The Ising model of a ferromagnet from 1920 to 2020 Stanislav
Smirnov
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Statistical physics (of lattice models of natural phenomena)
from 1D to 2D to 3D from 1920 to 2020 Stanislav Smirnov
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statistical physics: macroscopic effects of microscopic
interactions erosion simulation J.-F. Colonna, Ecole Polytechnique
DNA by atomic force microscopy Lawrence Livermore National
Laboratory
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2D statistical physics: macroscopic effects of microscopic
interactions erosion simulation J.-F. Colonna DNA by atomic force
microscopy Lawrence Livermore National Laboratory
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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
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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
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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
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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
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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.
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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
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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,
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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,
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1924: Ernst Ising thesis no phase transition in dimension 1 0
1................ n n+1 + + + + + + + + ?
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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
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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
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Domain (2n+1)(2n+1), boundary +. Z = (1+x) #edges ? How to
estmate P [()=-]? Does the Ising method work? Peierls argument
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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
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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
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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
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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
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Dobrushin BC: boundary is red below, blue above. x x crit Prob
x #{+-neighbors} Ising model: the phase transition
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1941 Kramers-Wannier Derive the critical temperature
1944 Lars Onsager A series of papers 1944-1950, some with
Bruria Kaufman. Partition function and other quantities derived. It
took a chemist!
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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
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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
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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]
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0 < x crit < x* crit three fixed points on [0,[
Renormalization in 2D Ising
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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)
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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
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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?
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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)= ?
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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
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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
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Time for a break!
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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
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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
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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]
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from Oded Schramms talk 1999
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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
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Draw the slit Schramm-Loewner Evolution
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Draw the slit Stop at capacity increments Schramm-Loewner
Evolution
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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
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Relation to lattice models
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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
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The simplest tiling of a square by squares of different sizes,
discovered by Brooks, Smith, Stone and Tutte
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Discrete holomorphic functions
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Defines angles but not distances on a planar graph Can be
constructed from discrete holomorphicity Example: circle packings
(non-linear relations) Discrete conformal geometry
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Defines angles but not distances on a planar graph Can be
constructed from discrete holomorphicity Example: square tilings
(linear relations) Discrete conformal geometry
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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
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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
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[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
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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]
Slide 65
2D statistical physics: macroscopic effects of microscopic
interactions erosion simulation J.-F. Colonna DNA by atomic force
microscopy Lawrence Livermore National Laboratory
Slide 66
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
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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
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Thank you for your attention!
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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
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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
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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
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Thank you for your attention!
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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?
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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]
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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
Slide 76
2D statistical physics: macroscopic effects of microscopic
interactions erosion simulation J.-F. Colonna DNA by atomic force
microscopy Lawrence Livermore National Laboratory