Quasi-Diffusion in a SUSY Hyperbolic Sigma Model

Transcription

1 Quasi-Diffusion in a SUSY Hyperbolic Sigma Model Joint work with: M. Disertori and M. Zirnbauer December 15, 2008

2 Outline of Talk A) Motivation: Study time evolution of quantum particle in a random environment using Statistical Mechanics. B) Results and Conjectures for Random Matrices: Quantum diffusion for 3D Random Schrödinger? C) Spectral properties equivalent to correlations in a Supersymmetric Statistical Mechanics Model: SU(1, 1 2), Hyperbolic symmetry, Spin = 4x4 super-matrix. Partition function 1. (F. Wegner, K. Efetov 1980 s)

3 D) Main Result in 3D: Diffusion for a simpler SUSY, Hyperbolic model. Equivalent to random walk in a random environment. E) Algebraic identities of SUSY: Why hyperbolic supersymmetry? F) Ideas of proof: Ward Identities - coming from symmetry + Estimates on nonuniformly elliptic equations. G) Edge Reinforced Random Walk (Merkl + Rolles)

4 Random Matrices Random Schrödinger matrix: H = + λv (j) j Z d, λ > 0 Random Band Matrix, H = H H = H ij i, j Z d Λ N, < H ij Hi j > = R(i j) δ(i, i ) δ(j, j ) 0 R(i j) e i j /W Λ N = cube side N, Conjecture: In 1D, if W >> N, W = band width. then H has GUE statistics.

5 Quantum Time Evolution on Z d Let H be random Schrödinger on Z d. Initial condition Ψ(0, x) supported near 0. Unitary Time evolution: Ψ(t, x) exp( ith)ψ 0 (x) Spread of Wave function: R 2 (t) x Ψ(t, x) 2 x 2 Main Conjectures: (for 0 < λ, small) : In 3D: Quantum Diffusion: < R 2 (t) > D t λ 2 t In 2D: Localization: < R 2 (t) > Const.

6 Quantum Green s Function G ɛ (E; x, y) = (H E + iɛ) 1 (x, y) Q-Diffusion Conjecture in 3D: < G ɛ (E; x, y) 2 > = ρ(e) (x, y) C x y 1 D + ɛ ρ(e) = Im < G ɛ (E; 0, 0) >= density of states D = D(E) > 0 is diffusion constant. Localization : < G ɛ (E; x, y) 2 > = ɛ 1 e x y /l

7 Example: SUSY Statistical Mechanics for GUE Let H be and N N Hermitian Gaussian matrix: Density of States = ρ(e): ρ(e) = 1/N Trace < (H E ɛ ) 1 > = N exp N(a 2 + b 2 ) (ib E ɛ) N (a E ɛ ) N (a E ɛ ) 1 (1 [(a E ɛ )(ib E ɛ )] 1 ) dadb Saddle (solve quadratic equation) gives Wigner semicircle law.

8 Key SUSY Identities - Gaussian Integrals: Let H = H be N N matrix, x C N, ɛ > 0 x j x k exp[i x (H + iɛ) x]d N x = C N (H + iɛ) 1 (j, k) det(h + iɛ) 1 Let ψ j, ψ j be Grassmann, j = 1,...N. exp[iψ (H + iɛ)ψ] d N ψ d N ψ = C 1 N det(h + iɛ) Determinants cancel and we can compute average over H.

9 To get (H + iɛ) 1 (j, k) 2, also need complex conjugate: Let y C N and χ, χ Grassmann. y j y k exp i[y (H iɛ)y] d N y = C N (H iɛ) 1 (j, k) det(h iɛ) 1. Example: N=1, H = h R, < h 2 >= a 2, x, y C < h iɛ 2 > h =< exp i h [ x 2 y 2 ] exp ɛ( x 2 + y 2 )) > h,x,y =< exp( a 2 [ x 2 y 2 ] 2 ɛ( x 2 + y 2 )) > x,y Hyperbolic Symmetry: SU(1,1)

10 Statistical Mechanics: SUSY Hyperbolic model This model, due to Zirnbauer (1991), is expected to qualitatively reflect the phenomenon of localization and delocalization for Random Band Matrices. For j Z d, S j = (t j, s j, ψ j, ψ j ) SUSY-hyperbolic plane. The t j, s j R; parametrize hyperboloid and ψ j, ψ j are Grassmann variables. Action is quadratic in s and ψ, ψ fields. Integrating these variables, produces an effective action A(t), with t j R A(t) is real but non local and nonconvex.

11 Effective action A(t) for the t-environment For t j R, and β jj > 0, j j = 1, define: (f, D β (t) f ) = Σ β jj e t j +t j (f (j) f (j )) 2 Effective Action A β (t) : (s, ψ integrated out) exp[ βa βɛ (t)] exp[ β[σ cosh (t j t j ) 1]] det 1/2 [D β (t) + ɛe t ] e ɛσ[cosh(t j) 1]. < F (t) > t (β, ɛ) F (t) exp[ βa βɛ (t)] j e t j dt j Z(β, ɛ) =< 1 > t (β, ɛ) 1

12 Quantum-Green s Function < G ɛ (E; x, y) 2 > RS = < [β D(t) + ɛe t ] 1 (0, x) > t (β, ɛ) D = e t De t = + V (t) 0 Renormalization of ɛe t j at Saddle Point t s ɛe ts = 1/β 1Dim ɛe ts = e β 2Dim t s = 0 3Dim for β large ɛe ts 1 3Dim for β small Saddle suggests that localization occurs in 1 and 2 dimensions and diffusion occurs in 3D. Zirnbauer proved localization in 1D.

