂D/∂ℓ = 0. The total derivative is dDℓ(Sℓ)/dℓ; it gives thechange of D′(S′)−D(S), where the partial derivative ∂D/∂ℓ gives thechange D′(S)−D(S).Solving eqn 12.5 for the flow of S, we find Sℓ = S0 exp(−cℓ). Here

S0 is both the initial condition for our differential equation at ℓ = 0,and is the physical avalanche size for which we want to know the prob-ability D0(S0). Similarly, Dℓ(Sℓ) = exp(aℓ)D0(S0). We want to solve



for the probability D0(S0) in terms of S0. We can get rid of the ex-ponential factor by noticing exp(aℓ) = exp(−cℓ)−a/c = (Sℓ/S0)−a/c,so Dℓ(Sℓ) = (Sℓ/S0)−a/cD0(S0). We can then choose to flow until ℓ∗

l*S =1

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r

Fig. 12.12 Disorder and avalanchesize: renormalization-group flows.Here S is the avalanche size and r =R − Rc is the change in the disor-der from the critical point. The dis-order grows under coarse-graining, andthe avalanche sizes shrink. The renor-malization group allows us to describeproperties of avalanches at any largesize S and small r (near the criticaldisorder) as rescaled versions of theavalanches along a single line (for con-venience, along S = 1). First, thesize distribution D(S,Rc) at criticalitycan be written in terms of the constantN = D(1, Rc) rescaled by a power law(eqn 12.6) (since all points on the blueaxis flow there). Later, we show thatD(S,Rc + r0) can be written as thesame power law times D(1, Rc + rℓ∗),where rℓ∗ (S0) is the intersection of theflow trajectory starting at (r0, S0).

such that Sℓ∗ = 1 (along the r = 0 axis in Fig. 12.12). Thus we findD0(S0) = Dℓ∗(1)S

−a/c0 , or

D(S) = NS−a/c. (12.6)

where N is the constant Dℓ∗(1). This argument is quite general. Notonly do we expect that avalanches (and earthquakes, Fig. 12.3) will showpower laws at critical points, but almost anything which rescales by aconstant factor under the renormalization group should exhibit a powerlaw.Because the properties shared in a universality class only hold up

to overall scales, the constant N is system dependent. (In this case,!∞

SsmallestD(S)dS = 1 because D is a probability distribution, so the

normalization factor N = (τ −1)S(1−τ)smallest.) However, the exponents a, c,

and a/c are universal—independent of experiment (with the universalityclass). Some of these exponents have standard names: the exponent cgiving the fractal dimension of the avalanche is usually called df or1/σν. The exponent a/c giving the size distribution law is called τ inpercolation and in most models of avalanches in magnets11 and is related

11Except ours, where we used τ to de-note the avalanche size law at the crit-ical field and disorder; integrated overthe hysteresis loop Dint ∝ S−τ withτ = τ + σβδ.

to the Gutenberg–Richter exponent for earthquakes12 (Fig. 12.3(b)).

12We must not pretend that we havefound the final explanation for theGutenberg–Richter law. There aremany different models that give expo-nents ≈ 2/3, but it remains controver-sial which of these, if any, are correctfor real-world earthquakes.

Most measured quantities depending on one variable will have sim-ilar power-law singularities at the critical point. Thus the correlationfunction of the Ising model at Tc (Fig. 10.4) decays with distance x indimension d as C(x) ∝ x−(d−2+η) and the distance versus time for ran-dom walks (Section 2.1) grows as t1/2, both because these systems areself-similar.13

13This is because power laws are theonly self-similar function. If f(x) =x−α, then on a new scale multiplyingx by B, f(Bx) = B−αx−α ∝ f(x).(See [102] for more on power laws.)

Self-similarity is also expected near to the critical point. Here as onecoarsens the length scale a system will be statistically similar to itself at adifferent set of parameters. Thus a system undergoing phase separation(Section 11.4.1, Exercise 12.17), when coarsened, is similar to itself at anearlier time (when the domains were smaller), and a percolation clusterjust above pc (Fig. 12.2 (bottom left)) when coarsened is similar to onegenerated further from pc (hence with smaller holes).For a magnet slightly below14 Tc, a system coarsened by a factor

14Thus increasing the distance t to Tc

decreases the temperature T .

B = 1+dℓ will be similar to one farther from Tc by a factor E = 1+edℓ.Here the standard Greek letter for the length rescaling exponent is ν =1/e. Similar to the case of the avalanche size distribution, the coarsenedsystem must have its magnetization rescaled upward by F = (1 + fdℓ),with f = β/ν) to match that of the lower-temperature original magnet(Fig. 12.13):

M ′(Tc − t) = FM(Tc − t) = M(Tc − Et),

(1 + fdℓ)M(Tc − t) = M"

Tc − t(1 + edℓ)#

,(12.7)

sodt/dℓ = et,

dM/dℓ = fM.(12.8)



Again, M ∝ tf/e = tβ, providing a rationale for the power laws we(M (T − t)cM

[4]M

[3]M

(T − t)cM’

T − E t)c

M’’

Fig. 12.13 Scaling near critical-ity. If two points in system spaceflow towards one another under coarse-graining, their behavior must be sim-ilar on long length scales. Here wemeasure the magnetization M(T ) forour system (top line) at two differ-ent temperatures, Tc − t and Tc − Et.The dots represent successive coarse-grainings by a factor B; under thisrenormalization group M → M ′ →

M ′′ → M [3] . . . . Here M(Tc − t) af-ter four coarse-grainings maps to nearlythe same system as M(Tc − Et) afterthree coarse-grainings. We thus know,on long length scales, that M ′(Tc − t)must agree withM(Tc−Et); the systemis similar to itself at a different set ofexternal parameters. All systems nearto criticality first are attracted near tothe fixed point, and then flow awayalong a common trajectory (here thehorizontal axis, in Fig. 12.8 the unsta-ble curve U). Their properties are uni-versal because they all escape along thesame path.

saw in magnetism and the liquid–gas transition (Fig. 12.6). Similarly,the specific heat, correlation length, correlation time, susceptibility, andsurface tension of an equilibrium system will have power-law divergences(T −Tc)−X , where by definition X is α, ν, zν, γ, and −2ν, respectively.One can also vary the field H away from the critical point and measurethe resulting magnetization, which varies as H1/δ.To specialists in critical phenomena, these exponents are central; whole

conversations often rotate around various combinations of Greek letters.We know how to calculate critical exponents from the various analyticalapproaches,15 and they are simple to measure (although hard to measure

15They can be derived from the eigen-values of the linearization of therenormalization-group flow around thefixed-point S∗ in Fig. 12.8 (see Exer-cises 12.1 and 12.24).

well, [88]).Critical exponents are not everything, however. Many other scaling

predictions are easily extracted from numerical simulations. Universal-ity should extend even to those properties that we have not been able towrite formulæ for. In particular, there are an abundance of functions oftwo and more variables that one can measure. Figure 12.14 shows thedistribution of avalanche sizesDint(S,R) in our model of hysteresis, inte-grated over the hysteresis loop (Fig. 8.16), at various disorders R aboveRc (Exercise 8.17). Notice that only at Rc ≈ 2.16 do we get a power-lawdistribution of avalanche sizes; at larger disorders there are extra smallavalanches, and a strong decrease in the number of avalanches beyonda certain size Smax(R).Let us derive the scaling form forDint(S,R). By using scale invariance,

we will be able to write this function of two variables as a power of oneof the variables times a universal, one-variable function of a combinedinvariant scaling combination. As in Fig. 12.8 we expect that systemswill flow away from criticality: a system at R = Rc + r after coarse-graining will be similar to a system further from the critical disorder,say at R′ = Rc +Er = Rc + (1+ edℓ)r. Together with our treatment ofthe avalanche sizes at Rc (eqns 12.5) we know that

dr/dℓ = er,

dS/dℓ = −cS,

dD/dℓ = aD

(12.9)

First, how does the deviation of the disorder r vary away from crit-icality? Solving dr/dℓ = er we find rℓ = r0 exp(eℓ). Similarly Sℓ =S0 exp(−cℓ), so we may write rℓ in terms of S0 as rℓ = r0(Sℓ/S0)−e/c.To derive the form for the avalanche size distribution at size S0, weshall again renormalize until the value ℓ∗ such that Sℓ∗ = 1 (Fig. 12.12).Hence

rℓ∗ = r0Se/c0 = r0S

σ0 (12.10)

where 1/σ = c/e is the exponent governing how the cutoff in the avalanchesize distribution varies with disorder (inset, Fig. 12.14).The combination X = rSσ in eqn 12.10 is invariant under the renor-

malization group flow. That is, each dashed curve in Fig 12.12 cor-responds to a different value of this invariant scaling combination. It



100 102 104 106

Avalanche size S

10-15

10-10

10-5

100

105

Din

t(S,R

)

0.0 0.5 1.0 1.5Sσr

0.1

0.2Sτ

+ σβδ

Din

t(Sσr)

R = 4R = 2.25

Fig. 12.14 Avalanche size distri-bution. The distribution of avalanchesizes in our model for hysteresis. No-tice the logarithmic scales. (We canmeasure a D(S) value of 10−14 by run-ning billions of spins and binning overranges ∆S ∼ 105.) (i) Although onlyat Rc ≈ 2.16 do we get a pure powerlaw (dashed line, D(S) ∝ S−τ ), wehave large avalanches with hundreds ofspins even a factor of two away fromthe critical point. (ii) The curves havethe wrong slope except very close to thecritical point; be warned that a powerlaw over two decades (although oftenpublishable [88]) may not yield a reli-able exponent. (iii) The scaling curves(thin lines) work well even far fromRc. Inset: We plot D(S)/S−τ versusSσ(R−Rc)/R to extract the universalscaling curve D(X) (eqn 12.11). Vary-ing the critical exponents and Rc toget a good collapse allows us to mea-sure the exponents far from Rc, wherepower-law fits are still unreliable (Ex-ercise 12.25(g)).

makes sense that such invariants would be important. The avalanchesof size S0 in a system with disorder shifted by r0 will have properties (du-ration, or their average shape, or the probability D0(S0, r0) we calculatehere), which are similar to other points on the same curve.Now, how does D renormalize? Solving dD/dℓ = aD, Dℓ(Sℓ, rℓ) =

exp(aℓ)D0(S0, r0) = (Sℓ/S0)−a/cD0(S0, r0), so D0 = S−a/c0 Dℓ∗(1, rℓ∗).

Using a/c = τ and eqn 12.10, we find for the ‘physical’ variablesD = D0,S = S0, and r = r0 that

D(S,R) = S−τD(rSσ) = S−τD((R −Rc)Sσ) (12.11)

for the scaling function D(X) = Dℓ∗(1, X). This scaling function isanother universal prediction of the theory (up to an overall choice ofunits for S, r and D). In Fig. 12.12, we have expressed the avalanchesize distribution D(S,Rc + r) for any point in the (r, S) plane in termsof the values D(1, Rc + rSσ). The emergent scale invariance allows usto write a function of two variables in terms of a universal function ofone variable.We can use a scaling collapse of the experimental or numerical data to

extract this universal function, by plotting D/S−τ against X = Sσ(R−Rc); the inset of Fig. 12.14 shows this scaling collapse.Similar universal scaling functions appear in many contexts. Consid-

ering just the equilibrium Ising model, there are scaling functions forthe magnetization M(H,T ) = (Tc−T )βM

$

H/(Tc − T )βδ%

, for the cor-relation function C(x, t, T ) = x−(d−2+η)C (x/|T − Tc|−ν , t/|T − Tc|−zν),and for finite-size effects M(T, L) = (Tc − T )βM (L/(Tc − T )−ν) in a



system confined to a box of size Ld.

12.3 Examples of critical points

Ideas from statistical mechanics have found broad applicability in sci-ences and intellectual endeavors far from their roots in equilibrium ther-mal systems. The scaling and renormalization-group methods intro-duced in this chapter have seen a particularly broad range of applica-tions; we will touch upon a few in this conclusion to our text.

12.3.1 Equilibrium criticality: energy versusentropy

0.0001 0.001 0.01t = 1-T/T

c

1.5

2.5

(ρs / ρ)

t-2

/3

0.05 bar7.27 bar12.13 bar18.06 bar24.10 bar29.09 bar

Fig. 12.15 Superfluid density inhelium: scaling plot. This classicexperiment [1,55] in 1980 measured thesuperfluid density ρs(T ) in helium togreat precision. Notice the logarithmicscale on the horizontal axis; the low-est pressure data (saturated vapor pres-sure ≈ 0.0504 bar) spans three decadesof temperature shift from Tc. This plotemphasizes the deviations from the ex-pected power law.

Scaling and renormalization-groupmethods have their roots in the studyof continuous phase transitions in equilibrium systems. Ising models,Potts models,16 Heisenberg models, phase transitions in liquid crys-

16Potts models are Ising-like modelswith N states per site rather than two.

tals, wetting transitions, equilibrium crystal shapes (Fig. 11.6), two-dimensional melting—these are the grindstones on which our renormali-zation-group tools were sharpened.The transition in all of these systems represents the competition be-

tween energy and entropy, with energy favoring order at low temper-atures and entropy destroying it at high temperatures. Figure 12.15shows the results of a classic, amazing experiment—the analysis of thesuperfluid transition in helium (the same order parameter, and also thesame universality class, as the XY model). The superfluid density isexpected to have the form

ρs ∝ (Tc − T )β(1 + d(Tc − T )x), (12.12)

where x is a universal, subdominant correction to scaling. Since β ≈ 2/3,they plot ρs/(T − Tc)2/3 so that deviations from the simple expectationare highlighted. The slope in the top, roughly straight curve reflects thedifference between their measured value of β = 0.6749±0.0007 and theirmultiplier 2/3. The other curves show the effects of the subdominant cor-rection, whose magnitude d increases with increasing pressure. Recentexperiments improving on these results were done on the space station,in order to reduce the effects of gravity.

12.3.2 Quantum criticality: zero-point fluctuationsversus energy

Thermal fluctuations do not exist at zero temperature, but there aremany well-studied quantum phase transitions which arise from the com-petition of potential energy and quantum fluctuations. Many of theearliest studies focused on the metal–insulator transition and the phe-nomenon of localization, where disorder can lead to insulators even whenthere are states at the Fermi surface. Scaling and renormalization-groupmethods played a central role in this early work; for example, the states


12.3 Examples of critical points 317

ν 1.2

T(K

)(Ω)

R

(K)T

9 Adc

t

d − d| c|t

(b)(a)

15 A

(A)d

(Ω)

R

R(Ω

)~~z

Fig. 12.16 The superconductor–insulator transition. (a) Thin filmsof amorphous bismuth are insulators(resistance grows to infinity at zerotemperature), while films above about12 A are superconducting (resistancegoes to zero at a temperature abovezero). (b) Scaling collapse. Resis-tance plotted against the scaled thick-ness for the superconductor–insulatortransition, with each thickness rescaledby an independent factor t to get a goodcollapse. The top scaling curve F− isfor the insulators d < dc, and the bot-tom one F+ is for the superconductorsd > dc. The inset shows t ∼ T−1/νz ,with νz ∼ 1.2. (From [91].)

near the mobility edge (separating localized from extended states) areself-similar and fractal. Other milestones include the Kondo effect,macroscopic quantum coherence (testing the fundamentals of quantummeasurement theory), transitions between quantum Hall plateaus, andsuperconductor–normal metal transitions. Figure 12.16 show a recentexperiment studying a transition directly from a superconductor to aninsulator, as the thickness of a film is varied. The resistance is expectedto have the scaling form

R(d, T ) = RcF±

"

(d− dc)T−1/νz

#

; (12.13)

the authors plot R(d, T ) versus t(d−dc), vary t until the curves collapse(main part of Fig. 12.16(b)), and read off 1/νz from the plot of t versus T(inset). While it is clear that scaling and renormalization-group ideasare applicable to this problem, we should note that as of the time thistext was written, no theory yet convincingly explains these particularobservations.

12.3.3 Dynamical systems and the onset of chaos µ

µ

2

1

x*( )µ

µ

Fig. 12.17 Self-similarity at theonset of chaos. The attractor as afunction of µ for the Feigenbaum lo-gistic map f(x) = 4µx(1 − x). Forsmall µ < µ1, repeatedly iterating fconverges to a fixed-point x∗(µ). Asµ is raised past µ1, the map convergesinto a two-cycle; then a four-cycleat µ2, an eight-cycle at µ3. . . Theseperiod-doubling bifurcations convergegeometrically: µ∞ − µn ∝ δ−n whereδ = 4.669201609102990 . . . is a univer-sal constant. At µ∞ the system goeschaotic. (Exercise 12.22).

