rmp_v080_p0379

Colloquium: Chaotic quantum dots with strongly correlated electrons

R. Shankar*

Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520, USA

�Published 1 April 2008�

Quantum dots pose a problem where one must confront three obstacles: randomness, interactions,and finite size. Yet it is this confluence that allows one to make some theoretical advances by invokingthree theoretical tools: random matrix theory, the renormalization group, and the 1/N expansion.Here the reader is introduced to these techniques and shown how they may be combined to answera set of questions pertaining to quantum dots.

DOI: 10.1103/RevModPhys.80.379 PACS number�s�: 73.21.La, 71.10.Ay

CONTENTS

I. Introduction 379II. The Renormalization Group 383

A. What is the RG? 383B. How is mode elimination actually done? 384C. Why do the RG? 384

III. The Problem of Interacting Fermions 385IV. RG Meets Dots 387V. The 1/N Approximation 389

VI. Summary and Conclusions 392Acknowledgments 393References 393

I. INTRODUCTION

This Colloquium is based on a lecture entitled “Dotsfor Dummies” I have frequently given. The title waschosen, not to offend, but to keep experts in the audi-ence from hijacking the lecture with minutiae while theintended goal was to introduce certain problems involv-ing quantum dots to a broad audience not necessarilyworking in this subfield. The present Colloquium at-tempts to do the same for the general readership of thisjournal. To follow this Colloquium you must be familiarwith the rudiments of second quantization and Feynmandiagrams.

The emphasis of this Colloquium is idiosyncratic. I ampartial to what I know best and like to talk about most:a frankly pedagogical tour of several useful techniquesand their applications to this problem. There is not acomparable emphasis on phenomenology, for which Iwill direct you to other excellent sources.

So let us begin by asking: “What are some of the ques-tions in the world of quantum dots, what are some of thetools brought to bear in addressing them, and what aresome of the answers?”

For our purposes, the dot is an island of size L �in thenanoscale� within which electrons are restricted to live.

If the dot is very small and the level spacing large

�compared to other scales like temperature and interac-tion strength�, we may be interested in just a few quan-tum states of the dot. For example, if the dot is essen-tially a two-state system, one may study it with theintention of using it as a qubit in quantum computation.Here one needs to understand in detail the level struc-ture of a particular dot so we can manipulate �program�it in a controlled way.

The focus here is on large dots with a very fine levelspacing. These dots are closer to the bulk system of in-teracting fermions in that a large number of energy lev-els are in play. However, the finite size and finite levelspacing are essential features. These dots are very muchlike nuclei—finite systems of interacting fermions—withthe main difference that the confining potential is exter-nally imposed and not internally generated. The hard-wall boundary of the dot is assumed to be sufficientlyirregular so that classical motion is chaotic at andaround the Fermi energy. The dot is otherwise dirt-freeand motion within is ballistic. The system is assumed tobe quantum coherent across the length of the dot.

The dot is juxtaposed with leads that act as source anddrain for electrons. Electrons are allowed to tunnel inand out of the dot. A gate voltage Vg allows one to varythe dot energy relative to the Fermi energy of electronsin the leads as in Fig. 1. A tiny voltage V is appliedbetween the source and drain to drive the current, andthe conductance G=I /V �in the limit of vanishing I and

*[email protected]

Source Drain

Vg IV

Gate

DOT

FIG. 1. A typical situation where leads bring in electronswhich tunnel into and out of a dot �opposite to the currentshown by arrows�. A gate voltage Vg is the control parameterand V is the vanishingly small source-drain voltage difference.

REVIEWS OF MODERN PHYSICS, VOLUME 80, APRIL–JUNE 2008

0034-6861/2008/80�2�/379�16� ©2008 The American Physical Society379

http://dx.doi.org/10.1103/RevModPhys.80.379

V� is measured as a function of the gate voltage Vg assketched in Fig. 2.1

The theoretical challenge is to describe the observedseries of peak positions and peak heights �Jalabert,Stone, and Alhassid, 1992; Chang et al., 1996; Folk et al.,1996; Guhr, Muller-Groeling, and Weidenmuller, 1998;Alhassid, 2000; Merlin, 2000; Aleiner et al., 2002�. Theseare decided by the shape of the empty dot, coupling tothe leads, the number of electrons in it, and the interac-tions between them. Now, no one disputes that givenenough such information, every peak may be describedin greatest detail since the underlying laws �nonrelativ-istic quantum mechanics and the Coulomb interaction�are thoroughly understood. However, this is not the goalhere. The goal is more akin to that set by Wigner indescribing excited states of nuclei. A nucleus, like a dot,is a finite many-body system with strong interactions be-tween its constituents. Wigner argued that while onecould, in principle, describe the specific levels of a par-ticular nucleus with its incredibly complicated Hamil-tonian, it would be more interesting to describe statisti-cal properties of the spectrum. More specifically hesuggested the following program. Consider a mathemati-cal ensemble of Hamiltonians which have the same sym-metries as the nuclear Hamiltonian �say, being real andHermitian� and have the same average level spacing.Calculate the statistics of level spacings in this ensemble.Compare this to the actual levels by averaging overmembers of the physical ensemble.2

Note the difference with statistical mechanics, as em-phasized by Guhr et al. �Guhr, Muller-Groeling, andWeidenmuller, 1998�. There we consider an ensemble ofsystems with the same Hamiltonian but different initialconditions, while here we consider an ensemble ofHamiltonians with the same symmetry properties as thegiven nuclear Hamiltonian.

In the case of dots, the members of the physical en-semble contain dots with the same density of states. One

can generate members of the ensemble by varying thenumber of electrons in one dot �which probes differentenergy regimes� or by manufacturing many similar dotsof the same area. If dots of different sizes are used, onecan rescale their energies so that the mean spacing is thesame. The mathematical ensemble consists of Hamilto-nians of the same symmetry class �e.g., real� and sameaverage level spacing. In cases where there is a magneticfield �and the ensemble contains complex unitary Hamil-tonians� one can also vary the field to move around theensemble. �A minimum change in field may be neededto generate an independent member.�

The following are some possible questions:

• If I measure �, the difference in gate voltage be-tween successive peaks, what will be the probabilitydistribution P��?

• If I measure G, the maximum of each conductancepeak, what will be the probability distribution P�G�?

There are of course other issues that we could discuss,but will not, such as the width of each peak as a functionof temperature or spin content.

To get a feeling for the results, let us see how wewould go about explaining the data, armed with just thetheory of noninteracting electrons �whose spin will beinitially ignored�.

To this end let us consider a particular dot of someparticular shape. We would naturally begin by solvingthe single-particle Schrödinger equation

H0��r� = ��r� , �1�

with ��r�=0 at the boundary. This would have to bedone on a computer for a generic dot. Imagine that wehave all the wave functions �� and energies ��. Let �i=�i+1−�i be the spacing between levels i and i+1 and let� stand for a generic one, as well as the average levelspacing �which is the inverse density of states�.

In the noninteracting theory, it is easy to see that at arandomly chosen value of gate voltage Vg there will beno conduction. This is because the Fermi energy of theelectrons in the lead will typically lie between occupiedand empty levels in the dot, so that transmission by theoccupied levels is forbidden by the Pauli principle andtransmission by the empty levels is forbidden by energyconservation. However, when the Fermi energy of theleads equals an energy level of the dot, there will betransmission. Note that during transmission the dot hasthe same energy with N or N+1 electrons. This featuresurvives even when interactions are included.

In this free-particle model, �i,i+1, the difference in thegate voltage Vg between peak i+1 and peak i times −e,the charge of the electron, will equal �i,i+1=�i+1−�i, thedifference in energy between the level that was respon-sible for peak i+1 and the one that was responsible forpeak i. It follows that �suppressing the constant −e�

P�� = P�� . �2�

In other words, the distribution of spacings in Vg be-tween successive peaks is the same as the distribution of

1There is a proportionality factor relating the actual gate volt-age to the applied gate voltage which we have set equal tounity.

