the ising model of a ferromagnet from 1920 to 2020 stanislav smirnov

The Ising model of a ferromagnet from 1920 to 2020 Stanislav Smirnov

Statistical physics (of lattice models of natural phenomena) from 1D to 2D to 3D from 1920 to 2020 Stanislav Smirnov

statistical physics: macroscopic effects of microscopic interactions erosion simulation J.-F. Colonna, Ecole Polytechnique DNA by atomic force microscopy Lawrence Livermore National Laboratory

2D statistical physics: macroscopic effects of microscopic interactions erosion simulation J.-F. Colonna DNA by atomic force microscopy Lawrence Livermore National Laboratory

Mathematical models of natural phenomena where simple microscopic interactions add up to exhibit complicated macroscopic effects Such nature was proposed heuristically for several physical phenomena by philosophers (Leucippus and Democritus 4-5 century BC) First scientific application to Brownian motion and heat transfer (Einstein 1905) We will discuss applications to magnetism statistical physics: macroscopic effects of microscopic interactions

Natural magnetism in lodestone (magnetite) known for millennia 6th century BC: first scientific discussions by Thales of Miletus and Shushruta of Varansi. Magnetism Name derived from a Greek province rich in iron ore Several practical uses, but poor understanding of its nature

1820 Hans Christian rsted discovers by accident that electric current induces magnetic field 1820-30 Andr-Marie Ampre; Carl Friedrich Gauss; Jean-Baptiste Biot & Flix Savart a formula 1831 Michael Faraday varying magnetic flux induces an electric current 1855-1873 James Clerk Maxwell synthesizes theory of electricity, magnetism and light, writes down Maxwell's equations Relation with electricity

1895 Pierre Curie in his doctoral thesis studies types of magnetism, discovers the effect of temperature a phase transition at the Curie point The phenomenon occurs at the atomic scale Ferromagnetism

1920 Wilhelm Lenz introduces a lattice model for ferromagnetism Lenz argued that atoms are dipoles which turn over between two positions: their free rotatability was incompatible with Borns theory of crystal structure; but they can perform turnovers as suggested by experiments on ferromagnetic materials; in a quantum-theoretical treatment they would by and large occupy two distinct positions.

There is a good reason why no explanation for magnetism was found before 20 th century: the Bohrvan Leeuwen theorem shows that magnetism cannot occur in purely classical solids The Lenz argument is flawed, but the model is basically correct the property of ferromagnetism is due to two quantum effects: the spin of an electron (hence it has a magnetic dipole moment) the Pauli exclusion principle (nearby electrons tend to have parallel spins) Quantum mechanics

who proposed a specific form of interaction Squares of two colors, representing spins s=1 Nearby spins want to be the same, parameter x : W(config)=x #{+-neighbors} Partition function Z = config W (config) Probability P(cfg) =W(cfg)/Z 1920-24: The Lenz-Ising model Lenz suggested the model to his student Ernst Ising,

who proposed a specific form of interaction Often written : W(config)=x #{+-neighbors} exp(- neighbors s(u)s(v)) Here spins s(u)=1 and x= exp(2 ) In magnetic field multiply by exp(- u s(u)) 1920-24: The Lenz-Ising model Lenz suggested the model to his student Ernst Ising,

1924: Ernst Ising thesis no phase transition in dimension 1 0 1................ n n+1 + + + + + + + + ?