Much of statistical mechanics focuses on systems with large numbers ofparticles, or systems connected to a large external environment. Con-tinuous transitions also arise in isolated or simply driven systems withonly a few important degrees of freedom, where they are called bifur-cations. A bifurcation is a qualitative change in behavior which ariseswhen a parameter in a set of differential equations passes through a crit-ical value. The study of these bifurcations is the theory of normal forms(Exercise 12.18). Bifurcation theory contains analogies to universalityclasses, critical exponents, and analytic corrections to scaling.Dynamical systems, even when they contain only a few degrees of free-

dom, can exhibit immensely complex, chaotic behavior. The mathemati-cal trajectories formed by chaotic systems at late times—the attractors—are often fractal in structure, and many concepts and methods from



statistical mechanics are useful in studying these sets.1717For example, statistical mechanicalensembles become invariant measures(Exercise 4.10), and the attractors arecharacterized using concepts related toentropy (Exercise 5.23).

It is in the study of the onset of chaos where renormalization-groupmethods have had a spectacular impact. Figure 12.17 shows a sim-ple dynamical system undergoing a series of bifurcations leading to achaotic state. Feigenbaum (Exercise 12.22) analyzed the series using arenormalization group, coarse-graining not in space but in time. Again,this behavior is universal—exactly the same series of bifurcations (upto smooth coordinate changes) arise in other maps and in real physicalsystems. Other renormalization-group calculations have been importantfor the study of the transition to chaos from quasiperiodic motion, andfor the breakdown of the last non-chaotic region in Hamiltonian systems(see Exercise 4.7).

12.3.4 Glassy systems: random but frozen

Let us conclude with a common continuous transition for which ourunderstanding remains incomplete: glass transitions.Glasses are out of equilibrium; their relaxation times diverge as they

are cooled, and they stop rearranging at a typical temperature knownas the glass transition temperature. Many other disordered systemsalso appear to be glassy, in that their relaxation times get very slow asthey are cooled, and they freeze into disordered configurations.18 This

18Glasses are different from disorderedsystems. The randomness in disorderedsystems is fixed, and occurs in boththe high- and low-temperature phases;the disorder in the traditional config-urational glasses freezes in as it cools.See also Section 5.2.2.

freezing process is sometimes described as developing long-range orderin time, or as a broken ergodicity (see Section 4.2).

F

F

A

Fig. 12.18 Frustration. A spin glasshas a collection of magnetic ions withinteractions of random sign. Here wesee a triangle of Ising ±1 spins withone antiferromagnetic bond—one of thethree bonds must be unsatisfied in anyspin configuration. Hence the system issaid to be frustrated.

The basic reason that many of the glassy systems freeze into ran-dom states is frustration. Frustration was defined first for spin glasses,which are formed by randomly substituting magnetic atoms into a non-magnetic host. The magnetic spins are coupled to one another at ran-dom; some pairs prefer to be parallel (ferromagnetic couplings) and someantiparallel (antiferromagnetic). Whenever strongly-interacting spinsform a loop with an odd number of antiferromagnetic bonds (Fig. 12.18)they are frustrated; one of the bonds will have to be left in an unhappystate, since there must be an even number of spin inversions around theloop (Fig. 12.18). It is believed in many cases that frustration is alsoimportant for configurational glasses (Fig. 12.19).The study of disordered magnetic systems is mathematically and com-

putationally sophisticated. The equilibrium ground state for the three-dimensional random-field Ising model,19 for example, has been rigorously

19Our model for hysteresis andavalanches (Figs 8.19, 12.5, 12.11,and 12.14; Exercises 8.17, 8.18,and 12.26) is this same random-fieldIsing model, but in a growing externalfield and out of equilibrium.

proven to be ferromagnetic (boring); nonetheless, when cooled in zero ex-ternal field we understand why it freezes into a disordered state, becausethe coarsening process develops diverging free energy barriers to relax-ation. Methods developed to study the spin glass transition have seenimportant applications in neural networks (which show a forgetting tran-sition as the memory becomes overloaded) and more recently in guidingalgorithms for solving computationally hard (NP–complete) problems(see Exercises 1.10 and 8.19). Some basic conceptual questions, however,remain unanswered. For example, we still do not know whether spinglasses have a finite or infinite number of equilibrium states—whether,


12.3 Examples of critical points 319

(b)(a)

Fig. 12.19 Frustration and curva-ture. One kind of frustration ariseswhen the energetically favorable localpacking of atoms or molecules is in-compatible with the demands of build-ing large structures. Here we show twoartistic renditions (courtesy of PamelaDavis Kivelson [70]). (a) The classicproblem faced by map-makers: the peelof an orange (or the crust of the Earth)cannot be mapped smoothly onto a flatspace without stretching and tearingit. (b) The analogous problem faced inmany metallic glasses, whose atoms lo-cally prefer to form nice compact tetra-hedra: twenty tetrahedra cannot beglued together to form an icosahedron.Just as the orange peel segments canbe nicely fit together on the sphere,the metallic glasses are unfrustrated incurved space [120].

upon infinitely slow cooling, one still has many glassy configurations.20 20There are ‘cluster’ theories which as-sume two (spin-flipped) ground states,competing with ‘replica’ and ‘cavity’methods applied to infinite-range mod-els which suggest many competingground states. Some rigorous resultsare known.

In real configurational glasses the viscosity and relaxation times growby ten to fifteen orders of magnitude in a relatively small temperaturerange, until the cooling rate out-paces the equilibration. We funda-mentally do not know why the viscosity diverges so rapidly in so manymaterials. There are at least three competing pictures for the glass tran-sition. (1) It reflects an underlying equilibrium transition to an ideal,zero-entropy glass state, which would be formed under infinitely slowcooling. (2) It is a purely dynamical transition (where the atoms ormolecules jam together). (3) It is not a transition at all, but just across-over where the liquid viscosity jumps rapidly (say, because of theformation of semipermanent covalent bonds).

12.3.5 Perspectives

Many of the physicists who read this text will spend their careers out-side of traditional physics. Physicists continue to play significant rolesin the financial world (econophysics, computational finance, derivativetrading), in biology (bioinformatics, models of ecology and evolution),computer science (traffic models, algorithms for solving hard problems),and to some extent in social science modeling (models of voting be-havior and consensus building). The tools and methods of statisticalmechanics (particularly the scaling methods used to study continuoustransitions) are perhaps the most useful tools that we bring to thesedisparate subjects. Conversely, I hope that this text will prove useful asan introduction of these tools and methods to computer scientists, biol-ogists, engineers, and finance professionals, as they continue to broadenand fertilize the field of statistical mechanics.



Exercises

We start by re-emphasizing the phenomena and conven-tions used in studying continuous phase transitions inIsing self-similarity, Scaling functions, and Scaling andcoarsening. We study solvable critical points in Bifur-cation theory, Mean-field theory and The onset of lasing.We illustrate how the renormalization-group flows deter-mine the critical behavior in Renormalization-group tra-jectories and Superconductivity and the renormalizationgroup; the latter explains schematically the fundamentalbasis for Fermi liquid theory. Period doubling and thetwo versions of The renormalization-group and the cen-tral limit theorem provide important applications wherethe reader may implement the renormalization group ex-plicitly and completely. We conclude with two numericalexercises, Percolation and universality and Hysteresis andavalanches: scaling, which mimic the entire experimentalanalysis from data collection to critical exponents andscaling functions.

(12.1) Renormalization-group trajectories. ⃝3This exercise provides an early introduction tohow we will derive power laws and universalscaling functions in Section 12.2 from univer-sality and coarse-graining.An Ising model near its critical temperature Tc

is described by two variables: the distance tothe critical temperature t = (T − Tc)/Tc, andthe external field h = H/J . Under coarse-graining, changing lengths to x′ = (1 − ϵ) x,the system is observed to be similar to itselfat a shifted temperature t′ = (1 + aϵ) t and ashifted external field h′ = (1 + bϵ)h, with ϵ in-finitesimal and a > b > 0 (so there are tworelevant eigendirections, with the temperaturemore strongly relevant than the external field).The curves shown below connect points that aresimilar up to some rescaling factor.(a) Which diagram below has curves consistentwith this flow, for a > b > 0? Is the flow un-der coarse graining inward or outward from theorigin? (No math should be required. Hint:After coarse-graining, how does h/t change?)

(A)

t

h

(B)

t

h

(C)

t

h

(D)

t

h

(E)

t

h


Exercises 321

The solid dots are at temperature t0; the opencircles are at temperature t = 2t0.(b) In terms of ϵ and a, by what factor mustx be rescaled by to relate the systems at t0 and2t0? (Algebraic tricks: Use (1 + δ) ≈ exp(δ)everywhere. If you rescale multiple times un-til exp(naϵ) = 2, you can solve for (1 − ϵ)n ≈exp(−nϵ) without solving for n.) If one of thesolid dots in the appropriate figure from part (a)is at (t0, h0), what is the field !h for the corre-sponding open circle, in terms of a, b, ϵ, and theoriginal coordinates? (You may use the rela-tion between !h and h0 to check your answer forpart (a).)The magnetization M(t, h) is observed torescale under this same coarse-grainingoperation to M ′ = (1 + cϵ)M , soM ((1 + aϵ) t, (1 + bϵ)h) = (1 + cϵ)M(t, h).(c) Suppose M(t, h) is known at (t0, h0), one ofthe solid dots. Give a formula for M(2t0,!h)at the corresponding open circle, in terms ofM(t0, h0), the original coordinates, a, b, c, andϵ. (Hint: Again, rescale n times.) Substituteyour formula for !h into the formula, and solvefor M(t0, h0).You have now basically derived the key resultof the renormalization group; the magnetiza-tion curve at t0 can be found from the magne-tization curve at 2t0. In Section 12.2, we shallcoarse-grain not to t = 2t0, but to t = 1. Weshall see that the magnetization everywhere canbe predicted from the magnetization where theinvariant curve crosses t = 1.(d) Substitute 2 → 1/t0 in your formula frompart (c). Show that M(t, h) = tβM(h/tβδ) (thestandard scaling form for the magnetization inthe Ising model). What are β and δ in terms ofa, b, and c? How is M related to M(t, h) wherethe curve crosses t = 1?

(12.2) Singular corrections to scaling. (Con-densed matter) ⃝3The renormalization group says that the num-ber of relevant directions at the fixed point insystem space is the number of parameters weneed to tune to see a critical point, and that thecritical exponents depend on the eigenvalues ofthese relevant directions. Do the irrelevant di-rections matter?

Let the Ising model in zero field be describedby flow equations

dtℓ/dℓ = tℓ/ν, duℓ/dℓ = −yuℓ (12.14)

where tℓ describes the renormalization of thereduced temperature t = (Tc − T )/Tc after acoarse-graining by a factor b = exp(ℓ), and uand uℓ represent a slowly-decaying irrelevantperturbation under the renormalization group.In Fig. 12.8, one may view t as the expand-ing eigendirection running roughly horizontally,and u as the contracting, irrelevant coordinaterunning roughly vertically. Thus our modelstarts with a value u associated to the distancein system space between Rc and S∗.(a) What is the invariant combination z = utω

that stays constant under the renormalizationgroup? What is ω in terms of the eigenvalues−y and 1/ν?Properties near critical points have universalpower law singularities, but the corrections tothese power laws also have universal propertiespredicted by the renormalization group. Thesecome in two types – analytic corrections to scal-ing and singular corrections to scaling.Let us consider corrections to the susceptibility.In analogy with other systems we have studied,we would expect that the susceptibility

χ(t, u) = t−γX(z) (12.15)

with X(z) a universal function of the invari-ant combination you found in part (a). As afunction of t, χ(t, u) has singularities at smallt. But we expect properties to be analytic as wevary u, since the irrelevant direction is not be-ing tuned to a special value, so we expect thata Taylor series of χ(t, u) in powers of u shouldmake sense. Since z ∝ u, we thus expect thatX(z) will be an analytic function of z for smallz.21

(b) Show that for small t, your z from part (a)goes to zero. Taylor expand X(z). What correc-tions do you predict for the susceptibility fromthe first and second-order terms in the series?These are the singular corrections to scaling dueto the irrelevant perturbation u.An Ising magnet on a sample holder is loadedinto a magnetometer, and the susceptibility ismeasured22 at zero external field as a function

21Had we used a scaling variable Z = tu1/ω , for example, we would not have expected the corresponding scaling function tobe analytic in small Z.22The accuracy of the quoted exponents is not experimentally realistic.



of reduced temperature t = (T − Tc)/Tc. It isfound to be well approximated by

χ(T ) = At−1.24 +Bt−0.83 +Ct0.42

+D + Et+ . . .(12.16)

You may ignore any errors due to the magne-tometer.(c) The exponent ω ≈ 0.407 for the 3D Isinguniversality class, and γ ≈ 1.237. Which termsare explained as singular corrections to scal-ing?(d) Can you provide a physical interpretationfor the terms in eqn 12.16 that are not explainedby your theory? For example, how do we expectthe susceptibility of the sample holder to dependon temperature?So far, we have relied on universality and rescal-ing to derive the universal power laws and scal-ing forms for properties near critical points.We can derive these in a mathematical wayby including the flows of the predictions alongwith the flows of the control parameters undercoarse-graining:

dχℓ/dℓ = −(γ/ν)χℓ,

dtℓ/dℓ = tℓ/ν,

duℓ/dℓ = −yuℓ

(12.17)

How do we derive the universal scaling functionX(z) from these renormalization group flows?Consider the flows illustrated in Fig. 12.8, ex-cept now with a third dimension involving theprediction χ. Consider a point (t0, u0,χ0)in the system space, and the invariant curvedefined by z = u0t

ω0 (dashed lines). Our

renormalization group allows us to calculateχℓ(tℓ, uℓ) along these curves – relating the be-havior everywhere near the critical manifold(vertical swath flowing toward S∗) to the prop-erties along the outgoing trajectories, which ap-proach closer and closer to the unstable man-ifold (the horizontal swath flowing away fromS∗.For example, we can define the universal scal-ing function X(z) (for positive time t) to be theχℓ∗ where the flow crosses tℓ∗ = 1.(e) Solve eqns 12.17 for uℓ and tℓ. Settingtℓ∗ = 1, what is uℓ∗ in terms of your invari-ant combination z?

So we label each invariant scaling curve bythe value of the vertical position uℓ∗ where itcrosses tℓ∗ = 1.(f) Solve eqns 12.17 for χℓ∗(1, uℓ∗), in termsof z, t0, and χ0(t0, u0). Use your solutionto solve for the physical behavior χ0(t0, u0) interms of t and X(z). Express X(z) in terms ofχℓ∗(1, uℓ∗).Remember the critical manifold is co-dimensionone (or two, if you include temperature and ex-ternal field), and the unstable manifold is di-mension one (or two) – so we get universal pre-dictions for a huge variety of systems, by ob-serving the outgoing trajectories near a narrowtube or surface emitted from the fixed point.

(12.3) Nonlinear flows, analytic corrections,

and hyperscaling.23 ⃝3We consider the effects of nonlinear terms in re-normalization group flows. The Ising model inzero field has one relevant variable (the devia-tion t of the temperature from Tc). To calculatethe specific heat, we shall also consider the flowof the free energy per spin f under the renor-malization group. Instead of a discrete coarse-graining by a factor b, here we use a continuouscoarse-graining measured by ℓ. One can thinkof one coarse-graining step by b = (1 + ϵ) asincrementing ℓ → ℓ + ϵ; equivalently, coarse-graining to ℓ changes length scales by exp(ℓ).Consider the particular flow equations24

dfℓ/dℓ = Dfℓ−atℓ2 dtℓ/dℓ = tℓ/ν, (12.18)

where D is the dimension of space and atℓ2 is

a nonlinear term that will be important in twodimensions.Notice there are several free energies here. Weshall call the free energy per spin of the ac-tual system either f or f0, and the tempera-ture of the actual system either t or t0. Thecoarse-grained free energy and temperature arefℓ and tℓ, after being coarse-grained by a fac-tor exp(ℓ). (Hence at ℓ = 0 we have not yetcoarse-grained, so f0 = f and t0 = t.) Noticehere that the free energy of our system is theinitial condition f0(t0) of this differential equa-tion, and fℓ = f(ℓ) and tℓ = t(ℓ) are the re-normalization group flows of the two variables.

23This exercise developed in collaboration with Colin Clement24Note that these are total derivatives. So the first equation tells us the total change in f after coarsening by a factor 1 + dℓ.f(t) then will coarse grain to fℓ(tℓ) without needing to worry about the chain rule df(t)/dℓ = ∂f/∂ℓ+ ∂f/∂tdt/dℓ.