2It can sometimes happen that any large segment of the spec-trum of almost every member of the physical ensemble exhib-its these statistical trends, in which case we can look at just onemember.

G

Vg∆

FIG. 2. A caricature of conductance G vs gate voltage Vg. Thepeaks have varying heights, widths, and spacings.

380 R. Shankar: Colloquium: Chaotic quantum dots with …

Rev. Mod. Phys., Vol. 80, No. 2, April–June 2008

level spacings in the dot. Consequently, to obtain thestatistical distribution P�� theoretically, we need tosolve for a large number of energy levels in one particu-lar dot, record the spacings �, and repeat within the en-semble, which means here other dots with similar den-sity of states but arbitrary shape. �In order to fold in theresults from many dots and many energy ranges, it willbe necessary to rescale energies so that �, the averagelevel spacing, is fixed at some value.�

As for the height of any conductance peak, it will begiven by the product of the probabilities for hopping onto the dot at the left lead L and hopping off on the rightlead R. These are determined ��L /R�, the value nearthe leads of the wave function �� corresponding to thestate which is responsible for the transmission. Onecould collect the statistics of these as well, from our nu-merical solutions to Eq. �1�.

Now it turns out we can spare ourselves a lot oftrouble in obtaining these probability distributions P��and P�G� by appealing to random matrix theory �RMT�,provided certain conditions apply. So we begin with acrash course on RMT.

The first step is to acquaint ourselves with ET=�vF /L, the Thouless energy, where vF is the Fermi ve-locity. Evidently ET stands for the uncertainty in energyof an electron that traverses the dot ballistically at theFermi velocity in a time �=L /vF. Thus if the dot is con-nected to big fat leads,

g �ET

�, �3�

single particle levels in this energy window will eachcontribute a maximum of e2 /h to conductance G. Thusin this case of a dot connected strongly to leads, g will bethe dimensional conductance, i.e., G measured in unitsof the quantum conductance e2 /h.

While this is one way to introduce ET, you could ob-ject that in the experiments we consider the leads areweakly coupled to the dot and the levels are sharp. Youwould be right, and it turns out we are interested in ETfor the following different reason.

If the dot has a generic shape with no conserved quan-tities except for energy, and the classical dynamics at theFermi energy is chaotic, the statistics of energy levelslying within a band of width ET and of the correspond-ing wave functions are determined by RMT �Mehta,1991�, given just general features like the average levelspacing � and symmetries of the Hamiltonian. This islike saying that the statistical properties of an isolatedbox of gas are determined by just gross features likevolume, number of particles, and energy. Such a statisti-cal approach to the energy levels of very complicatedsystems was first taken by Wigner in the case of nuclei,which are once again finite systems made up of a largenumber of particles subject to strong interactions thatdefy any direct treatment. Notice that to use RMT youjust need to know the symmetry class �real or complexmatrix elements, etc.� but no further details includingeven the spatial dimensionality of the problem.

The first RMT prediction is that P��, the distributionof exact energy-level differences �i+1−�i �for states lyingwith a band of width ET�, will be indistinguishable fromthat of an ensemble of random matrices of the samesymmetry class, which in our problem consists of all realHermitian matrices with the same �, the average levelspacing. �For a dot in a magnetic field, where time-reversal symmetry is broken, the ensemble will consistof complex Hermitian matrices.�

In other words, the results one person gets by pain-stakingly solving for the spectrum of an ensemble ofdots and compiling the level statistics is the same as thatcompiled by another person who picks real HermitianHamiltonians out of a hat, diagonalizes them, and plotstheir spacing distribution! Furthermore, this universaldistribution is known in analytic form as soon as we pro-vide �.

So, history repeats itself: when things get really bad�the dynamics goes from integrable to chaotic�, they be-come good again because statistical methods become ap-plicable.

The second RMT result bears on wave functions. Acommon quantity of interest is the ensemble average�denoted by �¯�� of products of wave functions such as

��*�r��r�� , �4�

defined as follows. Pick one realization of the dot andnumber the states by energy, with � and � being twosuch numerical labels. Pick two points r and r� inside thedot. Find ��

*�r��r��. Now change the dot smoothly,staying within the ensemble. Since there is no levelcrossing in a chaotic dot, the labels � and � retain theirintegrity. Now recompute ��

*�r��r��. Find the averageof such terms over the entire ensemble. This is what Eq.�4� means. Such correlations are of use in computingpeak-height distributions �Jalabert, Stone, and Alhassid,1992�.

In our problem we need the correlation of wave func-tions in momentum space rather than coordinate spaceand we shall use them to deal with electron-electron in-teractions rather than to calculate P�G�.

Let us begin by modifying the definition of “momen-tum space” as appropriate to the dot.

Consider a circular Fermi sea in k space and a con-centric annulus of width ET in energy. In the bulk, thisregion contains an infinite number of k states. If we nowgo to the dot of size L, the best we can do vis-à-vismomentum is wave packets centered at some k and ofwidth 1/L in both directions. It is readily verified thatwe can form g such “wheel-of-fortune” �WOF� states �asin Fig. 3� within this annulus �Murthy, Shankar, andMathur, 2005�. Suppose we expand the g exact eigen-states �labeled by �� of a dot within the Thouless band inthis WOF basis �labeled by k� via the functions ��k� asfollows:

381R. Shankar: Colloquium: Chaotic quantum dots with …


�� = �k

��k��k� . �5�

Then a typical RMT result invoked here is that as g→,

��*�k��k�� =

�kk��

g+ O� 1

g2 , �6�

where the �¯� denotes an average over an ensemble ofsimilar dots.

Note that Eq. �6� is the minimal correlation we musthave: in each sample, � and k label two orthonormalbases, so that if we set k=k� and sum over k we must get��, sample by sample and hence on average as well.�The same goes for setting �=� and summing over themto get �kk�.� Similar correlators exist for products of fourwave functions and these have the form of Wick’s theo-rem.

Unlike the result on level spacings, which can be dem-onstrated analytically, this one on wave functions is anassumption we will make. All we can say is that severalconsequences of this assumption have been verified inour numerical work �Murthy et al., 2004�.

Before proceeding, let me address a common ques-tion. Is there any reason to believe that the g exacteigenstates within ET can be expanded in terms of the gWOF states? We �Murthy, Shankar, and Mathur, 2005�have verified the following in a numerical study of a“billiard,” or dot. First we manufactured the g states ofmean momentum k by choosing g equally spaced pointson the Fermi circle and then chopping off the plane

waves of these momenta at the edges of the billiard. �Wechose g=37 in our study.� Then we verified that thesewave functions were orthonormal to an excellent accu-racy. Next we asked how good a basis these functionsformed for expanding the � states within ET. We foundthe exact eigenstate at the middle of the Thouless band,i.e., at the Fermi energy, retained more than 99.9% of itsnorm upon projection to the WOF basis. As we left thecenter of the band the overlap decreased and dropped toaround 50% at the edges. Thus our results get morereliable as we go deeper into the Thouless band cen-tered at the Fermi energy.

Let us turn to the data on P��, the distribution ofspacings in Vg between peaks. In the noninteracting casewe saw that P��=P��, and then we saw that P�� wasgiven by RMT.

So let us test the RMT results for level spacings P��,against P��, the measured distribution of spacings in Vgbetween successive peaks. These will match only if theassumptions of noninteracting electrons is correct.

The test fails. It is found that the peak spacings are anorder of magnitude larger than level spacings in the dot.The reason is of course electron-electron interactions.Even if we ignore the full quantum-mechanical treat-ment of interactions, we need to acknowledge the clas-sical fact that when we add an electron to a dot, it expe-riences Coulomb repulsion for the other occupants.Thus the energy cost of adding an electron is not justthat of going to the next empty level, it is the additionalrepulsive energy. This capacitive charging energy is pro-portional to N2 and we must add a term u0N2 to thesecond quantized Hamiltonian. By studying the data onecan come up with a good fit to u0.