Ising wrongly concluded that there is no phase transition in all dimensions. and Never returned to research, teaching physics at a college The paper was widely discussed (Pauli, Heisenberg, Dirac, ) with consensus that it is oversimplified Heisenberg introduced an XY model after writing Other difficulties are discussed in detail by Lenz, and Ising succeeded in showing that also the assumption of aligning sufficiently great forces between each of two neighboring atoms of a chain is not sufficient to create ferromagnetism

Unexpectedly proves that in dimension 2 the Lenz-Ising model undergoes a phase transition, which reignites the interest 1936 Rudolf Peierls His perhaps more influential work: 1940 memorandum with Otto Frisch often credited with starting the Manhattan project

Domain (2n+1)(2n+1), boundary +. Z = (1+x) #edges ? How to estmate P [()=-]? Does the Ising method work? Peierls argument

Domain (2n+1)(2n+1), boundary +. Z = (1+x) 8nn-4n ? No! After expanding many extra terms, not corresponding to spin configurations. How to estmate P [()=-]? Does the Ising method work? Peierls argument

Compare Z - and Z + term-wise Suppose the center spin is -, i.e. blue It is surrounded by at least one domain wall Change the colors inside the smallest loop, length l 4 Compare weights: W(left) = x l W(right) Not enough, but we get P(left)< x l Peierls argument

We have get P(left)< x l with l the length of the loop. The number of such loops is < l/2 3 l : take a point in ]0,l/2[ of the 0x axis and make l steps left/right/straight Conclusion: regardless of the box size, P[ (0)=] even l=4 l/2 3 l x l = (3x ) 4 / (1-(3x) 2 ) 2 1/9, if x 1/6. Peierls argument

Peirls argument: with + boundary conditions P[ (0)=] 1/9, when x 1/6. So magnetization at low temperatures, regardless of the box size! For x = 1 however P[ (0)=] = 1/2 Ising model: the phase transition

Dobrushin BC: boundary is red below, blue above. x x crit Prob x #{+-neighbors} Ising model: the phase transition

1941 Kramers-Wannier Derive the critical temperature

1941 Kramers-Wannier +++++ +++ +++ +++++ ++ ++ ++ +++++ +++ +++ +++++

1944 Lars Onsager A series of papers 1944-1950, some with Bruria Kaufman. Partition function and other quantities derived. It took a chemist!

2D Ising is exactly solvable From 1944 widely studied in mathematical physics, with many results by different methods: Kaufman, Onsager, Yang, Kac, Ward, Potts, Montroll, Hurst, Green, Kasteleyn, McCoy, Wu, Tracy, Widom, Vdovichenko, Fisher, Baxter, Only some results rigorous Limited applicability to other models, but still motivated much research Eventually regained prominence in physics, used in biology, economics, computer science

1951: Renormalization Group Petermann-Stueckelberg 1951, Kadanoff, Fisher, Wilson, 1963-1966, Block-spin renormalization rescaling + change of x Conclusion: At criticality the scaling limit is described by a massless field theory The Curie critical point is universal and hence translation, scale and rotation invariant

Renormalization Group A depiction of the space of Hamiltonians H showing initial or physical manifolds and the flows induced by repeated application of a discrete RG transformation Rb with a spatial rescaling factor b (or induced by a corresponding continuous or differential RG). Critical trajectories are shown bold: they all terminate, in the region of H shown here, at a fixed point H*. The full space contains, in general, other nontrivial (and trivial) critical fixed points, From [Michael Fisher,1983]

0 < x crit < x* crit three fixed points on [0,[ Renormalization in 2D Ising

1985: Conformal Field Theory Conformal transformations = those preserving angles = analytic maps Locally translation + + rotation + rescaling CFT [Belavin, Polyakov, Zamolodchikov 1985] : In the scaling limit, postulate conformal invariance Resulting Infinite symmetries allow to derive many quantities (unrigorously)

well-known example: 2D Brownian Motion is the scaling limit of the Random Walk Paul Lvy,1948: BM is conformally invariant The trajectory is preserved (up to speed change) by conformal maps. Not so in 3D!!! Conformal invariance

Beautiful algebra, but analytic and geometric parts missing or nonrigorous Spectacular predictions, e.g. by Den Nijs and Cardy: 2D CFT Percolation (Ising at infinite T or x=1): hexagons are coloured white or yellow independently with probability . Is there a top-bottom crossing of white hexagons?