Exercises 323

To derive the scaling behavior, we shall coarse-grain to ℓ∗ where tℓ∗ = 1, at which point thecoarse-grained free energy is fℓ∗ .Let us start with the linear case a = 0.(a) Solve for fℓ and tℓ for a = 0. Settingtℓ∗ = 1, solve for f0 in terms of fℓ∗ , t0, D,and ν. Solve for the specific heat per spinc = T∂2f/∂T 2, where t = t0 = (T −Tc)/Tc andf = f0. (Hint: Use the chain rule ∂f/∂T =(∂f/∂t)(dt/dT ).) Show that the specific heatnear Tc has a power-law singularity c ∝ t−α,with α = 2 − Dν. (For example, in D = 3,ν ≈ 0.63, so α = 2 − Dν ∼ 0.11; the specificheat diverges at Tc.) Writing

c = t−α(c0 + c1t+ c2t2 + . . . ), (12.19)

what is the first correction c1 to the specificheat near t = 0 in the absence of the nonlin-ear term?Why is the linear term in the the free energyflow equal to the dimension, df/dℓ = Df+ . . . ,where all other terms are hard-to-compute crit-ical exponents? There is no simple answer tothis question.25 Indeed, other models of disor-dered systems and glasses, and models abovethe upper critical dimension, the linear termis not given by the dimension. The relation2−α = Dν is called a hyperscaling relation (toemphasize they involve the dimension D), andthese other models are said to violate hyper-scaling.(b) In the case a = 0, show that the singular freeenergy f contained in a correlated volume ξD

near the critical temperature becomes indepen-dent of the distance to the critical point. (Hint:Look up the critical exponent describing howthe correlation length ξ diverges as t → 0.)Glassy and disordered systems become ex-tremely sluggish as they are cooled. In at leastsome cases, this is precisely because the en-ergy barriers needed to continue equilibrationdiverge as their correlation lengths grow – they

are glassy because their RG flows violate thehyperscaling relation.So much for the power law singularity – whatabout the correction term c1 in part (a)? Itis an analytic correction to scaling.26 Here itis subdominant – near the critical temperaturewhere t → 0, it is less singular than the lead-ing term. One expects in a real physical sys-tem that the microscopic bond free energy Jbetween spins will be some analytic function oftemperature, and the physical free energy andspecific heat will be multiplied by J . Expand-ing J in a Taylor series about t = 0 would alsogive us terms like those in eqn 12.19.Does the introduction of the higher-order non-linear term atℓ

2 in eqn 12.18 change the be-havior in an important way? Rather than ex-ercising your expertise in analytic solutions ofnonlinear differential equations, eqn 12.20 pro-vides not fℓ and tℓ as functions of ℓ, but therelation between the two:

fℓ(tℓ) =f0

"tℓt0

#Dν

− aνtℓ2

(2−Dν)(1− (tℓ/t0)

−(2−Dν)).

(12.20)(c) Show that fℓ(tℓ) in eqn 12.20 satisfies thedifferential equation given by eqn 12.18, using

dfℓdtℓ

=dfℓdℓ

$dtℓdℓ

(12.21)

Show that it has the correct initial conditions atℓ = 0. What is fℓ∗ at ℓ∗, where tℓ = 1? Showyour method.Again, it is important to remember that fℓ(tℓ)is not the free energy as a function of temper-ature – it is the coarse-grained free energy asa function of the coarse-grained temperature ofa system starting at a free energy f0 at a tem-perature t0. It is f0(t0) that we want to know.Since here we have only one relevant variable(in zero field), all the flows lead to the same27

final point fℓ∗(tℓ∗ = 1)

25The total free energy of a system stays the same under coarse-graining (tracing out some degrees of freedom), and one cansee that the free energy per spin thus must increase as dfcoarsened/dℓ = Df due to the reduction in the number of spins. Butthe RG has both coarse-graining and rescaling. Why is the total free energy not rescaled? Also, part of the coarse-grained freeenergy is left in an analytic background – why is the singular part governed by eqn 12.18 scaled to preserve the total singularfree energy?26These are distinct from singular corrections to scaling that arise, for example, from irrelevant terms under the renormalizationgroup, that would produce subdominant corrections to c with powers t−α+∆ where ∆ is not an integer, and is bigger than zero(hence subdominant).27Remember for systems with more than one variable (say t and h), the free energy depends on the invariant curve departingfrom the fixed point (labeled, say, by h/tβδ = hℓ∗(tℓ∗ = 1). We solve for f0(t0, h0) in terms of fℓ∗(1, hℓ∗ ) just as we do inpart (b).



(d) Solve for f0 in terms of fℓ∗ and t0. Solvefor the specific heat c = T∂2f/∂T 2, wheret = (T −Tc)/Tc. Show that it can be written inthe form

c = c+analytic(t) + t−αc∗analytic(t) (12.22)

where the additive correction c+analytic(t) andthe multiplicative correction c∗analytic(t) have asimple Taylor series about t = 0. Write thesetwo corrections, in terms of fℓ∗ , Tc, a, ν, andD.Here the nonlinear term a gives us not only asmooth multiplicative term in the specific heat,but also a smooth additive background. Thisclearly is also expected in a real physical sys-tem – other degrees of freedom uncoupled tothe magnetization, or even the box holding themagnet, will contribute a specific heat that isanalytic near Tc.

(12.4) Beyond power laws: Nonlinear flows and

logarithms in the 2D Ising model.28 ⃝3The two dimensional Ising model has a logarith-mic singularity in the specific heat. The exactresult shows that the specific heat per spin is

c(T ) =kB2π

"2J

kBTc

#2 %− log(1− T/Tc)

+ log(kBTc/(2J) − (1 + π/4)&

=− 8πkBT 2

clog

"t

1/2kBTc exp(−(1 + π/4))

#

= −c0 log

"tτ

#

(12.23)where t = (T − Tc)/Tc and we set J = 1. (Re-member log(t) is negative for small t.) Lin-earized flows around the renormalization groupfixed point predict c ∼ t−α, and when α → 0one often observes logarithmic corrections. Butsuch corrections are not predicted by the lin-earized flows. The key nonlinear term is theterm atℓ

2 of eqn 12.18 in Exercise 12.3.(a) Is the solution for fℓ(tℓ) in eqn 12.20 use-ful in D = 2? Why or why not? (Hint: Theexponent ν = 1 for the two-dimensional Isingmodel.)Again, we provide the solution of the nonlinearRG eqns 12.18 for D = 2

fℓ(tℓ) = f0(tℓ/t0)2 + atℓ

2 log(tℓ/t0). (12.24)

(b) Show that fℓ(tℓ) in eqn 12.24 satisfies thedifferential equation given by eqn 12.18, withthe correct initial conditions. Solve for f0 interms of fℓ∗ and t0, where tℓ∗ = 1. Solvefor the specific heat c = T∂2f/∂T 2, wheret = t0 = (T − Tc)/Tc and f = f0. (Remem-ber the chain rule: ∂f/∂T = (∂f/∂t)(dt/dT ).)Does it agree asymptotically with the exact re-sult in eqn 12.23? What are c0 and τ , in termsof a and fℓ∗?

(12.5) The Gutenberg Richter law. (Scaling) ⃝3Power laws often arise at continuous transitionsin non-equilibrium extended systems, particu-larly when disorder is important. We don’t yethave a complete understanding of earthquakes,but they seem clearly related to the transitionbetween pinned and sliding faults as the tec-tonic plates slide over one another.The size S of an earthquake (the energy radi-ated, shown in the upper axis of Figure 12.3b)is traditionally measured in terms of a ‘magni-tude’ M ∝ log S (lower axis). The Gutenberg-Richter law tells us that the number of earth-quakes of magnitude M ∝ log S goes downas their size S increases. Figure 12.3b showsthat the number of avalanches of magnitude be-tween M and M + 1 is proportional to S−B

with B ≈ 2/3. However, it is traditional in thephysics community to consider the probabilitydensity P (S) of having an avalanche of size S.If P (S) ∼ S−τ , give a formula for τ in termsof the Gutenberg-Richter exponent B. (Hint:The bins in the histogram have different rangesof size S. Use P (M) dM = P (S) dS.)

(12.6) Period Doubling. (Scaling) ⃝pMost of you will be familiar with the perioddoubling route to chaos, and the bifurcation di-agram shown below. (See also Section 12.3.3).

28This exercise developed in collaboration with Colin Clement


Exercises 325

δ

α

Fig. 12.20 Scaling in the period doubling bi-furcation diagram. Shown are the points x onthe attractor (vertical) as a function of the con-trol parameter µ (horizontal), for the logistic mapf(x) = 4µx(1 − x), near the transition to chaos.

The self-similarity here is not in space, but intime. It is discrete instead of continuous; thebehavior is the similar if one rescales time by afactor of two, but not by a factor 1 + ϵ. Henceinstead of power laws we find a discrete self-similarity as we approach the critical point µ∞.(a) From the diagram shown, roughly estimatethe values of the Feigenbaum numbers δ (gov-erning the rescaling of µ−µ∞) and α (goveringthe rescaling of x − xp, where xp = 1/2 is thepeak of the logistic map). (Hint: be sure tocheck the signs.)Remember that the relaxation time for the Isingmodel became long near the critical tempera-ture; it diverges as t−ζ where t measures thedistance to the critical temperature. Remem-ber that the correlation length diverges as t−ν .Can we define ζ and ν for period doubling?(b) If each rescaling shown doubles the periodT of the map, and T grows as T ∼ (µ∞ −µ)−ζ

near the onset of chaos, write ζ in terms of αand δ. If ξ is the smallest typical length scaleof the attractor, and we define ξ ∼ (µ∞ − µ)−ν

(as is traditional at thermodynamic phase tran-sitions), what is ν in terms of α and δ? (Hint:be sure to check the signs.)

(12.7) Random Walks. (Scaling) ⃝3Self-similar behavior also emerges withoutproximity to any obvious transition. Onemight say that some phases naturally have self-similarity and power laws. Mathematicianshave a technical term generic which roughlytranslates to ‘without tuning a parameter to

a special value’, and so this is termed genericscale invariance.The simplest example of generic scale invari-ance is that of a random walk. Figure 12.21shows that a random walk appears statisticallyself-similar.

Fig. 12.21 Random walk scaling. Each boxshows the first quarter of the random walk in theprevious box. While each figure looks different indetail, they are statistically self-similar. That is,an ensemble of medium-length random walks wouldbe indistinguishable from an ensemble of suitablyrescaled long random walks.

Let X(T ) ='T

t=1 ξt be a random walk oflength T , where ξt are independent randomvariables chosen from a distribution of meanzero and finite standard deviation. Derivethe exponent ν governing the growth of theroot-mean-square end-to-end distance d(T ) =(

⟨(X(T )−X(0))2⟩ with T . Explain the con-nection between this and the formula from fresh-man lab courses for the way the standard de-viation of the mean scales with the number ofmeasurements.



(12.8) Hysteresis and Barkhausen Noise. (Scal-ing) ⃝iHysteresis is associated with abrupt phase tran-sitions. Supercooling and superheating are ex-amples (as temperature crosses Tc). Magneticrecording, the classic place where hysteresis isstudied, is also governed by an abrupt phasetransition – here the hysteresis in the magne-tization, as the external field H is increased(to magnetize the system) and then decreasedagain to zero. Magnetic hysteresis is character-ized by crackling (Barkhausen) electromagneticnoise. This noise is due to avalanches of spinsflipping as the magnetic interfaces jerkily arepushed past defects by the external field (muchlike earthquake faults jerkily responding to thestresses from the tectonic plates). It is inter-esting that when dirt is added to this abruptmagnetic transition, it exhibits the power-lawscaling characteristic of continuous transitions.Our model of magnetic hysteresis (unlike theexperiments) has avalanches and scaling only ata special critical value of the disorder Rc ∼ 2.16(Figure 12.14). The integrated probability dis-tribution D(S,R) has a power law D(S,Rc) ∝S−τ at the critical point (where τ = τ + σβδfor our model) but away from the critical pointtakes the scaling form

D(S,R) ∝ S−τD(Sσ (R −Rc)). (12.25)

Note from eqn (12.25) that at the critical disor-der R = Rc the distribution of avalanche sizesis a power law D(S,Rc) = S−τ . The scalingform controls how this power law is altered asR moves away from the critical point. FromFigure 12.14 we see that the main effect of mov-ing above Rc is to cut off the largest avalanchesat a typical largest size Smax(R), and anotherimportant effect is to form a ‘bulge’ of extraavalanches just below the cut–off.Using the scaling form from eqn 12.25, withwhat exponent does Smax diverge as r = (Rc −R) → 0? (Hint: At what size S is D(S,R), say,one millionth of S−τ?) Given τ ≈ 2.03, howdoes the mean ⟨S⟩ and the mean-square ⟨S2⟩avalanche size scale with r = (Rc−R)? (Hint:Your integral for the moments should have alower cutoff S0, the smallest possible avalanche,but no upper cutoff, since that is provided bythe scaling function D. Assume D(0) > 0.

Change variables to Y = Sσr. Which momentsdiverge?)

(12.9) First to fail: Weibull.29 (Mathematics,Statistics, Engineering) ⃝3Suppose you have a brand-new supercomputerwith N = 1000 processors. Your parallelizedcode, which uses all the processors, cannot berestarted in mid-stream. How long a time t canyou expect to run your code before the firstprocessor fails?This is example of extreme value statistics (seealso exercises 12.10 and 12.11), where here weare looking for the smallest value of N randomvariables that are all bounded below by zero.For large N the probability distribution ρ(t)and survival probability S(t) =

)∞t

ρ(t′) dt′ areoften given by the Weibull distribution

S(t) = e−(t/α)γ ,

ρ(t) = −dSdt

=γα

"tα

#γ−1

e−(t/α)γ .(12.26)

Let us begin by assuming that the processorshave a constant rate Γ of failure, so the prob-ability density of a single processor failing attime t is ρ1(t) = Γ exp(−Γt) as t → 0), andthe survival probability for a single processorS1(t) = 1−

) t

0ρ1(t

′)dt′ ≈ 1−Γt for short times.(a) Using (1 − ϵ) ≈ exp(−ϵ) for small ϵ, showthat the the probability SN (t) at time t that allN processors are still running is of the Weibullform (eqn 12.26). What are α and γ?Often the probability of failure per unit timegoes to zero or infinity at short times, ratherthan to a constant. Suppose the probability offailure for one of our processors

ρ1(t) ∼ Btk (12.27)

with k > −1. (So, k < 0 might reflect abreaking-in period, where survival for the firstfew minutes increases the probability for latersurvival, and k > 0 would presume a dominantfailure mechanism that gets worse as the pro-cessors wear out.)(b) Show the survival probability for N identi-cal processors each with a power-law failure rate(eqn 12.27) is of the Weibull form for large N ,and give α and γ as a function of B and k.The parameter α in the Weibull distributionjust sets the scale or units for the variable t;only the exponent γ really changes the shape

29Developed with the assistance of Paul (Wash) Wawrzynek


Exercises 327

of the distribution. Thus the form of the fail-ure distribution at large N only depends uponthe power law k for the failure of the individualcomponents at short times, not on the behaviorof ρ1(t) at longer times. This is a type of uni-versality,30 which here has a physical interpre-tation; at large N the system will break downsoon, so only early times matter.The Weibull distribution, we must mention, isoften used in contexts not involving extremalstatistics. Wind speeds, for example, are nat-urally always positive, and are conveniently fitby Weibull distributions.

Advanced discussion: Weibull and fracturetoughnessWeibull developed his distribution when study-ing the fracture of materials under externalstress. Instead of asking how long a time t asystem will function, Weibull asked how big aload σ the material can support before it willsnap.31 Fracture in brittle materials often oc-curs due to pre-existing microcracks, typicallyon the surface of the material. Suppose wehave an isolated32 microcrack of length L in a(brittle) concrete pillar, lying perpendicular tothe external stress. It will start to grow whenthe stress on the beam reaches a critical valueroughly33 given by

σc(L) ≈ Kc/√πL. (12.28)

Here Kc is the critical stress intensity fac-tor, a material-dependent property which ishigh for steel and low for brittle materials likeglass. (Cracks concentrate the externally ap-plied stress σ at their tips into a square–rootsingularity; longer cracks have more stress toconcentrate, leading to eqn 12.28.)The failure stress for the material as a wholeis given by the critical stress for the longestpre-existing microcrack. Suppose there are Nmicrocracks in a beam. The length L of eachmicrocrack has a probability distribution ρ(L).