However, we can anticipate that this alone will notwork in the presence of spin, even in the noninteractingcase. With spin present, each single-particle orbital canbe doubly occupied. Thus adding an electron will �willnot� cost us a single-particle energy-level difference if itbrings the total occupancy to an odd �even� integer.Combined with the above-mentioned charging energy,we expect the peak spacing to have a bimodal distribu-tion as N varies from odd to even.

This is, however, not seen. The reason is based on theexchange interaction. While it makes sense to put twoelectrons in each orbital if one is minimizing kinetic en-ergy, this may be a bad idea in terms of potential energy:the electrons in any given orbital would have oppositespins and hence the Pauli principle would not keep themaway from each other, thereby raising the Coulomb re-pulsion. To minimize the potential energy we must oc-cupy each orbital once and place the electrons in thesame spin state so that the Pauli principle can keep themapart in space. To let the system decide what is best wemust give it an incentive for higher spins by adding aterm −JS2, where S is the total spin.

Thus we are led to the following �second-quantized�“universal” Hamiltonian �Andreev and Kamenev, 1998;Brouwer, Oreg, and Halperin, 1999; Baranger, Ullmo,and Glazman, 2000; Kurland, Aleiner, and Altshuler,

Fermi circle

Thoulessboundaries

FIG. 3. The wheel-of-fortune �WOF� states within a band ofenergy ET concentric with the Fermi circle. There are roughlyg such states of mean momenta k centered on g equally spacedpoints on the Fermi circle. The WOF states are obtained bychopping off plane waves of the desired mean momentum atthe edges of the dot. These states are very nearly orthonormal.



2000; Aleiner et al., 2002; Oreg et al., 2002�:

HU = ��

�†�� + u0N2 − J0S2, �7�

where � destroys an electron in state �. A third pos-sible interaction term pertaining to superconductingfluctuations has been dropped on the grounds that it isunimportant for the dots in question.

Some proponents of the universal Hamiltonian givethe following argument for why no other interactionsneed be considered. Suppose we take any other familiarinteraction and transcribe it to the exact basis. Sums ofproducts of random wave functions �� will appear andlead to terms with wildly fluctuating phases. These termscan be shown to have zero ensemble average as g→.Since deviations from the zero average will be down by1/g we can drop them at large g. By contrast, the twoterms kept �which commute with H0� are free of oscilla-tions and survive ensemble averaging.

While the success of HU in explaining a lot of data isunquestioned, the accompanying arguments are not per-suasive. In particular, ensemble averages should be per-formed not on the Hamiltonian but on calculated ob-servables. It is also not clear that a group of termsshould be dropped because they are small, since theycould ultimately prove important to physics at very lowenergies �low, even within the already small band ofwidth ET�.

Now there is a tried and tested way for determiningthe relative importance of possible interactions in thelow-energy limit. It is the renormalization group �RG�.Not only can it tell us how important any given interac-tion is in the low-energy sector, if there are competinginteractions, it can provide an unbiased answer in whichthey are all allowed to fight it out. As explained inShankar �1994�, the Luttinger liquid in d=1 is an ex-ample in which superconductivity and charge-densitywave formation compete and actually annul each otherwhile in d=2 the latter wins at half filling on a squarelattice.�

It is therefore natural to ask if the RG is applicable inthis problem and if so what its verdict is.

II. THE RENORMALIZATION GROUP

In order not to leave behind readers unfamiliar withthe RG, the following rapid review is provided. Expertscan skip to the next section.

A. What is the RG?

Imagine that you have some problem in the form of apartition function,

Z�a,b� =� dx� dye−a�x2+y2�e−b�x + y�4�8�

� dx� dye−S�x,y;a,b,. . .�, �9�

where a and b �along with other such possible terms� areconstant parameters and S is called the action. �In clas-sical statistical mechanics it would be called the energy.�

The parameters b, c, etc., are called couplings and themonomials they multiply are called interactions. The x2

term is called the kinetic or free-field term, and a is thecoefficient of the kinetic term, has no name, and is usu-ally set equal to 1

2 by rescaling x.The average of any f�x ,y� is given by

�f�x,y�� =� dx� dyf�x,y�e−a�x2+y2�e−b�x + y�4

� dx� dye−a�x2+y2�e−b�x + y�4. �10�

Suppose we are interested only in functions of just x.In this case

�f�x�� =� dxf�x� � dye−a�x2+y2�−b�x + y�4

� dx� dye−a�x2+y2�−b�x + y�4

� dxf�x�e−a�x2−b�x4+¯

� dxe−a�x2−b�x4+¯

, �11�

where

e−a�x2−b�x4¯ =� dye−a�x2+y2�−b�x + y�4

�12�

defines the parameters a� ,b� , . . .. In the general case wewould like to define an effective action S��x ;a� ,b� ,c� , . . . �by

e−S��x;a�,b�,c�,. . .� =� dye−S�x,y;a,b,c,. . .� �13�

and the corresponding partition function

Z�a�,b�, . . . � =� dxe−S��x;a�b�c�,. . .� �14�

which defines the effective theory for x. These param-eters a�, b�, etc., will reproduce exactly the same aver-ages for x as the original ones a ,b ,c , . . . did in the pres-ence of y. This evolution of parameters with theelimination of uninteresting degrees of freedom is calledrenormalization. It has nothing to do with infinities; youjust saw it happen in a problem with only two variables.

Note that even though y does not appear in the effec-tive theory, its effect has been fully incorporated in the



process of integrating it out to generate the renormal-ized parameters. We do not say “We are not interestedin y, so we will set it equal to zero everywhere it ap-pears,” instead we said, “What theory, involving just x,will give the same answers as the original theory thatinvolved x and y?”

The term “group” arises as follows. Let us say y is ashorthand for many variables as is always the case inreal life. If we eliminate one of them, say y1 which leadsto the renormalization �a ,b ,c , . . . �→ �a� ,b� ,c� . . . �, andthen we eliminate y2 so that now �a� ,b� ,c� , . . . �→ �a� ,b� ,c� . . . � the net result is equivalent to a singlerenormalization process in which �a ,b ,c , . . . �→ �a� ,b� ,c� . . . � under the elimination of y1 and y2. �TheRG is not a real group since we cannot define a uniqueinverse.�

B. How is mode elimination actually done?

Notice that to get the effective theory we need to do anon-Gaussian integral. This can only be done perturba-tively. At the simplest tree level, we simply drop y andfind b�=b. In other words, at tree level, the effectiveaction is found by simply setting the unwanted variablesto zero in the original action.

At higher orders, we bring down the nonquadratic ex-ponential and integrate in y term by term and generateeffective interactions for x.

Here is how it is done in our illustrative example:

e−S� = e−ax2−bx4� dye−ay2e−b�4xy3+4x3y+6x2y2+y4�

= e−ax2−bx4Z0�a�

� dye−ay2e−b�4xy3+4x3y+6x2y2+y4�

Z0�a�

= e−ax2−bx4Z0�a��e−b�4xy3+4x3y+6x2y2+y4��Z0

, �15�

where we have multiplied and divided by Z0�a�,

Z0�a� =� dye−ay2, �16�

the partition function for a Gaussian action e−ay2and

where

�e−b�4xy3+4x3y+6x2y2+y4��Z0�17�

stands for the average of the exponential with respect tothe partition function Z0�a�. Since Z0�a� is independentof x, we will simply ignore it in the effective actionS��x ;a� ,b� , . . . �, without altering any absolute probabil-ity. As for Eq. �17� we invoke the result valid for aver-ages over Gaussian actions:

�e−V�Z0= e−�V�−�1/2��V2�−�V�2�. . . . �18�

This is called the cumulant expansion and we see that inour example where V=b�4xy3+4x3y+6x2y2+y4�, the ex-ponent is a power series in b for contributions to S�.

