Quantum Diffusion and Delocalization for Random Band Matrices

Transcription

1 Quantum Diffusion and Delocalization for Random Band Matrices László Erdős Ludwig-Maximilians-Universität, Munich, Germany Montreal, Mar 22, 2012 Joint with Antti Knowles (Harvard University) 1

2 INTRODUCTION Universality conjecture for disordered quantum systems (vague): There are two regimes, depending on disorder strength: i) Strong disorder: localization and Poisson local spectral statistics ii) Weak disorder: delocalization and random matrix (GUE, GOE) local statistics. Local statistics depends only on symmetry but is otherwise independent of the details of the system. (Can be complemented by the conjectured relations to the classical counterparts of these systems: integrable vs. chaotic) 2

3 Two well studied models Random Schrödinger operators: (on-site randomness, short range hopping); represented by a narrow band matrix with nonzero elements at finite distance from the diagonal (E.g. d = 1, +λv is tridiagonal). Wigner random matrices: H = (H xy ) x,y Λ, with H xy centered i.i.d. up to symmetry constraint (H = H or H = H t ). They model a mean-field hopping mechanism with random quantum transition rates. No spatial structure. In this talk we consider an intermediate model, the random band matrices in d dimension with band width W. Here Λ Z d, and H xy are independent, centered, with variance σ 2 xy = E H xy 2 = 0 for x y W, y σ 2 xy = 1 y (W = O(1) Random Schrödinger; W = Λ, d = 1 is Wigner) 3

4 MAIN RESULTS (informally) For a random band matrix H in d dim with band width W we have (i) Quantum evolution e ith is diffusive up to times t W d/3 (ii) Localization length is at least l W t W 1+d/6. Note that (i) = (ii): Jump per unit time is O(W); after time t the displacement is W t. For an eigenfunction ψ ψ(x) = e ith ψ(x) spreads at distance W t First used by T. Chen to show that the localization length for the Anderson model + λv, λ 1, is at least λ 2, using that the quantum evolution up to time t λ 2 is kinetic (Boltzmann eq.) 4

5 Previous results on random Schrödinger The localization regime is (relatively) well understood: Localization (Fröhlich-Spencer, Aizenman-Molchanov,...) Poisson statistics (Minami,...) Much less is known in the delocalization regime: Bethe lattice (Klein, Aizenman-Sims-Warzel, Froese-Hasler-Spitzer) Scattering regime (Bourgain), Scaling limits (Spohn, E-Salmhofer-Yau, Chen). No result on the conjectured local statistics (apart from the Bethe lattice, where it is Poisson, despite the general conjecture [A-W]). 5

6 Previous results on Wigner matrices H = (h jk ) 1 j,k N hermitian N N matrix with h jk = h kj = 1 N (x jk +iy jk ) and h kk = 2 N x kk where x jk, y jk (for j < k) and x kk are i.i.d. with expectation zero, variance 1/2. [Gaussian GUE] 1 ρ (x) = 4 x 2 2π 2 2 Eigenvalues: λ 1 λ 2... λ N Typical eigenvalue spacing is λ i+1 λ i 1 N (in the bulk) 6

7 Local semicircle law on optimal η 1/N scale [E-Schlein-Yau-Yin] Local level correlation statistics (two point correlation function) lim N 1 [ρ sc (E)] 2 p(2) N ( E + x 1 Nρ sc (E),E + x 2 Nρ sc (E) = det { S(x i x j ) } 2 sinπx, S(x) = i,j=1 πx, E < 2 (k-point correlation functions are given by k k determinants). [Dyson, Mehta, Deift, Johansson, Pastur-Schcherbina, Tao-Vu...] Most general results: E-Schlein-Yau-Yin (+Peche, Ramirez) Eigenfunctions are always completely delocalized. [E-Schlein-Yau] Assume Ee x ij ε <. Fix E < 2, and 2 < p <, then for large M P v, v 2 = 1, Hv = xv, x E 1 N, v p MN 1 p 1 2 ) Ce c M Note v = ( N 1/2,N 1/2,... ) = v p = N 1 p 1 2 for all 2 p 7

8 ANDERSON TRANSITION FOR BAND MATRICES d = 1, W = O(1) [Random Schrödinger] is always localized. d = 1, W = Λ [Wigner ensemble] is always delocalized. Varying 1 W Λ = N can test the transition Conjecture: In d = 1: Localization length l W 2. Hence W N 1/2 localization (on scale W 2 N) and Poisson statistics; W N 1/2 delocalization (l N) and GOE/GUE statistics In d = 2: Localization length is exponential in W In d 3: Localization length is (equals to the system size) 8

9 RESULTS ON BAND MATRICES In d = 1 dimensions, the localization length l W 8 [Schenker] Global semicircle law in expectation with W 2 error for d = 3 via SUSY [Disertori-Pinson-Spencer] Edge universality (Tracy-Widom) for W N 5/6 in d = 1 [Sodin] Local semicircle law in probability on scale 1/W with error (Wη) 1 (in d = 1). [E-Yau-Yin] Localization length l W in d = 1 [E-Yau-Yin] Local semicircle law in expectation on scale W 0.99 with error W 1 (in d = 1) [Sodin] Our result on the localization length l W 1+d/6 is the counterpart of Schenker s result but from the delocalization side. Recall, the conjecture (in d = 1) is l W 2, and thus W 7/6 l W 8 is proven. 9

