Ferromagnets and the classical Heisenberg model. Kay Kirkpatrick, UIUC

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1 Ferromagnets and the classical Heisenberg model Kay Kirkpatrick, UIUC

2 Ferromagnets and the classical Heisenberg model: asymptotics for a mean-field phase transition Kay Kirkpatrick, Urbana-Champaign June 11, 2012

3 Ferromagnets and the classical Heisenberg model: asymptotics for a mean-field phase transition Kay Kirkpatrick, Urbana-Champaign June 11, 2012 Joint with Elizabeth Meckes (Case Western Reserve University).

4 The classical physics models of ferromagnets Simplest: Ising model on a periodic lattice of n sites has Hamiltonian energy for spin configuration σ { 1, +1} n H(σ) = J n σ i σ i+1 i=1

5 The classical physics models of ferromagnets Simplest: Ising model on a periodic lattice of n sites has Hamiltonian energy for spin configuration σ { 1, +1} n H(σ) = J n σ i σ i+1 Ising s 1925 solution in 1D. Onsager s 1944 solution in 2D. i=1

6 Main goals for spin models Gibbs measure Partition function 1 Z(β) e βh(σ). Z(β) = σ e βh(σ).

7 Main goals for spin models Gibbs measure Partition function 1 Z(β) e βh(σ). Z(β) = σ e βh(σ). May see a phase transition via the free energy 1 ϕ(β) = lim log Z(β). n βn

8 Main goals for spin models Gibbs measure Partition function 1 Z(β) e βh(σ). Z(β) = σ e βh(σ). May see a phase transition via the free energy 1 ϕ(β) = lim log Z(β). n βn Fruitful approach: Mean-field spin models.

9 Work on the mean-field Ising model = Curie-Weiss model Motivation: Curie-Weiss model is believed to approximate high-dimensional Ising model (d 4).

10 Work on the mean-field Ising model = Curie-Weiss model Motivation: Curie-Weiss model is believed to approximate high-dimensional Ising model (d 4). Ellis and Newman 78: Magnetization M n (σ) = 1 σ i n i of CW model has Gaussian law away from criticality, and at the critical temperature non-gaussian law like e x4 /12.

11 Work on the mean-field Ising model = Curie-Weiss model Motivation: Curie-Weiss model is believed to approximate high-dimensional Ising model (d 4). Ellis and Newman 78: Magnetization M n (σ) = 1 σ i n i of CW model has Gaussian law away from criticality, and at the critical temperature non-gaussian law like e x4 /12. Eichelsbacher and Martschink 10, Chatterjee and Shao 11: Rate of convergence and Berry-Esseen type error bound for magnetization at critical temperature.

12 Work on the mean-field Ising model = Curie-Weiss model Motivation: Curie-Weiss model is believed to approximate high-dimensional Ising model (d 4). Ellis and Newman 78: Magnetization M n (σ) = 1 σ i n i of CW model has Gaussian law away from criticality, and at the critical temperature non-gaussian law like e x4 /12. Eichelsbacher and Martschink 10, Chatterjee and Shao 11: Rate of convergence and Berry-Esseen type error bound for magnetization at critical temperature. Also Curie-Weiss-Potts model with finitely many discrete spins.

13 The classical Heisenberg model of ferromagnetism Spins are now in the sphere, and for spin configuration σ (S 2 ) n the Hamiltonian energy is: H(σ) = i,j J i,j σ i, σ j.

14 The classical Heisenberg model of ferromagnetism Spins are now in the sphere, and for spin configuration σ (S 2 ) n the Hamiltonian energy is: H(σ) = i,j J i,j σ i, σ j. Like Ising and Curie-Weiss models, two main cases: Nearest-neighbor: J i,j = J for nearest neighbors i, j, J i,j = 0 otherwise. Most interesting and challenging (and open) in 3D.

15 The classical Heisenberg model of ferromagnetism Spins are now in the sphere, and for spin configuration σ (S 2 ) n the Hamiltonian energy is: H(σ) = i,j J i,j σ i, σ j. Like Ising and Curie-Weiss models, two main cases: Nearest-neighbor: J i,j = J for nearest neighbors i, j, J i,j = 0 otherwise. Most interesting and challenging (and open) in 3D. Mean-field: averaged interaction J i,j = 1 2n for all i, j. Can be viewed as either sending the dimension or the number of vertices in a complete graph to infinity. (Mean-field theory is the starting point for phase transitions.)

