Why Hyperbolic Symmetry?

Transcription

1 Why Hyperbolic Symmetry? Consider the very trivial case when N = 1 and H = h is a real Gaussian variable of unit variance. To simplify notation set E ɛ = 0 iɛ and z, w C = G(E ɛ ) 2 h = [(h iɛ)(h + iɛ)] 1 h < exp[ih(z z w w) ɛ(z z + w w)] > h dz dw = exp[ 1 2 (z z w w) 2 ɛ(z z + w w)] dz dw. Note that ɛ breaks the hyperbolic symmetry to make the integral well defined.

2 Hyperbolic sigma models and Reinforced walk Quantum Scattering with random impurities - Manhattan Pinball (Beamond, Cardy, Owczarek and Gruzberg, Ludwig, Read) Linearly Edge Reinforced Random Walk - Persi Diaconis Zirnbauer s SUSY sigma model with a hyperbolic target. Anderson like phase transition in 3D. Cycle structure created by Random Adjacent Transpositions on Λ Z d. Universality of Mean Field for macroscopic cycle lengths? (Ueltschi, Chalker, Nahum, Schramm, Toth )

3 Manhattan Pinball Quantum Network Model with random scatterers Motivation: Chalker s network model Integer Quantum Hall. Particle moves on Z 2 along streets with alternating orientations scattered by random obstructions. Equivalent to Unitary evolution with SU(2) bond disorder. Beamond, Cardy, Owczarek, and Gruzberg, Ludwig, Read

4 obstruction Figure: Manhattan Lattice

5 Theorem If p > 1/2 then all trajectories are closed with probability 1. Localization. Proof (Chalker) by percolation. Conjecture: All trajectories are closed for any p > 0 Average loop diameter e c p 2 1. Thus in 2D any randomness produces Localization Mirror model: mirrors at vertices randomly placed at ±45. Fully packed mirrors equivalent to critical percolation. There is always an Extended channel. (Kozma-Sidovaricius)

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7 Linearly Edge Reinforced Random Walk History dependent walk W n Z d, n Z + : Walk takes nearest neighbor steps and favors edges j, k, Z d, j k = 1, it has visited in the past. Introduced by P. Diaconis while wandering the streets of Paris. He liked to return to streets he had visited in the past. Remarks: Not Markovian but is a superposition of Markov Processes Equivalent to a random walk in a correlated random environment given by statistical mechanics.

8 Definition of Reinforced Random Walk Let C jk (n) = number of times the walk has crossed edge jk up to time n and let β > 0. Prob{W n+1 = k W n = j} = 1 + C jk(n)/β N β, j k = 1. N is the normalization: N β = k (1 + C jk (n)/β), j k = 1 0 < β 1, strong reinforcement, (high temperature) β 1, weak reinforcement, (low temperature) Questions: Is ERRW localized? recurrent? transient?

9 P. Diaconis and D. Coppersmith (1986): ERRW random walk in a random environment. Environment: The rate at which an edge j, j is crossed w j,j > 0 are correlated random variables. (conductances) Distribution of w j,j > 0 is correlated, unbounded and given by an explicit statistical mechanics model. The generator for the RW is a weighted Laplacian D w s t D w s = w j,j (s j s j ) 2, s j R j j =1 Local conductance: w j,j

10 Quantum of particle on Z 3 scattered by impurities Quantum Time Evolution is analyzed via the Green s function: G(E + iɛ; j, k) = [H(ω) E iɛ] 1 (j, k). Average of G(j, k) 2 is the spin-spin correlation in SUSY Statistical mechanics. Spins s j, j Z d Λ are 4 4 super matrices in box Λ. Efetov sigma model target: U(1, 1 2)/[U(1 1) U(1 1)] Study a simpler vector version - Zirnbauer s model.

11 Zirnbauer s SUSY Hyperbolic Model - H (2 2) Sigma constraint : z 2 j x 2 j y 2 j 2η j ξ j = 1 Horospherical parametrization: s j, t j R : x = sinh t e t ( 1 2 s2 + ψψ ), y = e t s, ξ = e t ψ, η = e t ψ, S Λ = i j Λ S ij + ɛ k Λ S ij = cosh(t i t j ) (s i s j ) 2 e t i +t j + ( ψ i ψ j )(ψ i ψ j ) e t i +t j, z k z k = cosh t k + ( 1 2 s2 k + ψ k ψ k ) e t k. Local conductance: w j,j = e t j +t j.