To the leading order in b we have

S��x ;a�,b� . . . � = ax2 + bx4 + �b�4xy3 + 4x3y

+ 6x2y2 + y4��

= ax2 + bx4 + 6bx2�y2� + �y4�

= ax2 + bx4 + 6bx2

2a+

3

4a2 �19�

because �y�= �y3�=0, �y2�= 12a , and �y4�= 3

4a2 . Thus wehave our first RG result for renormalization:

a� = a +3b

a�20�

to leading order in b. Note that we do not care aboutx-independent constants like 3

4a2 . For readers who knowFeynman diagrams, the power series in b can be identi-fied with the Feynman graphical expansion for a �4

theory �since we kept up to quartic terms in the action�.In the case of the actual �4 field theory itself, when themomentum cutoff is reduced from � to � /s �where s�1�, x will denote low-momentum modes 0 k � /sand y denotes the high-momentum modes � /s k �,and the loop integration will be over the range � /s k �.

C. Why do the RG?

Why do we do this? Because the ultimate effect of anycoupling on the fate of x �the variable we care about� isnot so apparent when y �the variable in which we haveno direct interest� is around, but surfaces to the top onlyas we zero in on x. For example, we are going to con-sider a problem in which x stands for many low-energyvariables and y stands for for many high-energy vari-ables. As we integrate out high-energy variables andzoom in on the low-energy sector, an initially tiny cou-pling can grow in size, or an initially impressive one di-minish into oblivion.

This notion can be made more precise as follows.Consider the Gaussian model in which we have a�0.We have seen that this value does not change as y iseliminated since x and y do not talk to each other. This isan example of a fixed point of the RG since the couplingthat goes in comes out unchanged by elimination of un-wanted variables.3 Now turn on new couplings or “inter-actions” �corresponding to higher powers of x, y, etc.�with coefficients b, c, and so on. Let a�, b�, etc., be thenew couplings after y is eliminated. Let us in additionrescale x so that x2 has the same coefficient as before,

3There can be more complicated fixed points where this hap-pens despite the fact that the x’s and y’s talk to each other, andthe integrals do not factorize. While we will not encounterthem in our problem, the proposed strategy applies there aswell.



i.e., a�=a= 12say.4 Any of the renormalized couplings

�still called b� ,c� , . . .�, which are are bigger than the ini-tial ones, b ,c ,d , . . ., are called relevant while those thatare smaller are called irrelevant. This is because in real-ity y stands for many variables, and as they are elimi-nated one by one, the relevant coefficients will keepgrowing with each elimination and the irrelevant oneswill keep shrinking and ultimately disappear. If a cou-pling neither grows not shrinks it is called marginal.Thus the RG will tell us which couplings really matter inthe low-energy limit which controls the nature of theground state and its low-lying excitations.

There is another excellent reason for using the RG,and that is to understand the phenomenon of universal-ity in critical phenomena, an area I will not discuss.

Later we will apply these methods to quantum dots.For an interested reader, here is a glimpse how we willuse RG in the case of dots. First we will begin with allsingle-particle states in the Hilbert space of dots. Thenwe will zero in on states lying within a narrow band ofstates within an energy EL, the Fermi energy EF. �Allone needs is that EL�EF.� In other words, at this pointthe variables we called x �y� are states lying inside �out-side� this band. For the clean system in the bulk, theresult of such a mode elimination is known �Shankar,1994� to lead to the result that the system is described byan infinite number of Landau Fermi-liquid parametersum, where m is an integer. This process is briefly dis-cussed. The universal Hamiltonian contains just the u0term.5 We would like RG to tell us if and when we canignore all other um’s. To this end we will begin with atheory where only states within ET of EF are kept andthe starting Hamiltonian has all Landau interactionswritten in terms of the dot eigenfunctions ��. Then wewill ask what happens to them as we eliminate stateswithin ET, getting even closer to EF. To see what hap-pens when we do this �and how we do this� details areprovided below.

III. THE PROBLEM OF INTERACTING FERMIONS

We will now apply RG to a system of nonrelativisticspinless fermions of mass m and momentum K in twospace dimensions. The single-particle Hamiltonian is

H =K2

2m− � , �21�

where the chemical potential � is introduced to ensurewe have a finite density of particles in the ground state:all levels within the Fermi surface, a circle defined by

KF2

2m= � , �22�

are now occupied since occupying these levels lowersthe ground-state energy.

In second-quantization we start with the Hamiltonian

H0 =� d2K†�K��K2 − KF2

2m�K� , �23�

where †�K� creates a fermion of momentum K. Noticethat this system has gapless excitations above the groundstate. You can take an electron just below the Fermisurface and move it just above, and this costs little en-ergy. Such a system will carry a dc current in response toa dc voltage. An important question one asks is if thiswill be true when interactions are turned on. For ex-ample, the system could develop a gap and become aninsulator. Or it could become a superconductor. How dowe decide what the fermionic ground state and low-energy excitations will be, short of solving the problem?

In the noninteracting limit the ground state is simple:it is a single Slater determinant �antisymmetrized wavefunction� with all momentum states below KF occupied.If we turn on say quartic interactions, the new groundstate will have pieces in which two fermions from belowKF have been moved to two fermions above KF keepingthe total momentum at zero. Each such piece will comewith a numerator proportional to the interactionstrength and an energy denominator equal to the cost ofcreating this particle-hole pair. Clearly fermions deep inthe sea will not take part in this kind of process with anyserious likelihood as long as the interactions to be addedare weak. Likewise, if states far above KF were removedno one would be aware. Thus the fate of the system isgoing to be decided by states near the Fermi energy.This is the great difference between this problem andthe usual ones in relativistic field theory and statisticalmechanics. Whereas in the latter examples low energymeans small momentum, here it means small deviationsfrom the Fermi surface. Whereas in these older prob-lems we zero in on the origin in momentum space, herewe zero in on a surface. The low-energy region is shownin Fig. 4.

We are now going to add interactions and see whatthey can do. This is going to involve further reduction ofthe cutoff. Here we join the discussion of Shankar�1994�.

Let us begin then with a momentum band of width �on either side of the Fermi surface. In terms of energy,the band has a width EL, where L stands for Landau, forreasons that will follow.

Let us first learn how to do RG for noninteractingfermions. To apply our methods we need to cast theproblem in the form of a path integral. Following anynumber of sources see Shankar �1994��, we obtain thefollowing expression for the partition function of freefermions:

4We like to keep a fixed because what really matters is therelative size of a and the interaction terms. For example, if itturns out that b��b but also that a��a, it is not clear that therenormalized theory has stronger interactions.

5There are actually two sets of u’s, one for charge and one forspin. The two u0’s are the charging and exchange interactionsu0 and J0.



Z0 =� dd�eS0, �24�

where

S0 =� d2K�−

d��,K��i� −�K2 − KF

2�2m

��,K�

�25�

and

dd� = ��,K

d��,K�d��,K� �26�

and and are called Grassmann variables. They arereally strange objects one will understand after some fa-miliarity. There are some rules for doing integrals over

them. For now, I suggest you note only that �i� �� ,K�and �� ,K� are defined for each � and k in the annulus,�ii� Z is a product of Gaussian integrals, with the vari-ables at each �� ,K� not coupled with those at another.The interested reader can see Shankar �1994�.

We now adapt this general expression to a thin annu-lus to obtain

Z0 =� dd�eS0, �27�

where

S0 = �0

2�

d��−

d��−�

�

dk�i� − vFk� . �28�

We have approximated as follows:

K2 − KF2

2m�

KF

mk = vFk , �29�

where k−K−KF and vF is the Fermi velocity, hereafterset equal to unity. Thus � can be viewed as a momentumor energy cutoff EL measured from the Fermi circle. Wehave also replaced KdK by KFdk and absorbed KF in

and . It will be seen that neglecting k in relation to KFis irrelevant in the technical sense.

Let us now perform mode elimination and reduce thecutoff by a factor s. Since this is a Gaussian integral,mode elimination just leads to a multiplicative constantthat we are not interested in. So the result is just thesame action as above, but with �k�� /s. Consider thefollowing additional transformations:

��,k�� = s��,k� , �30�

„��,k��,��,k��… = s−3/2��

s,k�

s,��

s,k�

s� .