Beautiful algebra, but analytic and geometric parts missing or nonrigorous Spectacular predictions, e.g. by Den Nijs and Cardy: 2D CFT Percolation (Ising at infinite T or x=1): hexagons are coloured white or yellow independently with probability . Is there a top-bottom crossing of white hexagons? Difficult to see! Why? HDim (percolation cluster)= ?

Beautiful algebra, but analytic and geometric parts missing or nonrigorous Spectacular predictions, e.g. by Den Nijs and Cardy: 2D CFT Percolation (Ising at infinite T or x=1): hexagons are coloured white or yellow independently with probability . Connected white cluster touching the upper side is coloured in blue, it has HDim (percolation cluster)= 91/48

Last decade Two analytic and geometric approaches 1)Schramm-Loewner Evolution: a geometric description of the scaling limits at criticality SLE() random curve 2)Discrete analyticity: a way to rigorously establish existence and conformal invariance of the scaling limit New physical approaches and results Rigorous proofs Cross-fertilization with CFT

Time for a break!

Lawler, Schramm, Werner; Smirnov Schramms SLE(8/3) coincides with the boundary of the 2D Brownian motion the percolation cluster boundary (conjecturally) the self-avoiding walk ? Universality in lattice models

Last decade Two analytic and geometric approaches 1)Schramm-Loewner Evolution: a geometric description of the scaling limits at criticality SLE() random curve 2)Discrete analyticity: a way to rigorously establish existence and conformal invariance of the scaling limit New physical approaches and results Rigorous proofs Cross-fertilization with CFT

Schramm-Loewner Evolution A way to construct random conformally invariant fractal curves, introduced in 1999 by Oded Schramm (1961-2008) PercolationSLE(6) Uniform Spanning Tree SLE(8) [Smirnov, 2001] [Lawler-Schramm-Werner, 2001]

from Oded Schramms talk 1999

A tool to study variation of conformal maps and domains, introduced to attack Bieberbachs conjecture K. Lwner (1923), "Untersuchungen ber schlichte konforme Abbildungen des Einheitskreises. I", Math. Ann. 89 Instrumental in the eventual proof L. de Branges (1985), "A proof of the Bieberbach conjecture", Acta Mathematica 154 (1): 137152 Thm Let F(z)= a n z n be a map of unit disk into the plane. Then |a n | n, equality for a slit map. Wide open: find s.t. |a n | n for bounded maps. Loewner Evolution

Draw the slit Schramm-Loewner Evolution

Draw the slit Stop at capacity increments Schramm-Loewner Evolution

SLE=BM on the moduli space. Calculations reduce to It calculus, interesting fractal properties Lemma [Schramm] If an interface has a conformally invariant scaling limit, it is SLE() Schramm-Loewner Evolution

Relation to lattice models

New approach to 2D integrable models Find an observable F (edge density, spin correlation, exit probability,... ) which is discrete analytic (holomorphic) or harmonic and solves some BVP. Then in the scaling limit F converges to a holomorphic solution f of the same BVP. We conclude that F has a conformally invariant scaling limit Interfaces converge to Schramms SLEs Calculate dimensions and exponents with or without SLE Discrete holomorphic functions

The simplest tiling of a square by squares of different sizes, discovered by Brooks, Smith, Stone and Tutte

Discrete holomorphic functions

Defines angles but not distances on a planar graph Can be constructed from discrete holomorphicity Example: circle packings (non-linear relations) Discrete conformal geometry

Defines angles but not distances on a planar graph Can be constructed from discrete holomorphicity Example: square tilings (linear relations) Discrete conformal geometry

Several models were approached in this way: Random Walk [Courant, Friedrich & Lewy, 1928; .] Dimer model, UST [Kenyon, 1997-...] Critical percolation [Smirnov, 2001] Uniform Spanning Tree [Lawler, Schramm & Werner, 2003] Random cluster model with q = 2 and Ising model at criticality [Smirnov; Chelkak & Smirnov 2006-2010] Most observables are CFT correlations! Connection to SLE gives dimensions! Discrete holomorphic functions