16 Results for the mean-field Heisenberg model Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Previous work and set-up of Gibbs measures e βhn.

17 Results for the mean-field Heisenberg model Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Previous work and set-up of Gibbs measures e βhn. LDPs for the magnetization and empirical spin distribution, for any inverse temperature β.

18 Results for the mean-field Heisenberg model Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Previous work and set-up of Gibbs measures e βhn. LDPs for the magnetization and empirical spin distribution, for any inverse temperature β. Free energy, macrostates, second-order phase transition.

19 Results for the mean-field Heisenberg model Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Previous work and set-up of Gibbs measures e βhn. LDPs for the magnetization and empirical spin distribution, for any inverse temperature β. Free energy, macrostates, second-order phase transition. CLTs for magnetization above and below critical temperature.

20 Results for the mean-field Heisenberg model Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Previous work and set-up of Gibbs measures e βhn. LDPs for the magnetization and empirical spin distribution, for any inverse temperature β. Free energy, macrostates, second-order phase transition. CLTs for magnetization above and below critical temperature. Nonnormal limit theorem for magnetization at critical temperature.

21 Previous work on high-dimensional Heisenberg models Nearest-neighbor (NN) Heisenberg model in d-dimensions: H(σ) = J i j =1 Magnetization: normalized sum of spins σ i, σ j Kesten-Schonmann 88: approximation of the d-dimensional NN model by the mean-field behavior as dimension d, with critical temperature β c = 3 Magnetization is zero for all β < 3 and all dimensions d. Magnetization converges to the mean-field magnetization as d for β > 3.

22 The set-up and Gibbs measure Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Prob. measure P n the product of the uniforms on (S 2 ) n.

23 The set-up and Gibbs measure Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Prob. measure P n the product of the uniforms on (S 2 ) n. Gibbs measure P n,β, or canonical ensemble, has density: 1 Z exp β n σ i, σ j = 1 2n Z e βhn(σ). i,j=1

24 The set-up and Gibbs measure Classical Heisenberg model on the complete graph with n vertices: H n (σ) = 1 2n n σ i, σ j i,j=1 Prob. measure P n the product of the uniforms on (S 2 ) n. Gibbs measure P n,β, or canonical ensemble, has density: 1 Z exp β n σ i, σ j = 1 2n Z e βhn(σ). i,j=1 Partition function: Z = Z n (β) = (S 2 ) n e βhn(σ) dp n.

25 The Cramèr-type LDP at β = 0 (i.i.d. case) Empirical magnetization: M n (σ) = 1 n n i=1 σ i.

26 The Cramèr-type LDP at β = 0 (i.i.d. case) Empirical magnetization: M n (σ) = 1 n n i=1 σ i. Theorem (K.-Meckes 12): For i.i.d. uniform random points {σ i } n i=1 on S2 R 3, the magnetization laws µ n := L (M n ) satisfy an LDP (large deviations principle): P n (M n x) e ni (x),

27 The Cramèr-type LDP at β = 0 (i.i.d. case) Empirical magnetization: M n (σ) = 1 n n i=1 σ i. Theorem (K.-Meckes 12): For i.i.d. uniform random points {σ i } n i=1 on S2 R 3, the magnetization laws µ n := L (M n ) satisfy an LDP (large deviations principle): P n (M n x) e ni (x), I (c) = cg(c) log ( ) sinh(c) c, g(c) = coth(c) 1 c = x.

28 The Sanov-type LDP at β = 0 Empirical measure of spins: n µ n,σ = 1 n i=1 δ σi

29 The Sanov-type LDP at β = 0 Empirical measure of spins: n µ n,σ = 1 n i=1 δ σi Theorem (K.-Meckes 12): P n {µ n,σ B} exp{ n inf ν B H(ν µ)} where { S f log(f )dµ, f := dν H(ν µ) := 2 dµ exists;, otherwise. Here µ is uniform measure and B is a Borel subset of M 1 (S 2 ).

30 The Sanov-type LDP at β = 0 Empirical measure of spins: n µ n,σ = 1 n i=1 δ σi Theorem (K.-Meckes 12): P n {µ n,σ B} exp{ n inf ν B H(ν µ)} where { S f log(f )dµ, f := dν H(ν µ) := 2 dµ exists;, otherwise. Here µ is uniform measure and B is a Borel subset of M 1 (S 2 ). Now, how do the LDPs depend on temperature?