(c) What is the probability distribution ρ1(σ) forthe critical stress σc for a single microcrack, interms of ρ(L)? (Hint: Consider the populationin a small range dσ, and the same populationin the corresponding range dℓ.)The distribution of microcrack lengths dependson how the material has been processed. Thesimplest choice, an exponential decay ρ(L) ∼(1/L0) exp(−L/L0), perversely does not yielda Weibull distribution, since the probability ofa small critical stress does not vanish as a powerlaw Bσk (eqn 12.27).(d) Show that an exponential decay of microc-rack lengths leads to a probability distributionρ1(σ) that decays faster than any power law atσ = 0 (i.e., is zero to all orders in σ). (Hint:You may use the fact that ex grows faster thanxm for any m as x → ∞.)Analyzing the distribution of failure stressesfor a beam with N microcracks with this ex-ponentially decaying length distribution yieldsa Gumbel distribution [138, section 8.2], not aWeibull distribution.Many surface treatments, on the other hand,lead to power-law distributions of microcracksand other flaws, ρ(L) ∼ CL−η with η > 1. (Forexample, fractal surfaces with power-law corre-lations arise naturally in models of corrosion,and on surfaces exposed by previous fractures.)(e) Given this form for the length distribution ofmicrocracks, show that the distribution of frac-ture thresholds ρ1(σ) ∝ σk. What is k in termsof η?According to your calculation in part (b), thisimmediately implies a Weibull distribution offracture strengths as the number of microcracksin the beam becomes large.

(12.10) Biggest of bunch: Gumbel. (Mathematics,Statistics, Engineering) ⃝3Much of statistical mechanics focuses on the av-erage behavior in an ensemble, or the meansquare fluctuations about that average. In

distribution forms a one-parameter family of universality classes; see chapter 12.31Many properties of a steel beam are largely independent of which beam is chosen. The elastic constants, the thermal con-ductivity, and the the specific heat depends to some or large extent on the morphology and defects in the steel, but nonethelessvary little from beam to beam—they are self-averaging properties, where the fluctuations due to the disorder average outfor large systems. The fracture toughness of a given beam, however, will vary significantly from one steel beam to another.Self-averaging properties are dominated by the typical disordered regions in a material; fracture and failure are nucleated atthe extreme point where the disorder makes the material weakest.32The interactions between microcracks are often not small, and are a popular research topic.33This formula assumes a homogeneous, isotropic medium as well as a crack orientation perpendicular to the external stress.In concrete, the microcracks will usually associated with grain boundaries, second-phase particles, porosity. . .



many cases, however, we are far more interestedin the extremes of a distribution.Engineers planning dike systems are interestedin the highest flood level likely in the next hun-dred years. Let the high water mark in year j beHj . Ignoring long-term weather changes (likeglobal warming) and year-to-year correlations,let us assume that each Hj is an independentand identically distributed (IID) random vari-able with probability density ρ1(Hj). The cu-mulative distribution function (cdf) is the prob-ability that a random variable is less than agiven threshold. Let the cdf for a single year beF1(H) = P (H ′ < H) =

)Hρ1(H

′) dH ′.(a) Write the probability FN(H) that the high-est flood level (largest of the high-water marks)in the next N = 1000 years will be less than H,in terms of the probability F1(H) that the high-water mark in a single year is less than H.The distribution of the largest or smallest of Nrandom numbers is described by extreme valuestatistics [138]. Extreme value statistics is avaluable tool in engineering (reliability, disasterpreparation), in the insurance business, and re-cently in bioinformatics (where it is used to de-termine whether the best alignments of an un-known gene to known genes in other organismsare significantly better than that one wouldgenerate randomly).(b) Suppose that ρ1(H) = exp(−H/H0)/H0 de-cays as a simple exponential (H > 0). Using theformula

(1− A) ≈ exp(−A) small A (12.29)

show that the cumulative distribution functionFN for the highest flood after N years is

FN (H) ≈ exp

*− exp

"µ−H

β

#+. (12.30)

for large H. (Why is the probability FN(H)small when H is not large, at large N?) Whatare µ and β for this case?The constants β and µ just shift the scale andzero of the ruler used to measure the variableof interest. Thus, using a suitable ruler, thelargest of many events is given by a Gumbeldistribution

F (x) = exp(− exp(−x))

ρ(x) = ∂F/∂x = exp(−(x+ exp(−x))).(12.31)

How much does the probability distribution forthe largest of N IID random variables dependon the probability density of the individual ran-dom variables? Surprisingly little! It turns outthat the largest of N Gaussian random vari-ables also has the same Gumbel form that wefound for exponentials. Indeed, any probabilitydistribution that has unbounded possible valuesfor the variable, but that decays faster than anypower law, will have extreme value statisticsgoverned by the Gumbel distribution [96, sec-tion 8.3]. In particular, suppose

F1(H) ≈ 1− A exp(−BHδ) (12.32)

as H → ∞ for some positive constants A, B,and δ. It is in the region near H∗[N ], definedby F1(H

∗[N ]) = 1− 1/N , that FN varies in aninteresting range (because of eqn 12.29).(c) Show that the extreme value statisticsFN (H) for this distribution is of the Gumbelform (eqn 12.30) with µ = H∗[N ] and β =1/(BδH∗[N ]δ−1). (Hint: Taylor expand F1(H)at H∗ to first order.)The Gumbel distribution is universal. It de-scribes the extreme values for any unboundeddistribution whose tails decay faster than apower law.34 (This is quite analogous to thecentral limit theorem, which shows that thenormal or Gaussian distribution is the universalform for sums of large numbers of IID randomvariables, so long as the individual random vari-ables have non-infinite variance.)The Gaussian or standard normal distri-bution ρ1(H) = (1/

√2π) exp(−H2/2), for

example, has a cumulative distributionF1(H) = (1/2)(1 + erf(H/

√2)) which at

large H has asymptotic form F1(H) ∼ 1 −(1/

√2πH) exp(−H2/2). This is of the general

form of eqn 12.32 with B = 1/2 and δ = 2, ex-cept that A is a slowly varying function of H .This slow variation does not change the asymp-totics. Hints for the numerics are available inthe computer exercises section of the text Website [131].(d) Generate M = 10000 lists of N = 1000random numbers distributed with this Gaussianprobability distribution. Plot a normalized his-togram of the largest entries in each list. Plotalso the predicted form ρN(H) = dFN/dH

34The Gumbel distribution can also describe extreme values for a bounded distribution, if the probability density at theboundary goes to zero faster than a power law [138, section 8.2].


Exercises 329

from part (c). (Hint: H∗(N) ≈ 3.09023 forN = 1000; check this if it is convenient.)Other types of distributions can have extremevalue statistics in different universality classes(see Exercise 12.11). Distributions with power-law tails (like the distributions of earthquakesand avalanches described in Chapter 12) haveextreme value statistics described by Frechetdistributions. Distributions that have a strictupper or lower bound35 have extreme value dis-tributions that are described by Weibull statis-tics (see Exercise 12.9).

(12.11) Extreme value statistics: Gumbel,

Weibull, and Frechet. (Mathematics,Statistics, Engineering) ⃝3Extreme value statistics is the study of the max-imum or minimum of a collection of randomnumbers. It has obvious applications in theinsurance business (where one wants to knowthe biggest storm or flood in the next decades,see Exercise 12.10) and in the failure of largesystems (where the weakest component or flawleads to failure, see Exercise 12.9). Recently ex-treme value statistics has become of significantimportance in bioinformatics. (In guessing thefunction of a new gene, one often searches en-tire genomes for good matches (or alignments)to the gene, presuming that the two genes areevolutionary descendents of a common ances-tor and hence will have similar functions. Onemust understand extreme value statistics toevaluate whether the best matches are likely toarise simply at random.)The limiting distribution of the biggest orsmallest ofN random numbers as N → ∞ takesone of three universal forms, depending on theprobability distribution of the individual ran-dom numbers. In this exercise we understandthese forms as fixed points in a renormalizationgroup.Given a probability distribution ρ1(x), we de-fine the cumulative distribution function (CDF)as F1(x) =

) x−∞ ρ(x′) dx′. Let us define ρN (x)

to be the probability density that, out of Nrandom variables, the largest is equal to x. LetFN (x) to be the corresponding CDF.(a) Write a formula for F2N (x) in terms ofFN (x). If FN(x) = exp(−gN(x)), show thatg2N (x) = 2gN(x).

Our renormalization group coarse-graining op-eration will remove half of the variables, throw-ing away the smaller of every pair, and re-turning the resulting new probability distri-bution. In terms of the function g(x) =− log

) x

−∞ ρ(x′)dx′, it therefore will return arescaled version of the 2g(x). This rescalingis necessary because, as the sample size N in-creases, the maximum will drift upward—onlythe form of the probability distribution staysthe same, the mean and width can change.Our renormalization-group coarse-graining op-eration thus maps function space into itself, andis of the form

T [g](x) = 2g(ax+ b). (12.33)

(This renormalization group is the same as thatwe use for sums of random variables in Exer-cise 12.24 where g(k) is the logarithm of theFourier transform of the probability density.)There are three distinct types of fixed-point dis-tributions for this renormalization group trans-formation, which (with an appropriate linearrescaling of the variable x) describe most ex-treme value statistics. The Gumbel distribu-tion (Exercise 12.10) is of the form

Fgumbel(x) = exp(− exp(−x))

ρgumbel(x) = exp(−x) exp(− exp(−x)).

ggumbel(x) = exp(−x)

The Weibull distribution (Exercise 12.9) is ofthe form

Fweibull(x) =

,exp(−(−x)α) x < 0

1 x ≥ 0

gweibull(x) =

,(−x)α x < 0

0 x ≥ 0,

(12.34)

and the Frechet distribution is of the form

Ffrechet(x) =

,0 x ≤ 0

exp(−x−α) x > 0

gfrechet(x) =

,∞ x < 0

x−α x ≥ 0,

(12.35)

where α > 0 in each case.(b) Show that these distributions are fixedpoints for our renormalization-group transfor-mation eqn 12.33. What are a and b for eachdistribution, in terms of α?

ounded distributions that have power-law asymptotics have Weibull statistics; see note 34 and Exer-



In parts (c) and (d) you will show that thereare only these three fixed points g∗(x) forthe renormalization transformation, T [g∗](x) =2g∗(ax+ b), up to an overall linear rescaling ofthe variable x, with some caveats. . .(c) First, let us consider the case a = 1. Showthat the rescaling x → ax + b has a fixed pointx = µ. Show that the most general form for thefixed-point function is

g∗(µ± z) = zα′

p±(γ log z) (12.36)

for z > 0, where p± is periodic and α′ and γ areconstants such that p± has period equal to one.(Hint: Assume p(y) ≡ 1, find α′, and then showg∗/zα

′

is periodic.) What are α′ and γ? Whichchoice for a, p+, and p− gives the Weibull dis-tribution? The Frechet distribution?Normally the periodic function p(γ log(x − µ))is assumed or found to be a constant (some-times called 1/β, or 1/βα′

). If it is not constant,then the probability density must have an infi-nite number of oscillations as x → µ, forming aweird essential singularity.(d) Now let us consider the case a = 1. Showagain that the fixed-point function is

g∗(x) = e−x/βp(x/γ) (12.37)

with p periodic of period one, and with suitableconstants β and γ. What are the constants interms of b? What choice for p and β yields theGumbel distribution?Again, the periodic function p is often assumeda constant (eµ), for reasons which are not asobvious as in part (c).What are the domains of attraction of the threefixed points? If we want to study the maxi-mum of many samples, and the initial probabil-ity distribution has F (x) as its CDF, to whichuniversal form will the extreme value statis-tics converge? Mathematicians have sorted outthese questions. If ρ(x) has a power-law tail, so1−F (x) ∝ x−α, then the extreme value statis-tics will be of the Frechet type, with the same α.If the initial probability distribution is boundedabove at µ and if 1 − F (µ− y) ∝ yα, then theextreme value statistics will be of the Weibulltype. (More commonly, Weibull distributionsarise as the smallest value from a distributionof positive random numbers, Exercise 12.9.) If

the probability distribution decays faster thanany polynomial (say, exponentially) then theextreme value statistics will be of the Gum-bel form. (Gumbel extreme-value statistics canalso arise for bounded random variables if theprobability decays to zero faster than a powerlaw at the bound.)

(12.12) Diffusion equation and universal scaling

functions.36 ⃝2The diffusion equation universally describes mi-croscopic hopping systems at long length scales.We will investigate how to write the evolutionin a universal scaling form.The solution to a diffusion problem with anon-zero drift velocity is given by ρ(x, t) =1/

√4πDt exp

-−((x− vt)2/(4Dt)

.. We will

coarse grain by throwing away half the timepoints. We will then rescale the distributionso it looks like the original distribution. Wecan just write these two operations as t′ = t/2,x′ = x/

√2, ρ′ =

√2ρ.37 These three together

constitute our renormalization group operation.(a) Write an expression for ρ′(x′, t′) in termsof D, v, x′, and t′ (not in terms of D′ and v′).Use it to determine the new renormalized veloc-ity v′ and diffusion constant D′. Are v and Drelevant, irrelevant or marginal variables?Typically, whenever writing properties in ascaling function, there is some freedom in decid-ing which invariant combinations to use. Herelet us use the invariant combination of vari-ables, X = x/

√t and V =

√tv. We can then

write

ρ(x, t) = t−αP(X ,V, D), (12.38)

a power law times a universal scaling functionof invariant combination of variables.(b) Show that X and V are invariant under ourrenormalization group operation. What is α?Write an expression for P, in terms of X , V,and D (and not x, v, or t).(Note that we need to solve the diffusion equa-tion to find the universal scaling function P ,but we can learn a lot from just knowing thatit is a fixed point of the renormalization group.So, the universal exponent α and the invari-ant scaling combinations X , V, and D are de-termined just by the coarsening and rescaling

36This exercise was developed in collaboration with Archishman Raju.37Because ρ is a density, we need to rescale ρ′dx′ = ρdx.


Exercises 331

steps in the renormalization group. In experi-ments and simulations, one often uses data toextract the universal critical exponents and uni-versal scaling functions, relying on emergentscale invariance to tell us that a scaling formlike eqn (12.38) is expected.)

(12.13) Avalanche Size Distribution. (Scalingfunction) ⃝3One can develop a mean-field theory foravalanches in non-equilibrium disordered sys-tems by considering a system of N Ising spinscoupled to one another by an infinite-range in-teraction of strength J/N , with an externalfield H and each spin also having a local ran-dom field h:

H = −J0/N/

i,j

SiSj−/

i

(H+hi)Si. (12.39)

We assume that each spin flips over when it ispushed over; i.e., when its change in energy

H loci =

∂H∂Si

= J0/N/

j

Sj +H + hi

= J0m+H + hi

changes sign.38 Here m = (1/N)'

j Sj is theaverage magnetization of the system. All spinsstart by pointing down. A new avalanche islaunched when the least stable spin (the un-flipped spin of largest local field) is flippedby increasing the external field H . Each spinflip changes the magnetization by 2/N . If themagnetization change from the first spin flip isenough to trigger the next-least-stable spin, theavalanche will continue.We assume that the probability density for therandom field ρ(h) during our avalanche is a con-stant

ρ(h) = (1 + t)/(2J0). (12.40)

The constant t will measure how close the den-sity is to the critical density 1/(2J0).(a) Show that at t = 0 each spin flip will trig-ger on average one other spin to flip, for largeN . Can you qualitatively explain the differencebetween the two phases with t < 0 and t > 0?We can solve exactly for the probability D(S, t)of having an avalanche of size S. To have an

avalanche of size S triggered by a spin with ran-dom field h, you must have precisely S−1 spinswith random fields in the range h, h+2J0S/N(triggered when the magnetization changes by2S/N). The probability of this is given by thePoisson distribution. In addition, the randomfields must be arranged so that the first spintriggers the rest. The probability of this turnsout to be 1/S.(b) (Optional) By imagining putting periodicboundary conditions on the interval h, h +2J0S/N, argue that exactly one spin out of thegroup of S spins will trigger the rest as a singleavalanche. (Hint from Ben Machta: For sim-plicity, we may assume39 the avalanche startsat H = m = 0. Try plotting the local fieldH loc(h′) = J0m(h′) + h′ that a spin with ran-dom field h′ would feel if the spins between h′

and h were flipped. How would this functionchange if we shuffle all the random fields aroundthe periodic boundary conditions?)(c) Show that the distribution of avalanche sizesis thus

D(S, t) =SS−1

S!(t+ 1)S−1e−S(t+1). (12.41)

With t small (near to the critical density) andfor large avalanche sizes S we expect this tohave a scaling form:

D(S, t) = S−τD(S/t−x) (12.42)

for some mean-field exponent x. That is, tak-ing t → 0 and S → ∞ along a path with Stx

fixed, we can expand D(S, t) to find the scalingfunction.(d) Show that τ = 3/2 and x = 2. What isthe scaling function D? Hint: You’ll need touse Stirling’s formula S! ∼

√2πS(S/e)S for

large S, and that 1 + t = exp(log(1 + t)) ≈et−t2/2+t3/3....This is a bit tricky to get right. Let’s check itby doing the plots.(e) Plot SτD(S, t) versus Y = S/t−x for t =0.2, 0.1, and 0.05 in the range Y ∈ (0, 10). Doesit converge to D(Y )?(See “Hysteresis and hierarchies, dynamics ofdisorder-driven first-order phase transforma-tions”, J. P. Sethna, K. Dahmen, S. Kartha,

38We ignore the self-interaction, which is unimportant at large N39Equivalently, measure the random fields with respect to h0 of the triggering spin,and let m be the magnetization change since the avalanche started.