Consider a domain with 4 points on the boundary. Allow point z to move inside the domain. Let H a (z) be the probability of a crossing from ac to ab separating z from bc. Three functions H a (z), H b (z), H c (z) are approximately discrete harmonic conjugates. H a (z) vanishes on ac, also H a (b)=1 at bc. Unique solution, linear on equilateral triangle. Mapped to halfplane get Cardys formula: Percolation crossings b z c a

[Chelkak, Smirnov 2008-10] Partition function of the critical Ising model with a disorder operator is discrete holomorphic solution of the Riemann-Hilbert boundary value problem. Interface weakly converges to Schramms SLE(3) curve. Strong convergence follows with some work [Chelkak, Duminil-Copin, Hongler, Kemppainen, S] Preholomorphicity in Ising HDim = 11/8

Energy field in the Ising model Combination of 2 disorder operators is a discrete analytic Greens function solving a Riemann-Hilbert BVP, when fused gives the energy corellation: Theorem [Hongler Smirnov, 2013] At x c the correlation of neighboring spins satisfies ( is the lattice mesh; is the hyperbolic metric element; the sign depends on BC: + or free): Generalizations to multi spin and energy corellations: [Chelkak, Hongler, Izyurov]

Proposed as a model for a long unexplained phenomenon Deemed physically inaccurate and mathematically trivial Breakthrough by Onsager leads to much theoretical study Eventually retook its place in physics, biology, computer science Much fascinating mathematics, expect more The Lenz-Ising model

To do list : [Zamolodchikov, JETP 1987]: E8 symmetry in 2D Ising. [Coldea et al., Science 2011]: experimental evidence. A proof? Prove supercritical Ising = percolation (renormalization) Study random planr graphs weighted by Ising (= Quantum gravity) [Aizenman Duminil-Copin Sidoravicius 2013] In 3D no magnetization at criticality. Other results? The Lenz-Ising model

Thank you for your attention!

Lawler, Schramm, Werner; Smirnov Schramms SLE(8/3) coincides with the boundary of the 2D Brownian motion the percolation cluster boundary (conjecturally) the self-avoiding walk ? Other lattice models

Paul Flory, 1948: Proposed to model a polymer molecule by a self-avoiding walk (= random walk without self-intersections) How many length n walks? What is a typical walk? What is its fractal dimension? Flory: a fractal of dimension 4/3 The argument is wrong The answer is correct! Physical explanation by Nienhuis, later by Lawler, Schramm, Werner. Self-avoiding polymers

What is the number C(n) of length n walks? Nienhuis predictions: C(n) n n 11/32 11/32 is universal On hex lattice = Self-avoiding polymers Theorem [Duminil-Copin & Smirnov, 2010] On hexagonal lattice = x c -1 = Idea: for x=x c, = c discrete analyticity of F(z) = self-avoiding walks 0 z # turns x length

Thank you for your attention!

Thank you for your attention! DLA = Diffusion Limited Aggregation Particle repeatedly start Brownian motion from infinity until they stick to the aggregate. Is it a fractal?

Miermont, Le Gall 2011: Uniform random planar graph (taken as a metric space) has a universal scaling limit (a random metric space, topologically a plane) Duplantier-Sheffield, Sheffield, 2010: Proposed relation to SLE and Liouville Quantum Gravity (a random metric exp(G)|dz|) Quantum gravity from [Ambjorn-Barkeley-Budd]

Goals for next N years Prove conformal invariance for other models, establish universality Build rigorous renormalization theory Develop rigorous theory of 2D quantum gravity = random models on random planr graphs

the ising model of a ferromagnet from 1920 to 2020 stanislav smirnov

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