12 Partition function and Ward identity: H (2 2) Z Λ (β, ɛ) = C exp [ βs Λ ] Λ e t j dt j ds j dψ j d ψ j By supersymmetry Z Λ (β, ɛ) 1 all β, ɛ. Ward identity: e α t 0 = e (1 α) t 0, e t 0 = 1. β ρ(e) 2 λ 2, ρ(e) = density of states, λ = disorder

13 Effective Action: Integrate out Grassmann ψ, ψ and s variables. E SUSY ({t j }) = β j j cosh(t j t j ) 1/2 log det D β,ɛ (t) D β,ɛ (t) a finite difference elliptic operator: [s ; D β,ɛ (t) s] Λ = β (j j) et j +t j (s j s j ) 2 + ɛ k Λ et k s 2 k Local conductance = e t j +t j. Saddle Point t s of Effective Action: If t s = 0, as ɛ 0 - Conduction, 3D, when β large If ɛe t s e β as ɛ 0 localization, 2D

14 Spin-Spin correlation: y 0 y k =< s 0 e t 0 s k e t k > (β, ɛ) =< e t 0+t k D β,ɛ (t) 1 (0, k) > SUSY (β, ɛ) random walk in a random environment: In 3D if the local conductance e t j +t j does not fluctuate or t j 0 Then correlation < e t 0+t k D β,ɛ (t) 1 (0, k) > SUSY (β, ɛ) ( β + ɛ) 1 (0, k) is diffusive.

15 Phase Transition in 3D Theorem (Disertori, S., Zirnbauer 10) A) For β 1 and d 3 Q Diffusion: y 0 y k ( β + ɛ) 1 (0, k), k Z 3 B) For 0 < β 1, localization: y 0 y k (β, ɛ) ɛ 1 e k /l(β)

16 Remark: The Hyperbolic sigma model (No Fermions) has No phase transition in 3D - always delocalized. Effective action is convex for all β. (Sp, Zirnbauer, Brydges) Gruzberg-Mirlin (96) Replica analysis of H (2 2) on Bethe Lattice Conjectures: A)There is a multi-fractal transition in 3D B) In 2D exponential localization holds for all β.

17 SUSY Hyperbolic sigma model is essentially equivalent to ERRW Relation made precise by Sabot and Tarres following earlier observations of Kozma, Sznitman, Gawedzki

18 Phase Transition for ERRW in 3D Theorem (Sabot-Tarres) For strong reinforcement, ERRW is recurrent and we have Localization: Prob{ W (t) W (0) R} Ce R/l(β), E W 2 (t) Const and l(β) is the localization length. Theorem (Disertori-Sabot-Tarres) For weak reinforcement, β 1, and d 3, W (t) is Transient, quasi-diffusion.

19 Ideas of Proof in 3D: Ward identities To prove diffusion must show the t field has very small fluctuation for large β: cosh p (t j t k ) (β) Const all j, k, β 1 If F (t, s, ψ, ψ) is 0Sp(2 2) invariant, then. < F > SUSY (β, ɛ) = F (0) Example: < F m > = 1 all m 1. [ F = cosh(t j t k ) + e t j +t k 1 ] 2 (s j s k ) 2 + (ψ j ψ k )(ψ j ψ k )

20 Products of Random Transpositions on Z d. Let Λ = Λ L Z d be a box of side L, P Λ (0) be the Identity permutation of its vertices at time 0. Each adjacent edge (j, j ) Λ, has an independent Poisson clock which rings at rate 1. When the clock rings, we make a transposition j j. P Λ (T ) is the product of these transpositions up to time T. Question: After time T, what is the cycle structure of P Λ (T ) for large L? In 2D are these cycles like those of the Manhattan Pinball T p 2??

21 Theorem (Caputo, Liggett, Richthammer) In Λ L, if T L 2 then P Λ (T ) uniform weight on all permutations of Λ L. Conjecture: For d 3 there is a T independent of L, such that if T > T there are macroscopic cycles of length l 1 > l 2 > l 3... where l j c j L d. Moreover for T > T as L the length distribution is Mean Field - Poisson-Dirichlet (Schramm, Ueltschi) after scaling. Thus l 1 /l 2 reaches its Mean Field distribution long before T = L 2!

22 Theorem The partition function Z Λ (β) for the spin 1/2 Quantum Heisenberg Ferromagnet is Z Λ (β) = tr e β Λ j j S j S j = E Λ (β) [ 2 #cycles ] where E Λ (β) denotes the expectation of the independent Poisson clocks up to time T = β. Macroscopic cycles correspond to Bose-Einstein condensation. Theorem In 2D there are no macroscopic cycles - Mermin-Wagner Remark: PD(2) implies that Z(β) 1 tr e β Λ j j S j S j + Λ j Λ 1 h s j = S 2 e M(β)h S 0 dµ END