�31�

When you perform the exercise of making this changeof variables, you will find that the action and the phasespace all return to their old values. Our plan is to evalu-ate the role of quartic interactions in low-energy physicsas we do mode elimination. Now what really matters isnot the absolute size of the quartic term, but its sizerelative to the quadratic term. Keeping the quadraticterm identical before and after the RG action makes thecomparison easy: if the quartic coupling grows, it is rel-evant; if it decreases, it is irrelevant, and if it stays thesame it is marginal.

Let us now turn on a generic four Fermi interaction inpath-integral form:

S4 =� �4��3��2��1�u�4,3,2,1� , �32�

where � is a shorthand:

� �i=1

3 � d�i�−�

�

dki�−

d�i. �33�

At the tree level, we simply keep the modes within thenew cutoff, rescale fields, frequencies, and momenta,and read off the new coupling. We find

u��k�,��,�� = u�k�

s,��

s,� . �34�

This is the evolution of the coupling function. To dealwith coupling constants with which we are more familiar,we expand the functions in a Taylor series �schematic�,

u = u0 + ku1 + k2u2 + ¯ , �35�

where k stands for all the k’s and �’s and of course the�’s are fixed. An expansion of this kind is possible sincecouplings in the action are nonsingular for short-rangeinteractions. If we now make such an expansion andcompare coefficients in Eq. �34�, we find that u0 is mar-ginal and the rest are irrelevant, as is any coupling ofmore than four fields. For example,

u0� = u0, u1� =u1

s, un� =

un

sn . �36�

Suppose we start with some cutoff �0 and an initialcoupling we shall call un�0�. Let us now reduce � con-

KF

ΛKF +

ΛKF -

FIG. 4. The low-energy region for nonrelativistic fermions lieswithin the annulus concentric with the Fermi circle. It extends� in momentum and EL in energy from the Fermi circle. Inunits where the Fermi velocity vF is chosen to be unity, the twoare equal.



tinuously from �=�0e−t � /s so that the change in un iscontinuous in t. The relation un�=un /sn, Eq. �36�, can bewritten as un�t�=un�0�e−nt.

Sometimes this result is rewritten in terms of the �function:

��t� =dun�t�

dt= − nun�t�, where t = ln s . �37�

If you consider the �44 scalar field theory in four di-

mensions you find again an equation like Eq. �35� withthe difference that k is measured from the origin and notthe Fermi surface. Thus all we have left to worry aboutis one number u0, the first term in the Taylor seriesabout the origin, to which the low-energy region col-lapses. Here the low-energy manifold is a two-sphere nomatter how small � is and u0 will inevitably have depen-dence on the angles on the Fermi surface:

u0 = u��1,�2,�3,�4� .

Therefore in this theory we are going to get couplingfunctions and not a few coupling constants.

Let us analyze this function. Momentum conservationshould allow us to eliminate one angle. Actually it al-lows us more because of the fact that these momenta donot come from the entire plane, but a very thin annulusnear KF; see Fig. 5. Assuming that the cutoff has beenreduced to the thickness of the circle in the figure, it isclear that if two points K1 and K2 are chosen from it torepresent the incoming lines in a quartic coupling, theoutgoing ones are forced to be equal to them �not intheir sum, but individually� up to a permutation, which isirrelevant for spinless fermions. Thus we have in the endjust one function of two angles, and by rotational invari-ance, their difference:

u��1,�2,�1,�2� = F��1 − �2� u�� . �38�

About 50 years ago Landau came to the very same con-clusion �Landau, 1956�, that a Fermi system at low ener-gies would be described by one function defined on theFermi surface. He did this without the benefit of the RGand for that reason, some of the assumptions were hardto understand.

Since u is marginal at tree level �gets neither big norsmall as the RG process is iterated and modes are elimi-nated�, we have to go to higher orders in perturbationtheory to break the tie, to see if it ends up being relevantor irrelevant. The answer, given without proof, is that itremains marginal to all orders �Shankar, 1994�. This is

due to kinematical reasons. A brief review of how thehigher-order calculation will be done will be given whenwe discuss dots.

Often one writes

u�� = �m

um cos�m�� , �39�

where um are the Landau parameters. The correspond-ing interaction6 is, in schematic form with radial inte-grals over k suppressed,

HL = ��,��

n��n��um cosm�� − �� , �40�

where n�� is the number density at angle � on the Fermicircle.

Note that if u0 is the only nonvanishing Landau cou-pling, we get the u0N2 interaction of HU. If we includespin, there are spin-density–spin-density interactionsand J0 corresponds to keeping just the zeroth harmonic.So HU amounts to keeping just the lowest harmonic inthe Landau expansion.

We need to ask when and why m�0 terms can beignored.

IV. RG MEETS DOTS

The first step towards the goal of RG meets dots wastaken by Murthy and Mathur �2002�. Their strategy wasas follows.

• Step 1: Use the clean system RG described above tolearn that at low energies the important interactionsare the Landau interactions �of which u0 and J0 are asubset�.

• Step 2: Start at the energy scale ET and switch to theexact basis states of the chaotic dot, writing the ki-netic term and all the Landau terms in this basis.Run the RG by eliminating exact energy eigenstateswithin ET. See what happens to all Landau terms, dothey grow, fall, or stay the same?

I will discuss some of the subtleties here, referring thereader to Murthy et al. �2004� for more details.

A common concern is to ask if we can ignore the wallsof the dot and use momentum states at high energies.After all, momentum is not conserved in the dot andwhat meaning is there to using the Landau interactionwritten in the momentum basis? The point is this. Con-sider the collision of two particles in a box. As long asthe collision takes place quickly �in terms of the time to

6While this form of HL will suffice for the dot, in the cleanbulk system one must allow for small nonforward scatteringsince � is finite. Despite their small measure these terms arecrucial because the Fermi liquid has a very singular response atthe Fermi surface.

K1

K2

K

FIG. 5. Kinematical representation why momenta are indi-vidually conserved up to a permutation.



cross the box� the incoming and outgoing particles canbe labeled by momenta and this label will be useful�conserved� during the collision.7

However, this only brings us down to EL, the energybelow which Landau theory works. Although EL�EF, itis still finite for an infinite system while ET�1/L is evensmaller for large L. In the region between EL and ET,the flow of couplings is intractable and Landau param-eters could have evolved from their bulk values. To sum-marize, the universal Hamiltonian has just two of theinfinite Landau parameters �u0 ,J0� in it. Let us put in therest of them with any size, and see what their fate isunder further mode elimination.

While this looks like a reasonable plan, it is not clearhow it is going to be executed. There are at least twoobvious problems. To understand them you must under-stand in more detail how the RG is done at the one-looplevel. To illustrate this we use the more familiar scalartheory in d=4 with the u0�4 interaction, where u0, here-after called just u, is marginal and momentum indepen-dent. Suppose we want the scattering amplitude of fourscalar particles. The answer will be given by a perturba-tion series, depicted in Fig. 6. The second term is anintegral over the loop momentum K that goes from 0�K��. The integrand is the product of the two propa-gators, which go as 1/K2 each at large K. �We have as-sumed the external momenta are all zero and neglectedtwo other loop diagrams that behave the same way.� Theleft-hand side corresponds to a physical process and can-not depend on the cutoff �. It follows that u must ac-quire � dependence so as to make the right-hand side �independent. We can find u�� as follows. Suppose � isreduced by �d��. The loop will change by an amountequal to the integral in the region �− �d�� K �,which is now missing. This missing part must then comefrom the first term to keep the physical scattering ampli-tude fixed. A simple calculation shows that up to nu-merical factors

du = − u2 �d��

�41�

so that

��t� =du

dt= − u2. �42�

In summary, the change in u�� is given by the loopterms with internal lines restricted to the modes beingeliminated.

We run into the following problems if we try to do themode elimination for the dot.

First, knowledge of the exact eigenfunctions is neededto even write down the Landau interaction �the analogof u in the �4 example� in the disordered basis:

V�� = �kk�

u�� − ��*�k��

*�k�� − ��*�k��

*�k��

� ��k��k� − ��k��k�� , �43�

where k and k� take g possible values, � and �� are theangles associated with k and k� and the interaction hasbeen antisymmetrized to mate with the four-Fermi op-erators �

†�†��.