10 FORMULATION OF THE MAIN RESULT H = H xy hermitian (symmetric) matrix, with elements indexed by x,y Λ N = [ N,N] d Z d. Let H xy be independent (up to symmetry constraint), EH xy = 0 and with variance σxy 2 := E H xy 2 = 1 x y W df( W with a profile function f with f = 1 and covariance Σ ij := R d x i x j f(x)dx. Note that y Λ σ 2 xy = 1, x Λ (This normalization guarantees that Spec(H) lies essentially in [ 2, 2]) Assume H xy /σ xy is symmetric and has subexponential decay. Define the quantum transition probability from 0 to x in time t by (t,x) := E x e ith/2 0 2, clearly (t, ) is a probability density on Λ for all t. ) 10

11 Theorem (Quantum diffusion) [E-Knowles] Fix 0 < κ < 1/3. For any T 0 > 0 and any testfunction ϕ C b (R d ) we have lim W x Λ N ρ(w dκ T,x)ϕ ( x W 1+dκ/2 uniformly in N W 1+d/6 and 0 T T 0. Here ) = L(T,X)ϕ(X), (1) RddX L(T,X) := 1 0 dλ 4 π λ 1 λ 2 G(λT,X) (2) 2 is a superposition of heat kernels G(T,X) := 1 (2πT) d/2 detσ e 1 2T X Σ 1 X, λ [0,1] in (2) represents the fraction of the macroscopic time T that the particle spends moving effectively; the remaining fraction 1 λ of T represents the time the particle wastes in backtracking. Backtracking is due to a self-energy renormalization. 11

12 Let A ε,l be the index set of eigenvectors ψ α localized on scale l A ε,l := { α : x ψ α (x) P x,l ψ α ε }, P x,l (y) = 1( x y l) Theorem (Delocalization) [E-Knowles] For any ε > 0, κ < 1/3 we have 1 lim sup W Λ E A ε,l C ε with l = W 1+dκ/2, i.e. up to a negligible portion, all eigenvectors are delocalized on a scale at least W 1+dκ/2. Practically: delocalization on scale l W 1+d/6 in an typical sense. 12

13 Ideas of the proof: I. Chebyshev expansion Standard moment method (proof of Wigner semicircle) computes ETrH n = i 1,i 2,...,i n EH i1 i 2 H i2 i 3...H in i 1 and the nonzero terms have complete pairing (Gaussian case) plus higher order cumulants (general case). A sequence (path) i 1,i 2,...i n,i 1 is called non-backtracking if i k i k+2 for all k. It is called fully backtracking if it can be obtained from i 1 by successively applying the substitution rule x xyx. 13

14 ETrH n = EH i1 i 2 H i2 i 3...H in i 1 i 1,i 2,...,i n u v x,y,z,... EH xy H yz H zu...h yx z w xyzuvuzwzy... x y Fully backtracking paths have nonzero ( W dn/2 ) contribution (they naturally pair) and they have the largest entropy factor (n/2 indices can be chosen freely, i.e. the sum above has (W d ) n/2 terms). Wigner semicircle comes from counting the number of fully backtracking paths. 14

15 Taylor expansion for e ith/2 is unstable E δ x,e ith/2 δ y 2 = i n n t n+n n,n 2 n+n n!n! EHn xyh n Here EH n xy Hn yx 2n+n (as above), the main contribution comes from n+n t 1 to the sum. yx Large terms come from backtracking paths that need to be resummed into their non-backtracking stem. Similar to the two-legged subdiagram ( tadpole ) renormalization in diagrammatic perturbation theory. 15

16 e d c x a b f y x k u v w y H 23 xy = a,b,... H xa H ab H bc H cb H bd...h hy H (5) xy = k,u,... H xk H ku H uv H vw H wy Here prime means summations restricted to non-backtracking paths: x u, k v, u w, v y All paths with the same stem but different backtracking branches will be resummed. This resummation is done algebraically with the Chebyshev polynomials. 16

17 Define the k-th Chebyshev polynomial (of second kind), U k (ξ), via sin(k +1)θ U k (cosθ) = sinθ These are orthogonal polynomials on [ 1,1] wrt. π 2 1 ξ 2 dξ. Expand e itξ into this orthonormal system e itξ = k=0 α k (t)u k (ξ), α k (t) := 2 π ξ 2 e itξ U k (ξ)dξ The coefficients are explicitly computable (via Bessel functions): α k (t) = 2( i) kk +1 J k+1 (t) t The weights { α k (t) 2 } k=1,2,... form a prob. distribution for each t: k 0 α k (t) 2 = 1 α k (t) 2 is the weight of the k-th order expansion term to (t). 17

18 The function λ α [λt] (t) 2 for t = 150. The solid line is the weak limit as t : f(λ) = 4 π λ2 (1 λ 2 ) 1/2. The number of collisions k = [λt] do not concentrate around t (unlike for the usual quantum diffusion from weakly coupled random Schrödinger), due to the delay effect of the backtracking paths. 18