31 LDPs for arbitrary temperature Equivalence of ensembles approach (Ellis-Haven-Turkington 00): find hidden space, hidden process, interaction representation.

32 LDPs for arbitrary temperature Equivalence of ensembles approach (Ellis-Haven-Turkington 00): find hidden space, hidden process, interaction representation. Theorem (K.-Meckes 12): : LDP with respect to the Gibbs measures: P n,β {µ n,σ S} exp{ n inf I β(ν)}, ν S where I β (ν) = H(ν µ) β 2 2 xdν(x) ϕ(β), S 2 and free energy 1 ϕ(β) := lim n n log Z n(β) = inf ν [ H(ν µ) β 2 xdν(x) S 2 2 ]

33 Optimizing for the free energy Densities symmetric about the north pole maximize S 2 xdν(x).

34 Optimizing for the free energy Densities symmetric about the north pole maximize S 2 xdν(x). Consider measure ν g with density f (x, y, z) = g(z) increasing in z: = H(ν g µ) β 2 xdν g (x) S 2 ( 1 g(x) log[g(x)]dx β 1 2 ( g ) = h + log(2) β ( for increasing g : [ 1, 1] R + with = ) 2 xg(x) dx 2 xg(x) dx 2 ) g(x)dx = 1.

35 Optimizing for the free energy Densities symmetric about the north pole maximize S 2 xdν(x). Consider measure ν g with density f (x, y, z) = g(z) increasing in z: = H(ν g µ) β 2 xdν g (x) S 2 ( 1 g(x) log[g(x)]dx β 1 2 ( g ) = h + log(2) β ( for increasing g : [ 1, 1] R + with = ) 2 xg(x) dx 2 xg(x) dx 2 ) g(x)dx = 1. Usual entropy h, so constrained entropy optimization...

36 Free energy and phase transition... gives optimizing densities g(z) = ce kz, and the free energy ϕ(β) = inf k [0, ) Φ β (k), where ( ) k Φ β (k) := log + k coth k 1 β ( coth k 1 ) 2 sinh k 2 k

37 Free energy and phase transition... gives optimizing densities g(z) = ce kz, and the free energy ϕ(β) = inf k [0, ) Φ β (k), where ( ) k Φ β (k) := log + k coth k 1 β ( coth k 1 ) 2 sinh k 2 k Calculus: optimized at 0 for β β c := 3, and is given implicitly for k β > 3 by γ(k) = = β: coth k 1 k

38 Free energy and phase transition... gives optimizing densities g(z) = ce kz, and the free energy ϕ(β) = inf k [0, ) Φ β (k), where ( ) k Φ β (k) := log + k coth k 1 β ( coth k 1 ) 2 sinh k 2 k Calculus: optimized at 0 for β β c := 3, and is given implicitly for k β > 3 by γ(k) = = β: coth k 1 k

39 Free energy and phase transition... gives optimizing densities g(z) = ce kz, and the free energy ϕ(β) = inf k [0, ) Φ β (k), where ( ) k Φ β (k) := log + k coth k 1 β ( coth k 1 ) 2 sinh k 2 k Calculus: optimized at 0 for β β c := 3, and is given implicitly for k β > 3 by γ(k) = = β: coth k 1 k 2nd-order phase transition: ϕ and ϕ are continuous at β = 3.

40 The canonical macrostates across the phase transition First, note that β c = 3 matches Kesten-Schonmann.

41 The canonical macrostates across the phase transition First, note that β c = 3 matches Kesten-Schonmann. In the subcritical phase, β < 3, the canonical macrostates (zeroes of rate function I β ) are uniform on the sphere. Disordered.

42 The canonical macrostates across the phase transition First, note that β c = 3 matches Kesten-Schonmann. In the subcritical phase, β < 3, the canonical macrostates (zeroes of rate function I β ) are uniform on the sphere. Disordered. In the supercritical (ordered) phase, β > 3, the macrostates are rotations of the measure with density (x, y, z) ce kz, where c = k 2 sinh k, k = γ 1 (β).

43 The canonical macrostates across the phase transition First, note that β c = 3 matches Kesten-Schonmann. In the subcritical phase, β < 3, the canonical macrostates (zeroes of rate function I β ) are uniform on the sphere. Disordered. In the supercritical (ordered) phase, β > 3, the macrostates are rotations of the measure with density (x, y, z) ce kz, where c = k 2 sinh k, k = γ 1 (β). In particular, as β, these densities converge to delta functions (full alignment of the spins).