Next, additional information is needed on energy lev-els to do the higher-order �loop� calculation since thepropagator for state � is �i�−��−1. Remarkably, it ispossible to overcome ignorance of specific energy levelsand wave functions. Here are some, but not all, detailsof how this comes about.

First consider a specific realization. We want to reducethe cutoff by summing over some high-energy states�y’s�. If we write down the expression for the one-loopflow, fourfold products of the unknown wave functionsappear at each vertex. At the left vertex, two of thesewave functions correspond to external lines, which arefixed �and below the new cutoff, i.e., these are the xvariables� while the other two correspond to states to beeliminated and thus summed over. A similar thing hap-pens at the right vertex. In addition, there are the propa-gators for the two lines dependent on the single-particleenergies.

Thus what we have here is a product of four wavefunctions and an energy denominator summed overstates being eliminated. This sum will of course varyfrom dot to dot. Suppose we ignore this variation andreplace the sum by its ensemble average. �The productof wave functions and the energy denominators can beaveraged independently since to leading order in 1/g,RMT does not couple them.� The entire average willthen be just a function of just g as in Eq. �4�. But whatabout sample-to-sample variations? Here we invoke theresult �Murthy and Mathur, 2002� that deviations fromthe average are down by an extra power of g. Thus theflow is self-averaging, and every dot will have the sameflow as g→.8

7There will be a parametrically small number of cases �ratioof interaction range to dot size�, where collisions take placenear the walls, when this will be wrong.

8Experts should note that we are not averaging the � func-tion, it is self-averaging. There may also be concern that therewill not be enough states within d� to allow the averaging towork. This too can be handled: there is a way to define the �function �Shankar, 1994� in which the loop is first summed overall states inside � and then the � derivative is taken.

=+ +...

KG

uu

K

FIG. 6. Schematic of scattering amplitude in �4 theory. Thesolid dot is the full answer and the empty one is the couplingconstant that goes in. The loop is the first of many that enter.



Using such a device, these authors found the resultthat the renormalized V�� is itself equivalent to a Lan-dau interaction but with renormalized values of um flow-ing from

dum

dt= − um − cum

2 , m � 0, �44�

where c is independent of m and of order unity.Note that u0 does not flow and that just as in the BCS

flow of the clean system �Shankar, 1994�, different m’s donot mix to this order. If spin were included J0 would notflow either. This lack of flow is due the fact that thesecoefficients multiply operators that commute with H0,the noninteracting part of the Hamiltonian, and there-fore have no quantum fluctuations.

The flow �for m�0� implies that all positive um’s flowto zero, as do negative ones with um�u*, the fixed pointof the flow. Thus all points to the right of u* flow to HU,as shown in Fig. 7. If a modest amount of um, m�0 ofeither sign is turned on at the beginning �when �=ET�, itwill renormalize to zero as we lower energy. On theother hand, if we begin to the left of u*, we run off tolarge negative values.

The universal Hamiltonian is thus an RG fixed pointwith a domain of attraction of order unity, which is asatisfactory explanation of its success.

V. THE 1 ÕN APPROXIMATION

So far we have used the RG to understand the low-energy behavior of the quantum dot. We found that ifwe begin our analysis with states within ET of the Fermienergy EF and all possible Landau interactions um, andslowly reduce the energy cutoff, we end up with the uni-versal Hamiltonian as the low-energy fixed point for allpositive um’s and negative um’s that are not more nega-tive than some critical value um

* . From this knowledge atvery low energies we can conclude that the ground state�lowest of all energies� is determined by HU for um

�um* and that at um=um

* the system undergoes a second-order phase transition, which is what a zero of the betafunction or fixed point signifies. The RG does not tell usmuch about the nature of the new phase, other than thatit will be dominated by the um that runs off to strongcoupling.

While this is nice, there is clearly room for improve-ment. In particular, if the theory could be solved exactly,we would know even more. We would know if the phasetransition at um of order unity, found in the perturbativeone-loop calculation, really does occur. We would alsoknow to what state the system is driven after the transi-

tion. In other words, if we can solve a problem exactly,we do not need the RG. We could, after obtaining thesolution with some cutoff, easily ask how the coupling isto be modified if the cutoff is then reduced. We woulddo this by computing a physical quantity P„u�� ,�… as afunction of the input cutoff � and input coupling u��and then set dP /d�=0 to find du /d�. This would be theexact beta function that one could compare to any per-turbative result.

It turns out we can do all this in our problem of dots.Now, any controlled calculation is based on some smallquantity. If the small quantity vanishes the controlledcalculation gives exact results. The most common ex-ample is the coupling constant itself, and the exact solu-tion that emerges in the limit of zero coupling is a freetheory. However, there are other cases exhibiting non-trivial behavior, where the coupling is not small butsomething else is. For our dots the small parameterturns to be 1/g and the exact solution emerges in thelimit g→ as Murthy and Shankar �2003� discovered.Note that g→ corresponds to the limit of infinite dotsize. �Unfortunately this does not mean we understandbulk physics, since in this large dot the Thouless band,over which we have control using RMT, shrinks to zeroas 1/L.� What we assume is that what happens in thisinfinite dot will be seen also in large but finite dots, justas one assumes that genuine phase transitions, which areallowed only in infinite systems, will be well mimicked inlarge but finite systems. We could show that in this limitthere is indeed a transition at a negative coupling oforder unity and that to the left of it is a symmetry-broken phase which can be analyzed in some detail.

We showed that one did not have to rely on RG orperturbation theory in powers of u. Instead the theorycould be solved by saddle point methods for any u dueto the smallness of 1/g. The trick was to use a variant ofthe so-called 1/N expansion. So let us first get ac-quainted with the 1/N expansion. Only a few details willbe provided here, referring the interested reader to Mur-thy et al. �2004�.

Let us begin with the common situation wherein weexpand the answer in a power series in the coupling u,via Feynman diagrams of increasing complexity. Thismethod works provided the answer can be expanded ina series at u=0 and we limit ourselves to weak coupling�since we can typically compute only to some small or-der in u�. There are, however, cases where we need to goto all orders in u or pick up essential singularities beforethe correct physics can be found. Some of these prob-lems are tractable due to the 1/N expansion. Here is abrief summary of this method.

Suppose we have a theory of fermions with an internalisospin or flavor label that runs over N values. �Themethod works for bosons as well.� Let the theory bedefined by the following schematic path integral:

u =u*m u=0

FIG. 7. The flow of the coupling um for any m�0. The originof this flow is the universal Hamiltonian where every um=0except u0 which can have any value. All points to the right ofu* flow to this. We ask what happens if we begin to the left.



Z =� dd�eD+�u/2N��2, �45�

where D stands for the quadratic kinetic energy term,and the sum over flavors or integral over space-time issuppressed, so that, for example,

=� drd��i

N

ii.

The factor of 1/N in the quartic term is to ensure thatthe single sum over the flavor index in the kinetic termhas a chance against the double sum in the quartic term.In a theory with Dirac fermions D=�” while in a nonrel-ativistic problem it could be � /��−��.

If we now introduce a Hubbard-Stratonovic field � wecan rewrite Z as

Z =� dd�d�e�D+��−N�2/2u, �46�

the correctness of which is readily verified by doing theGaussian integral over �. If we now use the fact that foreach flavor

� dd�e�D+�� = det�D + �� = eTr ln�D+�� 47�

and that each flavor gives the same determinant, we ob-tain

Z =� dd�d�e�D+��−N�2/2u �48�

=� d�eNTr ln�D+��−N�2/2u. �49�

It is now clear that in the limit N→, we can do the �integral by saddle point. At large and finite N, we cancompute corrections in a series in 1/N. However, thecomplete and exact dependence on u is obtained to eachorder in 1/N. Often just the leading term at N= cap-tures all the novel physics. A celebrated example is the�1+1�-dimensional Gross-Neveu model �Gross andNeveu, 1974�, where there is a coupling constant u andan internal O�N� isovector index that runs from 1 to N.In the limit N→, the saddle point method allows oneto show that the � field spontaneously develops a non-zero average �which translates into a mass for the fer-mion� that goes as e−1/u, a result clearly outside the reachof traditional perturbation theory. Corrections aroundthe saddle point in powers of 1/N do not change themain features quoted above. In summary, the small pa-rameter that makes a calculation possible is not u but1/N.