J. A. Krumhansl, B. W. Roberts, and J. D.Shore, PRL 70, 3347 (1993) for more informa-tion.)

(12.14) Conformal Invariance.40 (Mathematics) ⃝3Emergence in physics describes new laws thatarise from complex underpinnings, and oftenexhibits a larger symmetry than the originalmodel. The diffusion equation emerges as acontinuum limit from complex random micro-scopic motion, and diffusion on a square lat-tice has circular symmetry. Critical phenomenaemerges near continuous phase transitions, andthe resultant symmetry under dilations is ex-ploited by the renormalization group to predictuniversal power laws and scaling functions.The Ising model on a square lattice at the criti-cal point, like the diffusion equation, also has anemergent circular symmetry: the complex pat-terns of up and down spins look the same onlong length scales also when rotated by an an-gle. Indeed, making use of the symmetries un-der changes of length scale, position, and angle(plus one spatially nonuniform transformation),systems at their critical points have a conformalsymmetry group.In two dimensions, the conformal symmetrygroup becomes huge. Roughly speaking, anycomplex analytic function f(z) = u(x + iy) +iv(x + iy) takes a snapshot of an Ising modelM(x, y) and warps it into a new magnetizationpattern at (u, v) that ‘looks the same’. (Hereu, v, x, and y are all real.)You may remember that most ordinary func-tions (like z2,

√z, sin(z), log(z), and exp(iz))

are analytic. All of them yield cool transfor-mations of the Ising model – weird and warpedwhen magnified until you see the pixels, butrecognizably Ising-like on long scales. This ex-ercise will generate an example.

Fig. 12.22 Two proteins in a critical mem-brane. The figure shows the pixels of a criti-cal Ising model simulated in a square, conformallywarping the square onto the exterior of two circles(representing two proteins in a cell membrane). Thewarped pixels vary in size – largest in the upper andlower left, smallest near the smaller circle. Theyalso locally rotate and translate the square lattice,but notice the pixels remain looking square – theangles and aspect ratios remain unchanged. Thepixels are gray rather than black and white, withonly the smallest pixels pure black and white; wemust not only warp the pixels conformally, but alsorescale the magnetization. Ignoring the pixelation,the different regions look statistically similar. Agray large pixel mimics the average color of similar-sized regions of tiny pixels.

(a) What analytic function shrinks a region uni-formly by a factor b, holding z = 0 fixed? Whatanalytic function translates the lattice by a vec-tor r0 = (u0, v0)? What analytic function ro-tates a region by an angle θ?(b) Expanding f(z + δ) = f(z) + δf ′(z), showthat an analytic function f transforms a localregion about z to linear order in δ by first ro-tating and dilating δ about z and then translat-ing. What complex number gives the net trans-lation?Figure 12.22 shows how one can use this con-formal symmetry to study the interactions be-tween circular ‘proteins’ embedded in a twodimensional membrane at an Ising criticalpoint.41

In the renormalization group, we first coarsegrain the system (shrinking by a factor b) andthen rescale the magnetization (by some powerbyM ) in order to return to statistically the samecritical state: 0M = byMM . This rescaling turns

40This exercise is based on a project of Benjamin Machta’s41We use this transformation to study the effective interaction between two circular ‘proteins’ in a two-dimensional cell mem-brane near its critical point. The energy of attraction between two ‘up’ proteins is derivable from the energy of the square-latticeIsing model with the two side boundaries set to ‘up’. (See B. B. Machta, S. L. Veatch, and J. P. Sethna, ‘Critical Casimirforces in cellular membranes, PRL 109, 138101 (2012).)


Exercises 333

the larger pixels in Fig. 12.22 more gray; amostly up-spin ‘white’ region with tiny pixels ismimicked by a large single pixel with the sta-tistically averaged gray color.We can discover the correct power for M byexamining the rescaling of the correlation func-tion C(r) = ⟨M(x)M(x + r)⟩. In the Isingmodel at its critical point the correlation func-tion C(r) ∼ r−(2−d+η). In dimension d = 2,η = 1/4. We expect that the correlation func-tion for the conformally transformed magneti-zation will be the same as the original correla-tion function.(c) If we coarse-grain by a constant factor b,what power of b must we multiply M by to makeC(r) = ⟨0M(f(z))0M(f(z) + r)⟩? Explain yourreasoning.When our conformal transformation takes apixel at M(z) to a warped pixel of area A atf(z), it rescales the magnetization by

0M(f(z)) = |A|−1/16M(z). (12.43)

The pixel area for a locally uniform compressionby b changes by |df/dz|2 = 1/b2. You may usethis to check your answer to part (c).Figure 12.23 illustrates the self-similarity forthe Ising model under rescaling, in analogy withFig. 12.11’s treatment of scale invariance in therandom-field avalanche model. Here, unlike inFig. 12.11, we incorporate the renormalizationof the magnetization by changing the grayscaleas we blow up successive lower–right–hand cor-ners.

Fig. 12.23 Ising model under conformalrescaling. The ‘powers of two’ rescaling of theavalanches in Fig. 12.11 ignored this rescaling ofM . Here we show the Ising model, again with thelowest right-hand quarter of each square inflated tofill the next – but now properly faded according toeqn 12.43.

Let us now explore the image of the Ising modelunder various analytic functions. Generate asnapshot of an Ising model, equilibrated at thecritical temperature.42

Notice that Mathematica’s notebook has trou-ble with plotting large numbers of polygons.Use Python. Or, if you prefer – save often, trynot using a second display (a known problem),try not opening other programs in the back-ground (including second Mathematica files),and avoid typing or otherwise disturbing theMathematica window while it is running a longtask. Debug everything with small systems,directly save your graphs of larger systems toa file, and display them outside Mathematica.

42You can get one from Matt Bierbaum’s Web simulation at http://mattbierbaum.github.io/ising.js/)of size at least 256x256, or download one from the course Web site (along with hints files) athttp://pages.physics.cornell.edu/∼sethna/teaching/562/HW.html.43If you work in Mathematica or Fortran, where the indices of arrays run from (1. . . L), zmn =

!(m− 1/2) + i(n− 1/2)

"/L.

http://mattbierbaum.github.io/ising.js/

http://pages.physics.cornell.edu/~sethna/teaching/562/HW.html



In either language, 512x512 will strain yourmachine; anything over 128x128 is acceptable,but larger systems will make the physics a bitclearer.The hints files will allow you to import an Isingimage, convert it into a two-dimensional arraySmn = ±1, and select an L×L subregion (if youwish, especially while you debug your code).We imagine these spins spread over the unitsquare in the complex plane; our code gener-ates a list of spins and square polygons associ-ated to each, with the spin Smn sitting at thecenter43 zmn =

-(m+ 1/2) + i(n+ 1/2)

./L of a

1/L × 1/L square. The code will allow you toprovide a function f(z) of your choice, and willreturn the deformed quadrilaterals44 centeredat f(zmn) with areas Amn, and their associatedrescaled spins A−1/16

mn Smn.45 The software will

also provide example routines showing how tooutput and shade in these quadrilaterals.(d) Generate a conformal transformation witha nonlinear analytic function of your choice,warping the Ising mode in an interesting way.Zoom in to regions occupied by lots of smallpixels, where the individual pixels are not ob-vious.46 Include both your entire plot and acropped figure showing the expanded zoom. Dis-cuss your zoomed plot critically – does it appearthat the Ising correlations and visual structureshave been faithfully retained by your analytictransformation?(e) Load an Ising model equilibrated above thecritical temperature at T = 100 (random noise),and one at T = 3 (short-range correlations).Distort and zoom each using your chosen con-formal transformation. Analyze each and in-clude your images. If you ‘blur your eyes’enough to ignore the individual pixels, can youtell how much the system has been dilated? Areyour conformally transformed images faithfullyretaining the correlations and visual structuresaway from the critical point? For T = 3, which

regions look qualitatively like Tc? Which regionslook like T = 100?(f) Invent a non-analytic function, and use it todistort your Ising model. (Warning: most func-tions you write down, like log(cosh4(z + 1/z2))will be analytic except at a few singularities.The author tried two methods: inventing real-valued functions u(x, y) and v(x, y) and form-ing f = u + iv, and picking two different ana-lytic functions g(z) and h(z) and using f(z) =Re(g(z)) + iIm(h(z)). Make sure your func-tion is non-analytic almost everywhere (e.g., vi-olates the Cauchy-Riemann equations), not justat a point.)47 Find an example that makes foran interesting picture; include your images, in-cluding a zoom into a region with many pixelsthat range in size. As above, examine the im-ages critically – does it appear that the Isingcorrelations and visual structures have beenfaithfully retained by your non-analytic trans-formation? Describe the distortions you see.(Are the pixels still approximately square?)Conformal symmetry in two dimensions wasstudied in an outgrowth of string theory. Therepresentations of the conformal group allowedthem to deduce the exact critical exponents forall the usual two dimensional statistical me-chanical models – reproducing Onsager’s re-sult for the 2D Ising model, known solutionsfor the 2D tri-critical Ising model, 2D per-colation, . . .More recently, field theorists [71]have used conformal invariance in higher di-mensions (with a strategy called ‘conformalbootstrap’) to produce bounds for critical ex-ponents. They now hold the record on accu-racy for the exponents of the 3D Ising model,giving β = 0.326419(3), ν = 0.629971(4), andδ = 4.78984(1).

(12.15) Ising self-similarity. ⃝iStart up the Ising model (computer exercisesportion of the book web site [131]). Run a large

44The routine will drop quadrilaterals that extend to infinity, and also will remove quadrilaterals with ‘negativlatter are associated with pixels which get inverted by f(z); plotting packages usually do not provide routinesexterior of a polygon.45The list of quadrilaterals and spins will be linear, not two-dimensional, since graphics routines plotting polygonsflattened lists.

46You can zoom either using the graphics software or (sometimes more efficient) by saving a vector-graphics figure (like pdf)and viewing it separately. Start with L = 64 or so to make plots and zooming efficient, but for your final plots use L as largeas feasible.47Warning: The program automatically drops quadrilaterals with ‘negative area’, which usually happen when an internal pointgoes to infinity (and the polygon should be shaded ‘outside’). This will also happen for the conjugate of an analytic function(e.g., f(x+ iy) = y+ ix = iz); you will get either errors about Transpose[] in Mathematica or an empty plot in Python. If thishappens for your choice of u(x, y) + iv(x, y), try using u− iv.


Exercises 335

system at zero external field and T = Tc =2/ log(1 +

√2) ≈ 2.26919. Set the refresh rate

low enough that graphics is not the bottle-neck,and run for at least a few hundred sweeps toequilibrate. You should see a fairly self-similarstructure, with fractal-looking up-spin clustersinside larger down-spin structures inside . . .Can you find a nested chain of three clusters?Four?

(12.16) Scaling and corrections to scaling. (Con-densed matter) ⃝pNear critical points, the self-similarity underrescaling leads to characteristic power-law sin-gularities. These dependences may be dis-guised, however, by less-singular corrections toscaling.An experiment measures the susceptibilityχ(T ) in a magnet for temperatures T slightlyabove the ferromagnetic transition temperatureTc. They find their data is fit well by the form

χ(T ) = A(T − Tc)−1.25 +B + C(T − Tc)

+D(T − Tc)1.77. (12.44)

(a) Assuming this is the correct dependencenear Tc, what is the critical exponent γ?When measuring functions of two variables nearcritical points, one finds universal scaling func-tions. The whole function is a prediction of thetheory.The pair correlation function C(r, T ) =⟨S(x)S(x + r)⟩ is measured in another, three-dimensional system just above Tc. It is foundto be spherically symmetric, and of the form

C(r, T ) = r−1.026C(r(T − Tc)0.65), (12.45)

where the function C(x) is found to be roughlyexp(−x).(b) What is the critical exponent ν? The expo-nent η?

(12.17) Scaling and coarsening. (Condensed mat-ter) ⃝pDuring coarsening, we found that the systemchanged with time, with a length scale thatgrows as a power of time: L(t) ∼ t1/2 for a non-conserved order parameter, and L(t) ∼ t1/3 fora conserved order parameter. These exponents,unlike critical exponents, are simple rationalnumbers that can be derived from argumentsakin to dimensional analysis (Section 11.4.1).Associated with these diverging length scales

there are scaling functions. Coarsening doesnot lead to a system which is self-similar to it-self at equal times, but it does lead to a systemwhich at two different times looks the same—apart from a shift of length scales.An Ising model with non-conserved magnetiza-tion is quenched to a temperature T well belowTc. After a long time t0, the correlation func-tion looks like Ccoar

t0 (r, T ) = c(r).Assume that the correlation function at shortdistances Ccoar

t (0, T, t) will be time indepen-dent, and that the correlation function at latertimes will have the same functional form apartfrom a rescaling of the length. Write the corre-lation function at time twice t0, C

coar2t0 (r, T ), in

terms of c(r). Write a scaling form

Ccoart (r, T ) = t−ωC(r/tρ, T ). (12.46)

Use the time independence of Ccoart (0, T ) and

the fact that the order parameter is not con-served (Section 11.4.1) to predict the numericalvalues of the exponents ω and ρ.It was only recently made clear that the scal-ing function C for coarsening does depend ontemperature (and is, in particular, anisotropicfor low temperature, with domain walls lin-ing up with lattice planes). Low-temperaturecoarsening is not as ‘universal’ as continuousphase transitions are (Section 11.4.1); even inone model, different temperatures have differ-ent scaling functions.

(12.18) Bifurcation theory. (Mathematics) ⃝iDynamical systems theory is the study of thetime evolution given by systems of differentialequations. Let x(t) be a vector of variablesevolving in time t, let λ be a vector of parame-ters governing the differential equation, and letFλ(x) be the differential equations

x ≡ ∂x∂t

= Fλ(x). (12.47)

The typical focus of the theory is not to solvethe differential equations for general initial con-ditions, but to study the qualitative behavior.In general, they focus on bifurcations—specialvalues of the parameters λ where the behaviorof the system changes qualitatively.(a) Consider the differential equation in onevariable x(t) with one parameter µ:

x = µx− x3. (12.48)

Show that there is a bifurcation at µc = 0, byshowing that an initial condition with small,



non-zero x(0) will evolve qualitatively differ-ently at late times for µ > 0 versus for µ < 0.Hint: Although you can solve this differentialequation explicitly, we recommend instead thatyou argue this qualitatively from the bifurca-tion diagram in Fig. 12.24; a few words shouldsuffice.Dynamical systems theory has much in com-mon with equilibrium statistical mechanics ofphases and phase transitions. The liquid–gastransition is characterized by external param-eters λ = (P, T,N), and has a current statedescribed by x = (V,E, µ). Equilibrium phasescorrespond to fixed-points (x∗(µ) with x∗ = 0)in the dynamics, and phase transitions cor-respond to bifurcations.48 For example, thepower laws we find near continuous phase tran-sitions have simpler analogues in the dynamicalsystems.

µ

x*(µ

)

Fig. 12.24 Pitchfork bifurcation diagram.The flow diagram for the pitchfork bifurcation(eqn 12.48). The dashed line represents unstablefixed-points, and the solid thick lines represent sta-ble fixed-points. The thin lines and arrows repre-sent the dynamical evolution directions. It is calleda pitchfork because of the three tines on the rightemerging from the handle on the left.