19 Ideas of the proof: II. Non-backtracking powers H (n) x 0 x n := under the restriction x i x i+2. x 1,x 2,...,x n 1 H x0 x 1 H x1 x 2...H xn 2 x n 1 H xn 1 x n Suppose that (later we will remove this condition) H xy 1 W d Unif(S1 ) then H (n) = HH (n 1) H (n 2) [Bai & Yin] Feldheim and Sodin observed that this is reminiscent to the recursion Ũ n (ξ) = ξũn 1(ξ) Ũn 2(ξ), Ũn(ξ) := U n (ξ/2) thus H (n) Ũn(H). Sodin used this to get Tracy-Widom. In our case, for the time evolution, the precise result is: 19

20 Lemma e ith/2 = a m (t)h (m), m 0 and therefore a m (t) := k 0 α m+2k (t) W dk (t,x) = a n (t)a n (t)eh (n) 0x H(n ) x0 n,n Top arc represents H (n) 0x = H 0x1 H x1 x 2...H xn 1 x, each bullet is a summation over x s (labels). 20

21 Since EH ab = 0, the expectation introduces pairings, and more general lumpings (higher moments), among the edges. EH (n) 0x H(n ) x0 = Val x (Γ) (3) Γ G n,n where Γ is a partition of the edge set, where each lump has even size. Examples for pairings: Main Lemma: The main contribution to (3) comes from the n = n ladder if t W d/3 : Γ G n,n Val x (Γ) = Val x (Ladder)+o(1), as W 21

22 A formula Γ G n,n Val x (Γ) Γ P n,n R x (Γ) where P is the set of pairings and ( ) 1 n R x (Γ) := W d Q x (x) Γ x γ Γ x ( γ) (for simplicity we assumed n = n ). Here Γ is a partition of the edges, γ Γ is a lump; Γ = { γ} γ Γ is an assignment of a label-pair to each γ. Q x (x) encodes the non-backtracking condition and x ( γ) encodes the compatibility between the pre-assigned label-pairs and the actual labelling. The main source of complication is that the edges have to be paired (or even lumped) since carry the randomness, but their endpoints, the vertices carry compatibility conditions and we have to (partially) decouple them to be able to sum up. 22

23 Basic philosophy: More complicated the pairing is, the more compatibility constraints it imposes on the labelling, hence the contribution is smaller. This compensates for the larger combinatorics of the complicated pairings. This balance has to be exploited to all orders of the complexity. E.g. (sub)ladders impose less conditions than non-ladders; by parallel pairing lines (called parallel bridges ) each new bridge gives rise to a new label x i, otherwise may not. 23

24 Ideas of the proof III. Subladder resummations Step 1: Collapsing parallel bridges to obtain the skeleton of the pairing, together with encoding the number l e of bridges collapsed The contribution of parallel bridges will be obtained via random walk (heat kernel) approximation where time corresponds to l e. Step 2: Bounding the number of labels in a skeleton: The 2/3 rule In a skeleton graph each label has an orbit of length at least three. Thus the number of different labels for a skeleton with 2m vertices (and m edges) is at most 2m/3 (compare with m for ladders). 24

25 Step 3: Encoding the skeleton by a multigraph Vertices are the orbits (generated by the skeleton); edges are the bridges. Within this new multigraph, the edges carry the random walk, i.e. effectively there is a heat kernel decay along each edge, where time is the number of parallel bridges l e. To sum up, we choose a spanning tree in the multigraph and apply L 1 -bound for the heat kernel for the tree edges, and L -bound for all other edges. This step uses t n W d/3 and precise heat kernel bounds. 25

26 Critical pairing So far we have completed the tadpole renormalization (by using Chebyshev polynomials) and the parallel bridge resummation (which is special four-legged subdiagram resummation). The following example shows that without further resummations, the threshold t W d/3 cannot be exceeded. Main reason: 6k edges and 4k labels, i.e. the 2/3 bound on the labels in a skeleton saturate. 26

27 Removing H = const condition. This condition was used in the algebraic identity on non-backtracking powers: H (n) = HH (n 1) H (n 2) (4) In general we have error terms in (4), essentially of two types: ( (Φ 2 ) xy = δ xy Hxy 2 σxy 2 ), σ 2 xy = E H xy 2 z (Φ 3 ) xy = H xy 2 H xy Very naively: Φ 2 is small by CLT, Φ 3 by power counting. The rigorous treatment requires to introduce more types of graphs and a delicate resummation of backtracking branches (since the Chebyshev transformation does not completely eliminate the tadpoles coming from the Φ s). The threshold t W d/3 is again necessary. The organization of the expansion is quite involved. 27

28 SUMMARY Quantum diffusion up to time t W d/3 for band matrices. The limiting equation is a delayed heat equation that is identified via Chebysev transform. Eigenvectors are delocalized on scale W 1+d/6. Chebyshev transformation is used for tadpole renormalization. Subladders are resummed by skeleton graphs. OPEN: Localization length W 2. 28