We are going to do the same thing here, with 1/gbeing the small parameter. You may object since in ourproblem D=� /��−��, the g different fermions are notrelated by symmetry, and the appearance of a largenumber �the analog of N� in front of the Tr ln is by nomeans assured. However, we found that if we went

ahead and evaluated the Tr ln order by order in � andexploited self-averaging as in the one-loop flow, a largenumber �g2� does indeed appear in front of the action,playing the role of N. Here is a brief summary of howthis occurs.

Let us first consider just one Landau term with cou-pling um, called u:

Z =� dd�d�e�D+��−�2/2u. �50�

Next let us make the change �→g� in Eq. �50�. Whilethis trivially brings a g2 in front of the �2 /2u term, whatis nontrivial is that every term in the Tr ln �developed inpowers of � � will also go as g2 to leading order. Thus wecan write the entire action as g2f�� where f has no gdependence to leading order and the saddle point of findeed gives exact answers as g→. I will now show thisjust for the quadratic term in the Tr ln. It will then beclear how it works for higher terms.

At a schematic level we can write

Tr ln�D + g�� = Tr ln D + Tr ln�1 + gD−1��

= C + gTrD−1� −g2

2TrD−1�D−1�

+ ¯ , �51�

where C is some constant independent of �. When allthe indices and details of the Landau interaction arefilled for this problem and an � integral is done, we find�dropping constants� the following quadratic term:

QT = g2 ��kk�

n� − n�

�� − ��

��12 cos m� cos m��

+ �22 sin m� sin m��

� ��*�k��

*�k��*�k��k�� , �52�

where � and �� refer to the angles of k and k� and n� andn� are the Fermi factors, and �1 and �2 refer to twocomponents of the � field.9

We will replace this term by its ensemble average,since deviations from the average are reduced an extrapower of 1/g and thus not significant as g→ �Murthyand Mathur, 2002�. As for the average, recall that toleading order in 1/g, averages of energy levels and wavefunctions factorize. Using

��*�k1��k2��

*�k3��k4�� =�k1k4

g

�k2k3

g+ O�1/g3�

�53�

for our case where k1=k2=k and k3=k4=k� and sum-ming over the g values of k and k� we find this average

9We need two components because n�k�n�k��cos m��−��=n�k�n�k��cos m� cos m��+sin m� sin m�� is a sum of twoterms and each needs to be factorized with its own Hubbard-Stratonovic field. We will refer to the two-component vector as�.



over wave functions goes as 1/g. �We are ignoring de-tails like a factor of 1/2 coming from the cos2 m� andsin2 m� in the momentum sums.�

As for the sum over energy denominators, considerone of the two similar possibilities, where n�=1 and n�

=0 so that �� ranges from −g� /2 and 0 while �� rangesfrom 0 to g� /2. Replacing the sum by an integral overdensity of states 1/� �valid for the ensemble average� wefind an integral of the form

1

�2�0

g�/2 �0

g�/2 dEdE�

E + E��

g

��54�

up to constants, so that

�QT� �g2

��2, �55�

where �2=�12+�2

2. If we consider higher powers of � weagain find that they go as g2

� �2n. Had we not scaled �initially, we would have found that each power of theunscaled � was accompanied by a different power of gand that no large g limit emerged.

It can be shown that had we included all um’s simulta-neously, they would not have interfered at the quadraticlevel. Since this is the term that controls symmetrybreaking, we get the correct picture taking just one um ata time.

I note that since we have a large N theory here, itfollows as in all large N theories that the one-loop flowand the new fixed point at strong coupling are parts ofthe final theory. However, the exact location of the criti-cal point cannot be predicted, as pointed out to us byBrower �2004�. The reason is that the Landau couplingsum are defined at a scale EL much higher than ET �butmuch smaller than EF� and their flow till we lower inenergy to ET, where our analysis begins, is not withinthe regime we can control. In other words, we can locateu* in terms of what couplings we begin with at ET, butthese are the Landau parameters renormalized in a non-universal way as we decrease from EL to ET.

What is the nature of the state for um�um* ? In the

strong-coupling region, � acquires an expectation valuein the ground state. The dynamics of the fermions isaffected by this variable in many ways: quasiparticlewidths become broad very quickly above the Fermi en-ergy, � �spacings in Vg between successive peaks� hasoccasionally very large values and can even benegative,10 and the system behaves like one with brokentime-reversal symmetry if m is odd �Murthy et al., 2004�.One example of the latter is as follows. Suppose we turnon a weak magnetic field B. All quantities—peak posi-

tions, spacing—will vary linearly in B and not quadrati-cally as in a time-reversal invariant system.11

Long ago Pomeranchuk �1958� found that if a Landauparameter um of a pure system exceeded a certain value,the Fermi surface underwent a shape transformationfrom a circle to a nonrotationally invariant form. Re-cently this transition has received a lot of attention�Varma, 1999; Oganesyan, Kivelson, and Fradkin, 2001�The transition in question is a disordered version of thesame. Details are given in Murthy and Shankar �2003�and Murthy et al. �2004�.

Details aside, there is another very interesting point:even if the coupling does not take us over to the strong-coupling phase, we can see vestiges of the critical pointum

* and associated critical phenomena. This is a generalfeature of many quantum critical points �Chakravarty etal., 1988; Sachdev, 1999�, i.e., points like um

* , where as avariable in a Hamiltonian is changed, the ground state ofthe system undergoes a phase transition �in contrast totransitions wherein temperature T is the control param-eter�.

Figure 8 shows what happens in a generic situation.On the x axis is a variable �um in our case� along whichthe quantum phase transition occurs. Along y is mea-sured a new variable, usually temperature T. Let us con-sider that case first. If we move from right to left at somevalue of T, we will first encounter physics of the weak-coupling phase determined by the weak-coupling fixedpoint at the origin. Then we cross into the critical fan�delineated by the V-shaped dotted lines�, where thephysics is controlled by the quantum critical point. Inother words, we can tell there is a critical point on the xaxis without actually traversing it. As we move furtherto the left, we reach the strongly coupled symmetry-broken phase, with a nonzero order parameter.

In our problem, 1/g2 plays the role of T since g2

stands in front of the effective action for �. �Here g alsoenters at a subdominant level inside the action, whichmakes it hard to predict the exact shape of the critical10How can the cost of adding one particle be negative �after

removing the charging energy�? The answer is that adding anew particle sometimes lowers the energy of the collectivevariable which has a life of its own. However, if we ignore itand attribute all the energy to the single-particle excitations, �can be negative.

11This result breaks down at exponentially small B, a regionwe ignore since the temperature in any experiment will bemuch higher.

1/g

uu*

weak coupling

quantum critical

bulk pomeranchuk

FIG. 8. The phase diagram in the u− 1g plane. The actual phase

transition takes place on the line 1/g=0. However, it can beperceived at finite g if one is within the V-shaped quantumcritical regime. In typical cases, the 1/g axis is replaced by theT axis, where T is the temperature. There a phase transition ofquantum-mechanical origin at T=0 influences the quantumcritical regime at finite T.



fan.� The bottom line is that we can see the critical pointat finite 1/g. In addition one can also raise the actualtemperature T or bias voltage to see the critical fan.

Subsequent work has shown, in more familiar ex-amples than Landau interactions, that the general pic-ture depicted here is true in the large g limit: upon add-ing sufficiently strong interactions the universalHamiltonian gives way to other descriptions with brokensymmetry �Murthy, 2004�.