(b) Find the critical exponent β for the pitch-fork bifurcation, defined by x∗(µ) ∝ (µ − µc)

β

as µ → µc.Bifurcation theory also predicts universal be-havior; all pitchfork bifurcations have the samescaling behavior near the transition.(c) At what value λc does the differential equa-tion

m = tanh (λm)−m (12.49)

have a bifurcation? Does the fixed-point valuem∗(λ) behave as a power law m∗ ∼ |λ − λc|β

near λc (up to corrections with higher powersof λ− λc)? Does the value of β agree with thatof the pitchfork bifurcation in eqn 12.48?Just as there are different universality classesfor continuous phase transitions with differentrenormalization-group fixed points, there aredifferent classes of bifurcations each with itsown normal form. Some of the other importantnormal forms include the saddle-node bifurca-tion,

x = µ− x2, (12.50)

transcritical exchange of stability,

x = µx− x2, (12.51)

and the Hopf bifurcation,

x = (µ− (x2 + y2))x− y,

y = (µ− (x2 + y2))y + x.(12.52)

(12.19) Mean-field theory. (Condensed matter) ⃝iIn Chapter 11 and Exercise 9.10, we make ref-erence to mean-field theories, a term which isoften loosely used for any theory which absorbsthe fluctuations of the order parameter fieldinto a single degree of freedom in an effectivefree energy. The original mean-field theory ac-tually used the mean value of the field on neigh-boring sites to approximate their effects.In the Ising model on a square lattice, thisamounts to assuming each spin sj = ±1 hasfour neighbors which are magnetized with theaverage magnetization m = ⟨sj⟩, leading to aone-spin mean-field Hamiltonian

H = −4Jmsj . (12.53)

(a) At temperature kBT , what is the value for⟨sj⟩ in eqn 12.53, given m? At what temper-ature Tc is the phase transition, in mean fieldtheory? (Hint: At what temperature is a non-zero m = ⟨s⟩ self-consistent?) Argue as inExercise 12.18 part (c) that m ∝ (Tc − T )β

near Tc. Is this value for the critical expo-nent β correct for the Ising model in eithertwo dimensions (β = 1/8) or three dimensions(β ≈ 0.325)?(b) Show that the mean-field solution you foundin part (a) is the minimum in an effective

48In Section 8.3, we noted that inside a phase all properties are analytic in the parameters. Similarly, bifurcations are valuesof λ where non-analyticities in the long-time dynamics are observed.


Exercises 337

temperature-dependent free energy

V (m) = kBT1m2

2

− log (cosh(4Jm/kBT ))kBT4J

2.

(12.54)

On a single graph, plot V (m) for 1/(kBT ) =0.1, 0.25, and 0.5, for −2 < m < 2, show-ing the continuous phase transition. Comparewith Fig. 9.27.(c) What would the mean-field Hamiltonian befor the square-lattice Ising model in an externalfield H? Show that the mean-field magnetiza-tion is given by the minima in49

V (m) = kBT1m2

2

− log (cosh((H + 4Jm)/kBT ))kBT4J

2.

(12.55)

On a single graph, plot V (m,H) for β = 0.5and H = 0, 0.5, 1.0, and 1.5, showing metasta-bility and an abrupt transition. At what value ofH does the metastable state become completelyunstable? Compare with Fig. 11.2(a).

(12.20) The onset of lasing.50 (Quantum, Optics,Mathematics) ⃝3Lasers represent a stationary, condensed state.It is different from a phase of matter not onlybecause it is made up out of energy, but also be-cause it is intrinsically a non-equilibrium state.In a laser entropy is not maximized, free ener-gies are not minimized—and yet the state has arobustness and integrity reminiscent of phasesin equilibrium systems.In this exercise, we will study a system of ex-cited atoms coupled to a photon mode just be-fore it begins to lase. We will see that it exhibitsthe diverging fluctuations and scaling that wehave studied near critical points.Let us consider a system of atoms weakly cou-pled to a photon mode. We assume that N1

atoms are in a state with energy E1, N2 atomsare in a higher energy E2, and that these atomsare strongly coupled to some environment that

keeps these populations fixed.51 Below the on-set of lasing, the probability ρn(t) that the pho-ton mode is occupied by n photons obeys

dρndt

= a-nρn−1N2 − nρnN1 − (n+ 1)ρnN2

+ (n+ 1)ρn+1N1

.. (12.56)

The first term on the right-hand side representsthe rate at which one of the N2 excited atomsexperiencing n− 1 photons will emit a photon;the second term represents the rate at whichone of the N1 lower-energy atoms will absorbone of n photons; the third term representsemission in an environment with n photons, andthe last represents absorption with n + 1 pho-tons. The fact that absorption in the presenceof m photons is proportional to m and emissionis proportional to m+1 is a property of bosons(Exercises 7.16(c) and 7.3). The constant a > 0depends on the lifetime of the transition, andis related to the Einstein A coefficient (Exer-cise 7.16).(a) Find a simple expression for d⟨n⟩/dt, where⟨n⟩ =

'∞m=0 mρm is the mean number of pho-

tons in the mode. (Hint: Collect all terms in-volving ρm.) Show for N2 > N1 that this meannumber grows indefinitely with time, leading toa macroscopic occupation of photons into thissingle state—a laser.52

Now, let us consider our system just before itbegins to lase. Let ϵ = (N2 − N1)/N1 be ourmeasure of how close we are to the lasing insta-bility. We might expect the value of ⟨n⟩ to di-verge as ϵ → 0 like ϵ−ν for small ϵ. Near a phasetransition, one also normally observes criticalslowing-down: to equilibrate, the phase mustcommunicate information over large distancesof the order of the correlation length, whichtakes a time which diverges as the correlationlength diverges. Let us define a critical-slowing-down exponent ζ for our lasing system, wherethe typical relaxation time is proportional to|ϵ|−ζ as ϵ → 0.(b) For ϵ < 0, below the instability, solve yourequation from part (a) for the long-time sta-tionary value of ⟨n⟩. What is ν for our sys-

49One must admit that it is a bit weird to have the external field H inside the effective potential, rather than coupled linearlyto m outside.50This exercise was developed with the help of Alex Gaeta and Al Sievers.51That is, we assume that the atoms are being pumped into state N2 to compensate for both decays into our photon modeand decays into other channels. This usually involves exciting atoms into additional atomic levels.52The number of photons will eventually stop growing when they begin to pull energy out of the N2 excited atoms faster thanthe pumping can replace them—invalidating our equations.



tem? For a general initial condition for themean number of photons, solve for the time evo-lution. It should decay to the long-time valueexponentially. Does the relaxation time divergeas ϵ → 0? What is ζ?(c) Solve for the stationary state ρ∗ for N2 <N1. (Your formula for ρ∗n should not involveρ∗.) If N2/N1 is given by a Boltzmann proba-bility at temperature T , is ρ∗ the thermal equi-librium distribution for the quantum harmonicoscillator at that temperature? Warning: Thenumber of bosons in a phonon mode is given bythe Bose–Einstein distribution, but the prob-ability of different occupations in a quantumharmonic oscillator is given by the Boltzmanndistribution (see Section 7.2 and Exercise 7.11).We might expect that near the instability theprobability of getting n photons might have ascaling form

ρ∗n(ϵ) ∼ n−τD(n|ϵ|ν). (12.57)

(d) Show for small ϵ that there is a scalingform for ρ∗, with corrections that go to zeroas ϵ → 0, using your answer to part (c). Whatis τ? What is the function D(x)? (Hint: Inderiving the form of D, ϵ is small, but nϵν is oforder one. If you were an experimentalist doingscaling collapses, you would plot nτρn versusx = n|ϵ|ν ; try changing variables in nτρn to re-place ϵ by x, and choose τ to eliminate n forsmall ϵ.)

(12.21) Superconductivity and the renormaliza-

tion group. (Condensed matter) ⃝iOrdinary superconductivity happens at arather low temperature; in contrast to phononenergies (hundreds of degrees Kelvin times kB)or electronic energies (tens of thousands of de-grees Kelvin), phonon-mediated superconduc-tivity in most materials happens below a fewKelvin. This is largely explained by the BCStheory of superconductivity, which predictsthat the transition temperature for weakly-coupled superconductors is

Tc = 1.764 !ωD exp (−1/V g(εF )) , (12.58)

where ωD is a characteristic phonon frequency,V is an attraction between electron pairs me-diated by the phonons, and g(εF ) is thedensity of states (DOS) of the electron gas

(eqn 7.74) at the Fermi energy. If V is small,exp (−1/V g(εF )) can be exponentially small,explaining why materials often have to be socold to go superconducting.Superconductivity was discovered decades be-fore it was explained. Many looked for expla-nations which would involve interactions withphonons, but there was a serious obstacle. Peo-ple had studied the interactions of phononswith electrons, and had shown that the systemstays metallic (no superconductivity) to all or-ders in perturbation theory.(a) Taylor expand Tc (eqn 12.58) about V =0+ (about infinitesimal positive V ). Guess thevalue of all the terms in the Taylor series. Canwe expect to explain superconductivity at pos-itive temperatures by perturbing in powers ofV ?There are two messages here.

• Proving something to all orders in perturba-tion theory does not make it true.

• Since phases are regions in which perturba-tion theory converges (see Section 8.3), thetheorem is not a surprise. It is a conditionfor a metallic phase with a Fermi surface toexist at all.

In recent times, people have developed arenormalization-group description of the Fermiliquid state and its instabilities53 (see note 23on p. 164). Discussing Fermi liquid theory,the BCS theory of superconductivity, or thisrenormalization-group description would takeus far into rather technical subjects. However,we can illustrate all three by analyzing a ratherunusual renormalization-group flow.Roughly speaking, the renormalization-grouptreatment of Fermi liquids says that the Fermisurface is a fixed-point of a coarse-graining inenergy. That is, they start with a systemspace consisting of a partially-filled band ofelectrons with an energy width W , including allkinds of possible electron–electron repulsionsand attractions. They coarse-grain by pertur-batively eliminating (integrating out) the elec-tronic states near the edges of the band,

W ′ = (1− δ)W, (12.59)

incorporating their interactions and effects intoaltered interaction strengths among the remain-

53There are also other instabilities of Fermi liquids. Charge-density waves, for example, also have the characteristic exp(−1/aV )dependence on the coupling V .


Exercises 339

ing electrons. These altered interactions givethe renormalization-group flow in the systemspace. The equation for W gives the changeunder one iteration (n = 1); we can pretend nis a continuous variable and take δn → 0, so(W ′ −W )/δ → dW/dn, and hence

dW/dn = −W. (12.60)

When they do this calculation, they find thefollowing.

• The non-interacting Fermi gas we studied inSection 7.7 is a fixed point of the renormali-zation group. All interactions are zero at thisfixed-point. Let V represent one of these in-teractions.54

• The fixed-point is unstable to an attractiveinteraction V > 0, but is stable to a repul-sive interaction V < 0.

• Attractive forces between electrons grow un-der coarse-graining and lead to new phases,but repulsive forces shrink under coarse-graining, leading back to the metallic freeFermi gas.

This is quite different from our renormalization-group treatment of phase transitions, where rel-evant directions like the temperature and fieldwere unstable under coarse-graining, whethershifted up or down from the fixed-point, andother directions were irrelevant and stable(Fig. 12.8). For example, the temperature ofour Fermi gas is a relevant variable, whichrescales under coarse-graining like

T ′ = (1 + aδ)T,

dT/dn = aT.(12.61)

Here a > 0, so the effective temperature be-comes larger as the system is coarse-grained.How can they get a variable V which grows forV > 0 and shrinks for V < 0?

• When they do the coarse-graining, they findthat the interaction V is marginal: to linearorder it neither increases nor decreases.

The next allowed term in the Taylor seriesnear the fixed-point gives us the coarse-grainedequation for the interaction:

V ′ = (1 + bδV )V,

dV/dn = bV 2.(12.62)

• They find b > 0.

-1 0 1g V, coupling strength times DOS

0

1

2

Tem

pera

ture

T

High-T Fermi gas

Supe

rcon

duct

or

Tc

Fig. 12.25 Fermi liquid theory renormali-zation-group flows. The renormalization flowsdefined by eqns 12.61 and 12.62. The temperatureT is relevant at the free Fermi gas fixed-point; thecoupling V is marginal. The distinguished curverepresents a phase transition boundary Tc(V ). Be-low Tc, for example, the system is superconducting;above Tc it is a (finite-temperature) metal.

(b) True or false? (See Fig. 12.25.)(T) (F) For V > 0 (attractive interactions), theinteractions get stronger with coarse-graining.(T) (F) For V < 0 (repulsive interactions),coarse-graining leads us back to the free Fermigas, explaining why the Fermi gas describesmetals (Section 7.7).(T) (F) Temperature is an irrelevant variable,but dangerous.(T) (F) The scaling variable

x = TV 1/βδ (12.63)

is unchanged by the coarse-graining (secondequations in 12.61 and 12.62), where β and δare universal critical exponents;55 hence x la-bels the progress along the curves in Fig. 12.25(increasing in the direction of the arrows).(T) (F) The scaling variable

y = T exp (a/(bV )) (12.64)

is unchanged by the coarse-graining, so eachcurve in Fig. 12.25 has a fixed value for y.

54V will be the pairing between opposite-spin electrons near the Fermi surface for superconductors.55Note that here δ is not the infinitesimal change in parameter.



Now, without knowing anything about super-conductivity, let us presume that our sys-tem goes superconducting at some tempera-ture Tc(V ) when the interactions are attrac-tive. When we coarse-grain a system that isat the superconducting transition temperature,we must get another system that is at its su-perconducting transition temperature.(c) What value for a/b must they calculatein order to get the BCS transition temper-ature (eqn 12.58) from this renormalizationgroup? What is the value of the scaling vari-able (whichever you found in part (b)) alongTc(V )?Thus the form of the BCS transition tempera-ture at small V , eqn 12.58, can be explained bystudying the Fermi gas without reference to thesuperconducting phase!

(12.22) Period doubling.56 (Mathematics, Com-plexity) ⃝4In this exercise, we use renormalization-group and scaling methods to study the on-set of chaos. There are several routes bywhich a dynamical system can start exhibitingchaotic motion; this exercise studies the period-doubling cascade, first extensively investigatedby Feigenbaum.Chaos is often associated with dynamics whichstretch and fold; when a batch of taffy is be-ing pulled, the motion of a speck in the taffydepends sensitively on the initial conditions. Asimple representation of this physics is providedby the map57

f(x) = 4µx(1− x) (12.65)

restricted to the domain (0, 1). It takes f(0) =f(1) = 0, and f(1/2) = µ. Thus, for µ = 1it precisely folds the unit interval in half, andstretches it to cover the original domain.

0 1x0

1

f(x)

Fig. 12.26 Period-eight cycle. Iterating aroundthe attractor of the Feigenbaum map at µ = 0.89.

The study of dynamical systems (e.g., differen-tial equations and maps like eqn 12.65) oftenfocuses on the behavior after long times, wherethe trajectory moves along the attractor. Wecan study the onset and behavior of chaos inour system by observing the evolution of theattractor as we change µ. For small enough µ,all points shrink to the origin; the origin is astable fixed-point which attracts the entire in-terval x ∈ (0, 1). For larger µ, we first get astable fixed-point inside the interval, and thenperiod doubling.(a) Iteration: Set µ = 0.2; iterate f for someinitial points x0 of your choosing, and convinceyourself that they all are attracted to zero. Plotf and the diagonal y = x on the same plot. Arethere any fixed-points other than x = 0? Repeatfor µ = 0.3, µ = 0.7, and 0.8. What happens?On the same graph, plot f , the diago-nal y = x, and the segments x0, x0,x0, f(x0), f(x0), f(x0), f(x0), f(f(x0)),. . . (representing the convergence of the trajec-tory to the attractor; see Fig. 12.26). See howµ = 0.7 and 0.8 differ. Try other values of µ.By iterating the map many times, find a pointa0 on the attractor. As above, then plot thesuccessive iterates of a0 for µ = 0.7, 0.8, 0.88,0.89, 0.9, and 1.0.You can see at higher µ that the system nolonger settles into a stationary state at longtimes. The fixed-point where f(x) = x ex-

56This exercise and the associated software were developed in collaboration with Christopher Myers.57We also study this map in Exercises 4.10, 5.18, and 5.23; parts (a) and (b) below overlap somewhat with Exercise 4.10.58In a continuous evolution, perturbations die away if the Jacobian of the derivative at the fixed-point has all negative eigen-values. For mappings, perturbations die away if all eigenvalues of the Jacobian have magnitude less than one.


Exercises 341

ists for all µ > 1/4, but for larger µ it isno longer stable. If x∗ is a fixed-point (sof(x∗) = x∗) we can add a small perturbationf(x∗ + ϵ) ≈ f(x∗)+ f ′(x∗)ϵ = x∗ + f ′(x∗)ϵ; thefixed-point is stable (perturbations die away) if|f ′(x∗)| < 1.58

In this particular case, once the fixed-point goesunstable the motion after many iterations be-comes periodic, repeating itself after two iter-ations of the map—so f(f(x)) has two newfixed-points. This is called period doubling.Notice that by the chain rule d f(f(x))/dx =f ′(x)f ′(f(x)), and indeed

d f [n]

dx=

d f(f(. . . f(x) . . . ))dx

(12.66)

= f ′(x)f ′(f(x)) . . . f ′(f(. . . f(x) . . . )),

so the stability of a period-N orbit is deter-mined by the product of the derivatives of f ateach point along the orbit.(b) Analytics: Find the fixed-point x∗(µ) of themap 12.65, and show that it exists and is sta-ble for 1/4 < µ < 3/4. If you are ambitiousor have a computer algebra program, show thatthe period-two cycle is stable for 3/4 < µ <(1 +

√6)/4.