44 The canonical macrostates across the phase transition First, note that β c = 3 matches Kesten-Schonmann. In the subcritical phase, β < 3, the canonical macrostates (zeroes of rate function I β ) are uniform on the sphere. Disordered. In the supercritical (ordered) phase, β > 3, the macrostates are rotations of the measure with density (x, y, z) ce kz, where c = k 2 sinh k, k = γ 1 (β). In particular, as β, these densities converge to delta functions (full alignment of the spins). Now, what about asymptotics of the magnetization in each phase? Central and non-central limit theorems...

45 The subcritical (disordered) phase Scaling of magnetization for β < 3: 3 β W := n n σ i. i=1

46 The subcritical (disordered) phase Scaling of magnetization for β < 3: 3 β W := n n σ i. i=1 Theorem (K.-Meckes 12): There exists c β such that sup Eh(W ) Eh(Z) c β log(n) h:m 1 (h),m 2 (h) 1 n M 1 (h) is the Lipschitz constant of h M 2 (h) is the maximum operator norm of the Hessian of h Z is a standard Gaussian random vector in R 3.

47 The supercritical (ordered) phase, β > 3 Scaled magnetization: W := n β2 n n 2 k 2 j=1 σ j 2 1.

48 The supercritical (ordered) phase, β > 3 Scaled magnetization: W := n β2 n n 2 k 2 j=1 σ j 2 1. Theorem (K.-Meckes 12): There exists c β such that sup h: h 1, h 1 ( ) log(n) 1/4 Eh(W ) Eh(Z) c β, n

49 The supercritical (ordered) phase, β > 3 Scaled magnetization: W := n β2 n n 2 k 2 j=1 σ j 2 1. Theorem (K.-Meckes 12): There exists c β such that sup h: h 1, h 1 ( ) log(n) 1/4 Eh(W ) Eh(Z) c β, n where Z is a centered Gaussian random variable with variance σ 2 4β 2 [ ] 1 := (1 βg (k)) k 2 k 2 1 sinh 2, (k) for g(x) = coth x 1 x.

50 At the critical temperature β = 3 W := c 3 n j=1 σ j 2 n 3/2, where c 3 : EW = 1.

51 At the critical temperature β = 3 W := c 3 n j=1 σ j 2 n 3/2, where c 3 : EW = 1. Theorem (K.-Meckes 12): There exists C such that sup Eh(W ) Eh(X ) C log(n), h 1, h 1 n h 1

52 At the critical temperature β = 3 W := c 3 n j=1 σ j 2 n 3/2, where c 3 : EW = 1. Theorem (K.-Meckes 12): There exists C such that sup Eh(W ) Eh(X ) C log(n), h 1, h 1 n h 1 where X has density with c = 1 5c 3 p(t) = { 1 z t5 e 3ct2 t 0; 0 t < 0, and z a normalizing factor.

53 The main ideas of the proofs and the upshot Stein s method.

54 The main ideas of the proofs and the upshot Stein s method. Special non-normal limit theorem using Stein s method.

55 The main ideas of the proofs and the upshot Stein s method. Special non-normal limit theorem using Stein s method. The mean-field Heisenberg model is exactly solvable. Asymptotics for magnetization above, below, and at (non-gaussian) the critical temperature.

56 What s next 3D nearest-neighbor Heisenberg model is the big challenge. Other spin models, e.g., for superconductors with a double phase transition.

What s next 3D nearest-neighbor Heisenberg model is the big challenge. Other spin models, e.g., for superconductors with a double phase transition.

57 What s next 3D nearest-neighbor Heisenberg model is the big challenge. Other spin models, e.g., for superconductors with a double phase transition. Figure: Arrays of 87-nm-thick Nb islands (red) on 10-nm-thick Au layer (yellow). Edge-to-edge spacing of 140nm (a) and 340nm (b). Courtesy of N. Mason at UIUC.

58 Thank you arxiv:

Ferromagnets and superconductors. Kay Kirkpatrick, UIUC

Ferromagnets and superconductors Kay Kirkpatrick, UIUC Ferromagnet and superconductor models: Phase transitions and asymptotics Kay Kirkpatrick, Urbana-Champaign October 2012 Ferromagnet and superconductor