It was mentioned earlier that the critical coupling u*

�a nonuniversal quantity� cannot be reliably predicted inthe large g limit. It has become clear from numericalwork �Adam, Brouwer, and Sharma, 2003� that the criti-cal doping coincides with the bulk coupling for the Po-meranchuk transition. In other words, when we crossover to the left u*, the size of the order parameter rap-idly grows from the mesoscopic scale of order ET toorder of the Fermi energy EF. However, the physics inthe critical fan as well as the weak-coupling side is asdescribed by the RMT+RG analysis. The strong-coupling side has to be reworked since the Fermi surfaceassumed in the RG that decreased to ET has sufferedlarge deformations �on the scale of EF�.

VI. SUMMARY AND CONCLUSIONS

Our goal was to understand the transport propertiesof a quantum dot—an island that electrons could tunnelon to and tunnel out of. As one varies the gate voltageVg between the leads and the dot, the conductance Gexhibits isolated peaks of varying location and height.What we wanted was a statistical description of thesefeatures as exhibited by an ensemble of similar dots. Thedot was assumed to be so irregular and the classical mo-tion so chaotic, that the only conserved quantity wasenergy.

To begin with, we asked how we would go about ad-dressing the problem if electron-electron interactionscould be neglected. It was seen that peak heights andpositions could be determined from the wave functionsand energy levels. These in turn could be determined byRMT given just the mean level spacing. In the noninter-acting theory, P��, the distribution of differences in Vgbetween successive peaks, was the same as P��, the dis-tribution of spacings between successive levels in thedot.

Actual comparison of these two distributions showedclear disagreement, which represented a failure, not ofRMT, but of assuming noninteracting electrons. As aresult interactions had to be included.

The Coulomb interaction implied that adding an elec-tron to the dot would cost not just the gap to the nextempty level, but an additional energy due to repulsionby the N electrons already in the dot. This charging en-ergy could be accounted for by adding a term u0N2 tothe second quantized Hamiltonian. In the presence ofspin, we also needed to add a term −J0S2 to representthe exchange interaction. The final result was HU, theuniversal Hamiltonian. While the success of this model

is unquestioned at moderate values of interactionstrength, some arguments for why this had to be theright answer, and why other interactions could be ne-glected because their ensemble averages vanished, werenot persuasive. We asked if there was there a better wayto understand the success of HU.

It was pointed out that the RG was such a way since itoffered an unbiased procedure for determining whichinteractions were really important in deciding low-energy properties like the ground state and its low-energy excitations. In the RG approach one divided thevariables into two sets: x, which we cared about, and y,which we did not care about. In our example x was thelow-energy region �near the Fermi energy� and y every-thing else. One then eliminated or integrated out the y’sto obtain an effective theory of just x, which gave thesame answers in the x domain as the original one. In thisprocess some initial couplings could fade into oblivion�irrelevant� while small ones could grow in size �rel-evant� and some could remain fixed �marginal�. In anyevent, we showed which couplings really mattered. Weshowed that while the RG concept itself was nonpertur-bative, mode elimination was typically done perturba-tively in the interaction.

The application of RG to our problem requires a two-stage process as developed by Murthy and Mathur. Firstone ignores disorder and finite size and eliminates high-energy modes outside the Landau band, a region ofwidth EL measured from the Fermi surface. Here weknow from past RG work on clean bulk systems that wemust invariably end up with the Landau interaction u��.But we are not done yet. Although the Landau scale ELis much smaller than EF, the Fermi energy, it does notvanish for infinite system size. So ET, which vanishes as1/L, lies even closer to EF. So we need to renormalizedown from EL to ET. During this process the Landauparameters could renormalize in a way we cannot deter-mine. �Though in a clean system um are strictly marginal,once we approach ET and take disorder seriously, a non-zero flow is guaranteed.� So one begins with ET by writ-ing the �renormalized� Landau interaction in the disor-dered single-particle basis � and eliminating the � statesto determine the fate of the Landau interactions. It wasfound that only u0 and J0, the zeroth harmonics on theFermi circle of the Landau interactions �for charge andspin densities�, survived at the lowest energies if thestarting value of um was either positive or not below anegative coupling um

* , of order unity. Thus the low-energy fixed point for this range of initial coupling wasjust HU, the universal Hamiltonian. In addition to pro-viding this justification of the emergence of HU, the cal-culation also suggested that for um um

* the system un-derwent a phase transition in the g→ limit. Beforediscussing the phase transition, let us recall how the cal-culation of the � function was done.

In any interacting field theory with some coupling u,one computes a physical quantity like a scattering am-plitude in a power series in u, starting with u itself, fol-lowed by loop diagrams of increasing complexity. These



loops involve momentum or energy sums �or integrals�up to some cutoff �. If the sum of all these terms has tobe independent of � �as is the physical scattering ampli-tude� then u itself must become u�� and vary with � insuch a way as to keep the series as a whole � indepen-dent. Conversely it is possible to determine u�� bydrawing diagrams to some order in u making this de-mand. In our problem, the computation of ��u�=−�du /d� for any one specific dot required knowledgeof the wave functions and energy levels lying within ET.But due to self-averaging, one replaces the � functionfor the given dot by its ensemble average. The zeros ofthis � function are what showed HU to be a fixed point�at the origin� and um

* to be the critical point for thephase transition.

This calculation was nonetheless a weak-couplinganalysis predicting a phase transition at strong coupling.Was the transition real and if so, what was on the strong-coupling side? The answer is that the 1/N techniquecame in. In the large g limit one showed that all physicscould be extracted in a saddle point calculation of the1/N type for any coupling um. In contrast to theorieswhere there are N equivalent species, here we had gfermions with different energies and matrix elements.However, due to disorder self-averaging, one pulls out ag2 in front of the action for the Hubbard-Stratonovicfield �. The saddle point theory confirmed the fixedpoint nature of HU for um

* um and the transition atum

* . Furthermore, we see beyond the transition to theother side: here � acquired an average �a disorderedversion of the Pomeranchuk transition in clean systems�and produced many attendant consequences like time-reversal breaking. However, the “exact” critical pointum=um

* of this calculation was not a directly measurablequantity since the saddle point theory had as its inputnot the Landau interaction defined at EL, but what ithad evolved into, between EL and ET. The region be-tween EL and ET is where disorder was too strong to beignored, but not strong enough to use RMT since wewere not within ET making the flow intractable. How-ever, arguments from Adam et al. showed that the tran-sition occurs at the bulk critical value of um=1.

Since the phase transition occurs only at infinite g �afinite system always has finite g and cannot have a tran-sition�, it might seem that our study of it was academic.This is not so, and the reason is the same as in quantumcritical phenomena. Recall that there, a quantum phasetransition at T=0 as a function of some coupling u=u*

can be felt even at T�0 inside a V-shaped region calledthe quantum critical region. Here g2 plays the role of1/T as a prefactor in the action. Thus we can see theeffects of the critical point even at finite g and over awide range of coupling.

I began with the remark that three obstacles are in-volved here: randomness, strong interactions, and finitesize. Yet these obstacles ended up being benign: ran-domness and finite size led to a finite Thouless bandwithin which we used RMT and ply our trade, while

strong interactions led to an interesting phase transition.As for the RG, it had to be invoked in the first stage aswe came down to EL using the clean system RG to endup with the Landau interactions. At this point we couldeither use perturbative self-averaged RG inside ET or,better still, use the self-averaged 1/N method to solvethe model by saddle point.

This Colloquium has emphasized the following prob-lem: the confluence of physical complications and theinterplay of diverse techniques that lead to a solution.Of necessity it has been sparse on phenomenology.However, armed with the ideas explained here one isready to remedy this, following any number of the ref-erences mentioned in the text.

It will be very interesting if experimentalists un-earthed the phenomena described here by studying dotswith strongly interacting electrons, a possibility morereadily realized than in the bulk since electron density indots can be controlled by gates.

ACKNOWLEDGMENTS

I am grateful to the National Science Foundation forGrant No. DMR-0354517 that made this research pos-sible. I thank my constant collaborator Ganpathy Mur-thy for so many shared insights including those pertain-ing to this colloquium.

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