(c) Bifurcation diagram: Plot the attractor asa function of µ, for 0 < µ < 1; comparewith Fig. 12.17. (Pick regularly-spaced δµ, runntransient steps, record ncycles steps, and plot.After the routine is working, you should be ableto push ntransient and ncycles both larger than100, and δµ < 0.01.) Also plot the attractor foranother one-humped map

fsin(x) = B sin(πx), (12.67)

for 0 < B < 1. Do the bifurcation diagramsappear similar to one another?

δ

α

Fig. 12.27 Self-similarity in period-doublingbifurcations. The period doublings occur atgeometrically-spaced values of the control param-eter µ∞ − µn ∝ δ−n, and the attractor during theperiod-2n cycle is similar to one-half of the attractorduring the 2n+1-cycle, except inverted and larger,rescaling x by a factor of α and µ by a factor of δ.The boxes shown in the diagram illustrate this self-similarity; each box looks like the next, except ex-panded by δ along the horizontal µ axis and flippedand expanded by α along the vertical axis.

Notice the complex, structured, chaotic regionfor large µ (which we study in Exercise 4.10).How do we get from a stable fixed-point µ < 3/4to chaos? The onset of chaos in this sys-tem occurs through a cascade of period dou-blings. There is the sequence of bifurcationsas µ increases—the period-two cycle starting atµ1 = 3/4, followed by a period-four cycle start-ing at µ2, period-eight at µ3—a whole period-doubling cascade. The convergence appears ge-ometrical, to a fixed-point µ∞:

µn ≈ µ∞ − Aδ−n, (12.68)

so

δ = limn→∞

(µn−1 − µn−2)/(µn − µn−1) (12.69)

and there is a similar geometrical self-similarityalong the x axis, with a (negative) scale factor αrelating each generation of the tree (Fig. 12.27).In Exercise 4.10, we explained the boundariesin the chaotic region as images of x = 1/2. Thesespecial points are also convenient for studyingperiod-doubling. Since x = 1/2 is the maximumin the curve, f ′(1/2) = 0. If it were a fixed-point (as it is for µ = 1/2), it would not onlybe stable, but unusually so: a shift by ϵ awayfrom the fixed point converges after one step ofthe map to a distance ϵf ′(1/2) + ϵ2/2f ′′(1/2) =O(ϵ2). We say that such a fixed-point is su-perstable. If we have a period-N orbit thatpasses through x = 1/2, so that the Nth iter-ate fN (1/2) ≡ f(. . . f(1/2) . . . ) =

1/2, then the or-bit is also superstable, since (by eqn 12.66) thederivative of the iterated map is the productof the derivatives along the orbit, and hence isalso zero.These superstable points happen roughly half-way between the period-doubling bifurcations,and are easier to locate, since we know that



x = 1/2 is on the orbit. Let us use them to in-vestigate the geometrical convergence and self-similarity of the period-doubling bifurcation di-agram from part (d). For this part and part (h),you will need a routine that finds the rootsG(y) = 0 for functions G of one variable y.(d) The Feigenbaum numbers and universality:Numerically, find the values of µs

n at which the2n-cycle is superstable, for the first few val-ues of n. (Hint: Define a function G(µ) =f [2n]µ (1/2) − 1/2, and find the root as a functionof µ. In searching for µs

n+1, you will want tosearch in a range (µs

n + ϵ, µsn + (µs

n − µsn−1)/A)

where A ∼ 3 works pretty well. Calculateµ0 and µ1 by hand.) Calculate the ratios(µs

n−1 − µsn−2)/(µ

sn − µs

n−1); do they appearto converge to the Feigenbaum number δ =4.6692016091029909 . . . ? Extrapolate the seriesto µ∞ by using your last two reliable valuesof µs

n and eqn 12.69. In the superstable orbitwith 2n points, the nearest point to x = 1/2 isf [2n−1](1/2).

59 Calculate the ratios of the am-plitudes f [2n−1](1/2) − 1/2 at successive values ofn; do they appear to converge to the univer-sal value α = −2.50290787509589284 . . . ? Cal-culate the same ratios for the map f2(x) =B sin(πx); do α and δ appear to be universal(independent of the mapping)?The limits α and δ are independent of the map,so long as it folds (one hump) with a quadraticmaximum. They are the same, also, for ex-perimental systems with many degrees of free-dom which undergo the period-doubling cas-cade. This self-similarity and universality sug-gests that we should look for a renormalization-group explanation.

0 1x0

1

f (f

(x))

= f

[2] (x

)

Fig. 12.28 Renormalization-group transfor-mation. The renormalization-group transforma-tion takes g(g(x)) in the small window with uppercorner x∗ and inverts and stretches it to fill thewhole initial domain and range (0, 1) × (0, 1).

(e) Coarse-graining in time. Plot f(f(x)) vs. xfor µ = 0.8, together with the line y = x (or seeFig. 12.28). Notice that the period-two cycle off becomes a pair of stable fixed-points for f [2].(We are coarse-graining in time—removing ev-ery other point in the time series, by study-ing f(f(x)) rather than f .) Compare the plotwith that for f(x) vs. x for µ = 0.5. Noticethat the region zoomed in around x = 1/2 forf [2] = f(f(x)) looks quite a bit like the en-tire map f at the smaller value µ = 0.5. Plotf [4](x) at µ = 0.875; notice again the small one-humped map near x = 1/2.The fact that the one-humped map reappearsin smaller form just after the period-doublingbifurcation is the basic reason that succeedingbifurcations so often follow one another. Thefact that many things are universal is due tothe fact that the little one-humped maps havea shape which becomes independent of the orig-inal map after several period-doublings.Let us define this renormalization-group trans-formation T , taking function space into itself.Roughly speaking, T will take the small upside-down hump in f(f(x)) (Fig. 12.28), invert it,and stretch it to cover the interval from (0, 1).Notice in your graphs for part (g) that theline y = x crosses the plot f(f(x)) not onlyat the two points on the period-two attrac-

59This is true because, at the previous superstable orbit, 2n−1 iterates returned us to the original point x = 1/2.60For asymmetric maps, we would need to locate this other corner f(f(xc)) = x∗ numerically. As it happens, breaking thissymmetry is irrelevant at the fixed-point.


Exercises 343

tor, but also (naturally) at the old fixed-pointx∗[f ] for f(x). This unstable fixed-point playsthe role for f [2] that the origin played for f ;our renormalization-group rescaling must map(x∗[f ], f(x∗)) = (x∗, x∗) to the origin. The cor-ner of the window that maps to (1, 0) is conve-niently located at 1−x∗, since our map happensto be symmetric60 about x = 1/2. For a generalone-humped map g(x) with fixed-point x∗[g]the side of the window is thus of length 2(x∗[g]−1/2). To invert and stretch, we must thus rescaleby a factor α[g] = −1/(2(x∗[g] − 1/2)). Ourrenormalization-group transformation is thus amapping T [g] taking function space into itself,where

T [g](x) = α[g] (g (g(x/α[g] + x∗[g]))− x∗[g]) .(12.70)

(This is just rescaling x to squeeze into the win-dow, applying g twice, shifting the corner of thewindow to the origin, and then rescaling by αto fill the original range (0, 1)× (0, 1).)(f) Scaling and the renormalization group:Write routines that calculate x∗[g] and α[g],and define the renormalization-group transfor-mation T [g]. Plot T [f ], T [T [f ]],. . . and com-pare them. Are we approaching a fixed-point f∗

in function space?This explains the self-similarity; in particular,the value of α[g] as g iterates to f∗ becomes theFeigenbaum number α = −2.5029 . . .(g) Universality and the renormalization group:Using the sine function of eqn 12.67, compareT [T [fsin]] to T [T [f ]] at their onsets of chaos.Are they approaching the same fixed-point?By using this rapid convergence in functionspace, one can prove both that there will (of-ten) be an infinite geometrical series of period-doubling bifurcations leading to chaos, and thatthis series will share universal features (expo-nents α and δ and features) that are indepen-dent of the original dynamics.

(12.23) The renormalization group and the cen-

tral limit theorem: short. (Mathemat-ics) ⃝4If you are familiar with the renormalizationgroup and Fourier transforms, this problem canbe stated very quickly. If not, you are prob-ably better off doing the long version (Exer-cise 12.24).Write a renormalization-group transformationT taking the space of probability distributions

into itself, that takes two random variables,adds them, and rescales the width by the squareroot of two [28]. Show that the Gaussian ofwidth σ is a fixed-point. Find the eigenfunc-tions fn and eigenvectors λn of the lineariza-tion of T at the fixed-point. (Hint: It is easierin Fourier space.) Describe physically what therelevant and marginal eigenfunctions represent.By subtracting the fixed-point distribution froma binomial distribution, find the leading correc-tion to scaling, as a function of x. Which eigen-function does it represent? Why is the leadingirrelevant eigenvalue not dominant here?

(12.24) The renormalization group and the cen-

tral limit theorem: long. (Mathematics) ⃝4

In this exercise, we will develop a renormal-ization group in function space to derive thecentral limit theorem [28]. We will be usingmaps (like our renormalization transformationT ) that take a function ρ of x into another func-tion of x; we will write T [ρ] as the new func-tion, and T [ρ](x) as the function evaluated at x.We will also make use of the Fourier transform(eqn A.6)

F [ρ](k) =

3 ∞

−∞e−ikxρ(x) dx; (12.71)

F maps functions of x into functions of k.When convenient, we will also use the tilde no-tation: 4ρ = F [ρ], so for example (eqn A.7)

ρ(x) =12π

3 ∞

−∞eikx4ρ(k) dk. (12.72)

The central limit theorem states that the sumof many independent random variables tends toa Gaussian, whatever the original distributionmight have looked like. That is, the Gaussiandistribution is the fixed-point function for largesums. When summing many random numbers,the details of the distributions of the individualrandom variables becomes unimportant; sim-ple behavior emerges. We will study this usingthe renormalization group, giving an examplewhere we can explicitly implement the coarse-graining transformation. Here our system spaceis the space of probability distributions ρ(x).There are four steps in the procedure.1. Coarse-grain. Remove some fraction (usuallyhalf) of the degrees of freedom. Here, we willadd pairs of random variables; the probabilitydistribution for sums of N independent random



variables of distribution f is the same as the dis-tribution for sums of N/2 random variables ofdistribution f ∗f , where ∗ denotes convolution.(a) Argue that if ρ(x) is the probability that arandom variable has value x, then the probabil-ity distribution of the sum of two random vari-ables drawn from this distribution is the convo-lution

C[ρ](x) = (ρ ∗ ρ)(x) =3 ∞

−∞ρ(x− y)ρ(y) dy.

(12.73)Remember (eqn A.23) the Fourier transformof the convolution is the product of the Fouriertransforms, so

F [C[ρ]](k) = (4ρ(k))2 . (12.74)

2. Rescale. The behavior at larger lengths willtypically be similar to that of smaller lengths,but some of the constants will shift (or renor-malize). Here the mean and width of the dis-tributions will increase as we coarse-grain. Weconfine our main attention to distributions ofzero mean. Remember that the width (stan-dard deviation) of the sum of two random vari-ables drawn from ρ will be

√2 times the width

of one variable drawn from ρ, and that the over-all height will have to shrink by

√2 to stay nor-

malized. We define a rescaling operator S√2

which reverses this spreading of the probabilitydistribution:

S√2[ρ](x) =

√2ρ(

√2x). (12.75)

(b) Show that if ρ is normalized (integrates toone), so is S√

2[ρ]. Show that the Fourier trans-form is

F [S√2[ρ]](k) = 4ρ(k/

√2). (12.76)

Our renormalization-group transformation isthe composition of these two operations,

T [ρ](x) = S√2[C[ρ]](x)

=√2

3 ∞

−∞ρ(√2x− y)ρ(y) dy.

(12.77)

Adding two Gaussian random variables (con-volving their distributions) and rescaling thewidth back should give the original Gaussian

distribution; the Gaussian should be a fixed-point.(c) Show that the Gaussian distribution

ρ∗(x) = (1/√2πσ) exp(−x2/2σ2) (12.78)

is indeed a fixed-point in function space underthe operation T . You can do this either by directintegration, or by using the known properties ofthe Gaussian under convolution.(d) Use eqns 12.74 and 12.76 to show that

F [T [ρ]](k) = 4T [4ρ](k) = 4ρ(k/√2)2. (12.79)

Calculate the Fourier transform of the fixed-point 4ρ∗(k) (or see Exercise A.4). Usingeqn 12.79, show that 4ρ∗(k) is a fixed-point inFourier space under our coarse-graining opera-tor 4T .61These properties of T and ρ∗ should allow youto do most of the rest of the exercise withoutany messy integrals.The central limit theorem tells us that sumsof random variables have probability distri-butions that approach Gaussians. In ourrenormalization-group framework, to prove thiswe might try to show that our Gaussian fixed-point is attracting: that all nearby probabilitydistributions flow under iterations of T to ρ∗.3. Linearize about the fixed point. Consider afunction near the fixed point: ρ(x) = ρ∗(x) +ϵf(x). In Fourier space, 4ρ(k) = 4ρ∗(k) + ϵ 4f(k).We want to find the eigenvalues λn and eigen-functions fn of the derivative of the mappingT . That is, they must satisfy

T [ρ∗ + ϵfn] = ρ∗ + λnϵfn +O(ϵ2),

4T [4ρ∗ + ϵ 4fn] = 4ρ∗ + λnϵ 4fn +O(ϵ2).(12.80)

(e) Show using eqns 12.79 and 12.80 that thetransforms of the eigenfunctions satisfy

5fn(k) = (2/λn) 4ρ∗(k/√2)5fn(k/

√2). (12.81)

4. Find the eigenvalues and calculate the uni-versal critical exponents.(f) Show that

4fn(k) = (ik)n 4ρ∗(k) (12.82)

is the Fourier transform of an eigenfunction(i.e., that it satisfies eqn 12.81.) What is theeigenvalue λn?

61To be explicit, the operator #T = F T F−1 is a renormalization-group transformation that maps Fourier space into itself.


Exercises 345

Our fixed-point actually does not attract all dis-tributions near it. The directions with eigen-values greater than one are called relevant;they are dangerous, corresponding to devia-tions from our fixed-point that grow undercoarse-graining. The directions with eigenval-ues equal to one are called marginal; they do notget smaller (to linear order) and are thus alsopotentially dangerous. When you find relevantand marginal operators, you always need to un-derstand each of them on physical grounds.(g) The eigenfunction f0(x) with the biggesteigenvalue corresponds to an unphysical pertur-bation; why? (Hint: Probability distributionsmust be normalized to one.) The next twoeigenfunctions f1 and f2 have important physi-cal interpretations. Show that ρ∗+ϵf1 to lowestorder is equivalent to a shift in the mean of ρ,and ρ∗+ ϵf2 is a shift in the standard deviationσ of ρ∗.In this case, the relevant perturbations do nottake us to qualitatively new phases—just toother Gaussians with different means and vari-ances. All other eigenfunctions should haveeigenvalues λn less than one. This means that aperturbation in that direction will shrink underthe renormalization-group transformation:

TN(ρ∗ + ϵfn)− ρ∗ ∼ λNn ϵfn. (12.83)

Corrections to scaling and coin flips. Does any-thing really new come from all this analysis?One nice thing that comes out is the leadingcorrections to scaling. The fixed-point of therenormalization group explains the Gaussianshape of the distribution of N coin flips in thelimit N → ∞, but the linearization about thefixed-point gives a systematic understanding ofthe corrections to the Gaussian distribution forlarge but not infinite N .Usually, the largest eigenvalues are the oneswhich dominate. In our problem, consideradding a small perturbation to the fixed-pointf∗ along the two leading irrelevant directionsf3 and f4:

ρ(x) = ρ∗(x) + ϵ3f3(x) + ϵ4f4(x). (12.84)

These two eigenfunctions can be inverse-transformed from their k-space form

(eqn 12.82):

f3(x) ∝ ρ∗(x)(3x/σ − x3/σ3),

f4(x) ∝ ρ∗(x)(3− 6x2/σ2 + x4/σ4).(12.85)

What happens to these perturbations undermultiple applications of our renormalization-group transformation T ? After ℓ applications(corresponding to adding together 2ℓ of our ran-dom variables), the new distribution should begiven by

T ℓ(ρ)(x) ∼ ρ∗(x) + λℓ3ϵ3f3(x) + λℓ

4ϵ4f4(x).(12.86)

Since 1 > λ3 > λ4 . . . , the leading correctionshould be dominated by the perturbation withthe largest eigenvalue.(h) Plot the difference between the binomial dis-tribution giving the probability of m heads in Ncoin flips, and a Gaussian of the same meanand width, for N = 10 and N = 20. (TheGaussian has mean of N/2 and standard devi-ation

√N/2, as you can extrapolate from the

case N = 1.) Does it approach one of the eigen-functions f3 or f4 (eqns 12.85)?(i) Why did a perturbation along f3(x) not dom-inate the asymptotics? What symmetry forcedϵ3 = 0? Should flips of a biased coin break thissymmetry?Using the renormalization group to demon-strate the central limit theorem might not bethe most efficient route to the theorem, butit provides quantitative insights into how andwhy the probability distributions approach theasymptotic Gaussian form.