13 Main Theorem: Diffusion in 3D (Di-Sp-Zi): If β is large, then t 0 and we have: < e t 0+t x [βd(t) + ɛe t ] 1 (0, x) > t (β) ( β + ɛ) 1 (0, x) C x α with α 1. Hence, the evolution is Diffusive. Remark: For small β, localization should occur in 3D. Conjecture: the set [j : t j M] percolates for large M.

14 Ward Identities are Key to Proof If F (t, s, ψ, ψ) is 0Sp(2 2) invariant, then Example: For l Z d, let < F > SUSY (β, ɛ) = F (0) [ F l = cosh(t 0 t l ) + e t 0+t l 1 ] 2 (s 0 s l ) 2 + (ψ 0 ψ l )(ψ 0 ψ l ) Then < F m l > = 1 all m 1.

15 After integrating over the ψ and s variables The Ward identity < F m l >= 1 yields: < cosh m (t 0 t l )(1 m β G l(t)) >= 1 where G l is the Green s function of D(t) : G l (t) e t 0+t l ((δ 0 δ l ), {βd(t) + ɛe t } 1 (δ 0 δ l )) If 0 G l (t) C, then for Cm < β < cosh m (t 0 t l ) > (1 Cm β ) 1 Implies that t - fluctuations at scale l are small for large β.

16 Fluctuations Suppressed for l = 1 Because for all t, 0 G 1 (t) 1. Hence, C=1 and for 2m β Prob ( t 0 t 1 a) 2cosh m (a) This is very small for a > 0 and m large.

17 When l > 1, no uniform estimates on G l (t). To estimate G 2, can use fluctuation estimates for l = 1. By Ward identity, we then estimate fluctuation for l = 2. In general: We use Induction on l and Ward Identities to prove: Then for p = ±1, 2, 3 we prove < cosh m (t 0 t l ) > 2. < exp(pt 0 ) > Const. i.e. t 0, and Theorem follows.

18 Conclusions and open Problems Quantum Dynamics SUSY hyperbolic model = RWRE. In our Hyperbolic SUSY model, localization and diffusion are revealed using a simple saddle analysis. But fluctuations need to be estimated. We do this in 3D for large β. Once the Ward identities are established, analysis is classical: Induction on length scales + estimates on elliptic Greens functions. Problem: Prove Localization in 2D and Phase Transition in 3D.

19 Edge reinforced random walk ERRW is a history dependent walk which tends to visit previously visited edges. It can be expressed (Diaconis) as a random walk in a random environment much like ours. ERRW is known to be localized in 1D and some localization results for D=2, (Merkl+Rolles). There is a phase transition on the Cayley tree (Pemantle) from recurrent to transient. Relation between Quantum and classical dynamics? Special class of 2D Quantum Nework models are equivalent to 2D Percolation. (Gruzberg, Ludwig, Read, Cardy...) The End

20 Background - Evolution Equations The Heat Equation for x in Z d or R d is U(t, x)/ t = U(t, x) = D w U(t, x). The Schrödinger equation looks similar: i Ψ(t, x)/ t = Ψ(t, x) + λv (x)ψ(t, x). The Laplacian on the lattice is defined by the quadratic form: [f, f ] = j j f (j) f (j ) 2 where j, j Z d [f, D w f ] = j j f (j) f (j ) 2 exp w j,j V(x) = random impurities, exp w jj = random conductances.

21 Properties of time Evolution U(t, x) = probability of finding a random walk at x at time t. Ψ(t, x) 2 = probability of finding quantum particle at x at time t. Σ x U(t, x) and Σ x Ψ(t, x) 2 independent of time If V (x) = w jj = 0, then Σ x U(t, x)x 2 = D t 1 but Σ x Ψ(t, x) 2 x 2 = C t 2 Heat diffusion spreads more slowly than wave propagation. What happens when we add V(x) 0, x Z d, random iid?

22 Eigenfunctions Eigenfunction, f α, is localized if: f α (j) 2 C exp( c α j /l) all j Λ N. Theorem 1 On Z 1, for λ > 0, all eigenstates of Random Schrödinger are localized with probability 1. Theorem 2 On Z d, localization occurs for (large λ) or for large E. Thus there is No diffusion and no conduction. Conjecture: In 3D, for small λ > 0, quantum particles diffuse. Absolutely continuous spectrum and conduction.

23 Equivalent to random walk in a random environment: with w jj = t j + t j, generated by the t j : [f, D w f ] = j j f (j) f (j ) 2 exp(t j + t j ). Remarks on t - Environment: (s and ψ integrated out) The t j are highly correlated. D is not uniformly elliptic: - conductance exp(t j + t j ) very small for t 0. Localization may be produced! The Effective action describing t is Non Local and Non Convex. If t 0 then we have diffusion because D

24 SUSY Hyperbolic Sigma model Sigma model constraint at each lattice site: z 2 + x 2 + y 2 + 2ψψ = 1. Action A, parameterized in horospherical coordinates: + t j, s j R, and ψ j, ψ j Grassmann A ɛ = Σ (cosh(t j t j ) 1) + ɛ cosh t j β <j, j > j Λ [ e (t j+t j ) 1 ] 2 (s j s j ) 2 + (ψ j ψ j )(ψ j ψ j )