(12.25) Percolation and universality.62 (Complex-ity) ⃝4Cluster size distribution: power laws at pc. Asystem at its percolation threshold pc is self-similar. When looked at on a longer lengthscale (say, with a ruler with notches spaced1+ ϵ farther apart, for infinitesimal ϵ), the sta-tistical behavior of the large percolation clus-ters should be unchanged, if we simultaneouslyrescale various measured properties accordingto certain rules. Let x be the length and S

62This exercise and the associated software were developed in collaboration with Christopher Myers.



be the size (number of nodes) in a percolationcluster, and let n(S) be the probability thata given cluster will be of size S at pc.

63 Thecluster measured with the new ruler will have alength x′ = x/ (1− ϵ), a size S′ = S/ (1 + cϵ),and will occur with probability n′ = (1 + aϵ)n.(a) In precise analogy to our analysis of theavalanche size distribution (eqns 12.5–12.6),show that the probability is a power law, n(S) ∝S−τ . What is τ , in terms of a and c?In two dimensions, there are exact resultsknown for many properties of percolation. Inparticular, it is known that64 τ = 187/91. Youcan test this numerically, either with the codeyou developed for Exercise 2.20, or by using thesoftware at our web site [131].(b) Calculate the cluster size distribution n(S),both for bond percolation on the square latticeand for site percolation on the triangular lat-tice, for a large system size (perhaps L × Lwith L = 400) at p = pc.

65 At some moderatesize S you will begin occasionally to not haveany avalanches; plot log(n(S)) versus log(S)for both bond and site percolation, together withthe power law n(S) ∝ S−187/91 predicted by theexact result. To make better use of the data, oneshould bin the avalanches into larger groups, es-pecially for larger sizes where the data is sparse.It is a bit tricky to do this nicely, and you canget software to do this at our web site [131]. Dothe plots again, now with all the data included,using bins that start at size ranges 1 ≤ S < 2and grow by a factor of 1.2 for each bin. Youshould see clear evidence that the distributionof clusters does look like a power law (a straightline on your log–log plot), and fairly convincingevidence that the power law is converging to theexact result at large S and large system sizes.

The size of the infinite cluster: power laws nearpc. Much of the physics of percolation above pcrevolves around the connected piece left afterthe small clusters fall out, often called the per-colation cluster. For p > pc this largest clusteroccupies a fraction of the whole system, oftencalled P (p).66 The fraction of nodes in this

largest cluster for p > pc is closely analogousto the T < Tc magnetization M(T ) in mag-nets (Fig. 12.6(b)) and the density differenceρl(T )− ρg(T ) near the liquid–gas critical point(Fig. 12.6(a)). In particular, the value P (p)goes to zero continuously as p → pc.Systems that are not at pc are not self-similar.However, there is a scaling relation between sys-tems at differing values of p − pc: a systemcoarsened by a factor 1 + ϵ will be similar toone farther from pc by a factor 1 + ϵ/ν, exceptthat the percolation cluster fraction P must berescaled upward by 1+βϵ/ν.67 This last rescal-ing reflects the fact that the percolation clusterbecomes more dense as you coarse-grain, fillingin or blurring away the smaller holes. You maycheck, just as for the magnetization (eqn 12.7),that

P (p) ∼ (pc − p)β. (12.87)

In two dimensions, β = 5/36 and ν = 4/3.(c) Calculate the fraction of nodes P (p) in thelargest cluster, for both bond and site percola-tion, at a series of points p = pc + 2−n for aslarge a percolation lattice as is convenient, anda good range of n. (Once you get your methoddebugged, n = 10 on an L × L lattice withL = 200 should be numerically feasible.) Doa log–log plot of P (p) versus p − pc, and com-pare along with the theory prediction, eqn 12.87with β = 5/36.You should find that the numerics in part (c)are not compelling, even for rather large sys-tem sizes. The two curves look a bit like powerlaws, but the slopes βeff on the log–log plot donot agree with one another or with the theory.Worse, as you get close to pc the curves, al-though noisy, definitely are not going to zero.This is natural; there will always be a largestcluster, and it is only as the system size L → ∞that the largest cluster can vanish as a fractionof the system size.Finite-size scaling (advanced). We can extractbetter values for β from small simulations by

63Hence the probability that a given node is in a cluster of size S is proportional to Sn(S).64A non-obvious result!65Conveniently, the critical probability pc = 1/2 for both these systems, see Exercise 2.20, part(c). This enormously simplifiesthe scaling analysis, since we do not need to estimate pc as well as the critical exponents.66For p < pc, there will still be a largest cluster, but it will not grow much bigger as the system size grows and the fractionP (p) → 0 for p < pc as the system length L → ∞.67We again assure the reader that these particular combinations of Greek letters are just chosen to give the conventional namesfor the critical exponents.


Exercises 347

explicitly including the length L into our analy-sis. Let P (p,L) be the mean fraction of nodes68

in the largest cluster for a system of size L.(d) On a single graph, plot P (p,L) versus p forbond percolation L = 5, 10, 20, 50, and 100,focusing on the region around p = pc wherethey differ from one another. (At L = 10 youwill want p to range from 0.25 to 0.75; forL = 50 the range should be from 0.45 to 0.55or so.) Five or ten points will be fine. Youwill discover that the sample-to-sample varia-tions are large (another finite-size effect), soaverage each curve over perhaps ten or twentyrealizations.Each curve P (p,L) is rounded near pc, as thecharacteristic cluster lengths reach the systembox length L. Thus this rounding is itself asymptom of the universal long-distance behav-ior, and we can study the dependence of therounding on L to extract better values of thecritical exponent β. We will do this using a scal-ing collapse, rescaling the horizontal and verti-cal axes so as to make all the curves fall onto asingle scaling function.First, we must derive the scaling function forP (p, L). We know that

L′ = L/(1 + ϵ),

(pc − p)′ = (1 + ϵ/ν)(pc − p),(12.88)

since the system box length L rescales like anyother length. It is convenient to change vari-ables from p to X = (pc−p)L1/ν ; let P (p,L) =P (L, (pc − p)L1/ν).(e) Show that X is unchanged under coarse-graining (eqn 12.88). (You can either showX ′ = X up to terms of order ϵ2, or you canshow dX/dϵ = 0.)The combination X = (pc − p)L1/ν is an-other scaling variable. The combination ξ =|p − pc|−ν is the way in which lengths divergeat the critical point, and is called the corre-lation length. Two systems of different lengthsand different values of p should be similar if thelengths are the same when measured in unitsof ξ. L in units of ξ is L/ξ = Xν , so differ-ent systems with the same value of the scal-ing variable X are statistically similar. We canturn this verbal assertion into a mathematical

scaling form by studying how P (L,X) coarse-grains.(f) Using eqns 12.88 and the fact that P rescalesupward by (1 + βϵ/ν) under coarse-graining,write the similarity relationship for P . Follow-ing our derivation of the scaling form for theavalanche size distribution (through eqn 12.11),show that P (L,X) = L−β/νP(X) for somefunction P(X), and hence

P (p,L) ∝ L−β/νP((p− pc)L1/ν). (12.89)

Presuming that P(X) goes to a finite value asX → 0, derive the power law giving the per-colation cluster size L2P (pc, L) as a functionof L. Derive the power-law variation of P(X)as X → ∞ using the fact that P (p,∞) ∝(p− pc)

β.Now, we can use eqn 12.89 to deduce how torescale our data. We can find the finite-sizedscaling function P by plotting Lβ/νP (p,L) ver-sus X = (p − pc)L

1/ν , again with ν = 4/3 andβ = 5/36.(g) Plot Lβ/νP (p,L) versus X for X ∈[−0.8,+0.8], plotting perhaps five points foreach curve, for both site percolation and bondpercolation. Use system sizes L = 5, 10, 20,and 50. Average over many clusters for thesmaller sizes (perhaps 400 for L = 5), and overat least ten even for the largest.Your curves should collapse onto two scalingcurves, one for bond percolation and one forsite percolation.69 Notice here that the finite-sized scaling curves collapse well for small L,while we would need to go to much larger L tosee good power laws in P (p) directly (part (c)).Notice also that both site percolation and bondpercolation collapse for the same value of β,even though the rough power laws from part (c)seemed to differ. In an experiment (or a theoryfor which exact results were not available), onecan use these scaling collapses to estimate pc,β, and ν.

68You can take a microcanonical-style ensemble over all systems with exactly L2p sites or 2L2p bonds, but it is simpler just todo an ensemble average over random number seeds.69These two curves should also have collapsed onto one another, given a suitable rescaling of the horizontal and vertical axes,had we done the triangular lattice in a square box instead of a rectangular box (which we got from shearing an L× L lattice).The finite-size scaling function will in general depend on the boundary condition, and in particular on the shape of the box.



(12.26) Hysteresis and avalanches: scaling.

(Complexity) ⃝3For this exercise, either download Matt Kuntz’shysteresis simulation code from the book website [131], or make use of the software you de-veloped in Exercise 8.17 or 8.18.Run the simulation in two dimensions on a1000 × 1000 lattice with disorder R = 0.9, ora three-dimensional simulation on a 1003 lat-tice at R = 2.16.70 The simulation is a sim-plified model of magnetic hysteresis, describedin [130]; see also [129]. The spins si begin allpointing down, and flip upward as the externalfield H grows from minus infinity, depending onthe spins of their neighbors and a local randomfield hi. The flipped spins are colored as theyflip, with spins in the same avalanche sharingthe same color. An avalanche is a collection ofspins which flip together, all triggered from thesame original spin. The disorder is the ratioR of the root-mean-square width

(⟨h2

i ⟩ to theferromagnetic coupling J between spins:

R =(

⟨h2⟩/J. (12.90)

Examine the M(H) curve for our model andthe dM/dH curve. The individual avalanchesshould be visible on the first graph as jumps,and on the second graph as spikes. This kindof time series (a set of spikes or pulses with abroad range of sizes) we hear as crackling noise.You can go to our site [72] to hear the noiseresulting from our model, as well as cracklingnoise we have assembled from crumpling paper,from fires and Rice KrispiesTM, and from theEarth (earthquakes in 1995, sped up to audiofrequencies).Examine the avalanche size distribution. The(unlabeled) vertical axis on the log–log plotgives the number of avalanches D(S,R); thehorizontal axis gives the size S (with S = 1 onthe left-hand side). Equivalently, D(S,R) is theprobability distribution that a given avalancheduring the simulation will have size S. Thegraph is created as a histogram, and the curvechanges color after the first bin with zero en-tries (after which the data becomes much lessuseful, and should be ignored).If available, examine the spin–spin correlationfunction C(x,R). It shows a log–log plot of the

probability (vertical axis) that an avalanche ini-tiated at a point x0 will extend to include a spinx1 a distance x =

((x1 − x0)2 away.

Two dimensions is fun to watch, but the scal-ing behavior is not yet understood. In threedimensions we have good evidence for scalingand criticality at a phase transition in the dy-namical evolution. There is a phase transitionin the dynamics at Rc ∼ 2.16 on the three-dimensional cubic lattice. Well below Rc onelarge avalanche flips most of the spins. Wellabove Rc all avalanches are fairly small; at veryhigh disorder each spin flips individually. Thecritical disorder is the point, as L → ∞, whereone first finds spanning avalanches, which ex-tend from one side of the simulation to theother.Simulate a 3D system at R = Rc = 2.16 withL = 100 (one million spins, or larger, if youhave a fast machine). It will be fastest if you usethe sorted list algorithm (Exercise 8.18). Thedisplay will show an L× L cross-section of the3D avalanches. Notice that there are many tinyavalanches, and a few large ones. Below Rc

you will find one large colored region formingthe background for the others; this is the span-ning, or infinite avalanche. Look at the M(H)curve (the bottom half of the hysteresis loop).It has many small vertical jumps (avalanches),and one large one (corresponding to the span-ning avalanche).(a) What fraction of the system is flipped bythe one largest avalanche, in your simulation?Compare this with the hysteresis curve at R =2.4 > Rc. Does it have a similar big jump, oris it continuous?Below Rc we get a big jump; above Rc allavalanches are small compared to the systemsize. If the system size were large enough, webelieve the fraction of spins flipped by the span-ning avalanche at Rc would go to zero. Thelargest avalanche would nonetheless span thesystem—just like the percolation cluster at pcspans the system but occupies zero volume inthe limit of large systems.The other avalanches form a nice power-law sizedistribution; let us measure it carefully. Do a

70If you are using the brute-force algorithm, you are likely to need to run all of the three-dimensional simulationssystem size, perhaps 503. If you have a fast computer, you may wish to run at a larger size, but make surewatch.


Exercises 349

set of 10 runs (# Runs 10) at L = 100 andR = Rc = 2.16.71

Watch the avalanches. Notice that sometimesthe second-largest avalanche in the view (thelargest being the ‘background color’) is some-times pretty small; this is often because thecross-section we view missed it. Look at theavalanche size distribution. (You can watch itas it averages over simulations.) Print it outwhen the simulations finish. Notice that at Rc

you find a pretty good power-law distribution(a straight line on the log–log plot). We denotethis critical exponent τ = τ + σβδ:

D(S,Rc) ∼ S−τ = S−(τ+σβδ). (12.91)

(b) From your plot, measure this exponent com-bination from your simulation. It should beclose to two. Is your estimate larger or smallerthan two?This power-law distribution is to magnets whatthe Gutenberg–Richter law (Fig. 12.3(b)) is toearthquakes. The power law stems naturallyfrom the self-similarity.We want to explore how the avalanche size dis-tribution changes as we move above Rc. Wewill do a series of three or four runs at differentvalues of R, and then graph the avalanche sizedistributions after various transformations.Do a run at R = 6 and R = 4 with L = 100,and make sure your data files are properly out-put. Do runs at R = 3, R = 2.5, and R = 2.16at L = 200.(c) Copy and edit your avalanche size distribu-tion files, removing the data after the first binwith zero avalanches in it. Start up a graphicsprogram, and plot the curves on a log–log plot;they should look like power laws for small S,

and cut off exponentially at larger S. Enclosea copy of your plot.We expect the avalanche size distribution tohave the scaling form

D(S,R) = S−(τ+σβδ)D(S(R−Rc)1/σ) (12.92)

sufficiently close to Rc. This reflects the sim-ilarity of the system to itself at a differentset of parameters; a system at 2(R − Rc) hasthe same distribution as a system at R − Rc

except for an overall change A in probabilityand B in the size scale of the avalanches, soD(S,R −Rc) ≈ AD(BS, 2(R −Rc)).(d) What are A and B in this equation for thescaling form given by eqn 12.92?At R = 4 and 6 we should expect substantialcorrections! Let us see how well the collapseworks anyhow.(e) Multiply the vertical axis of each curve bySτ+σβδ. This then should give four curvesD(S(R −Rc)

1/σ) which are (on a log–log plot)roughly the same shape, just shifted sidewayshorizontally (rescaled in S by the typical largestavalanche size, proportional to 1/(R−Rc)

1/σ).Measure the peak of each curve. Make a ta-ble with columns R, Speak, and R − Rc (withRc ∼ 2.16). Do a log–log plot of R−Rc versusSpeak, and estimate σ in the expected power lawSpeak ∼ (R −Rc)

−1/σ.(f) Do a scaling collapse: plot Sτ+σβδD(S,R)versus (R−Rc)

1/σ S for the avalanche size dis-tributions with R > Rc. How well do they col-lapse onto a single curve?The collapses become compelling only near Rc,where you need very large systems to get goodcurves.

71If your machine is slow, do fewer. If your machine is fast, use a larger system. Make sure you do not run out of RAM,though (lots of noise from your hard disk swapping); if you do, shift to the bits algorithm if its available. Bits will use muchless memory for large simulations, and will start up faster than sorted list, but it will take a long time searching for the lastfew spins. Both are much faster than the brute-force method.

continuous phasetransitions 12 - cornell...

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