Quantum Diffusion and Delocalization for Random Band Matrices Antti Knowles Harvard University Warwick – 12 January 2012 Joint work with L´ aszl´ o Erd˝ os
Two standard models of quantum disorder Consider the two random Hamiltonians on C N (one-dimensional lattice). Random Schr¨ odinger operator. On-site randomness + short-range hopping. v 1 1 1 v 2 1 � ... ... H = − ∆ + v x = 1 ... x v N − 1 1 1 v N Eigenvectors are localized, local spectral statistics are Poisson. Wigner random matrix. H = ( H xy ) N x,y =1 with random centred entries, i.i.d. up to the constraint H = H T or H = H ∗ . This is a mean-field model with no spatial structure. Eigenvectors are delocalized, local spectral statistics are GOE/GUE. Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 1
Band matrices Intermediate model: random band matrix. The elements H xy are centred, independent (up to H = H ∗ ), and satisfy H xy = 0 for | x − y | > W . Here W is the band width. Summary: If W = O (1) then H ∼ random Schr¨ odinger operator. If W = O ( N ) then H ∼ Wigner matrix. Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 2
Anderson transition for band matrices • W = O (1) = ⇒ eigenvectors are localized. • W = O ( N ) = ⇒ eigenvectors are delocalized. Varying 1 ≪ W ≪ N provides a means to test the Anderson transition. Conjecture (numerics, nonrigorous SUSY arguments) The Anderson transition occurs at W ∼ N 1 / 2 . Let ℓ denote the typical localization length of the eigenvectors of H . Then the conjecture means that ℓ ∼ W 2 . Rigorous results: • ℓ/W � W 7 (Schenker). • ℓ/W � W 1 / 6 (Erd˝ os, K). Conjecture for higher dimensions If d = 2 then ℓ is exponential in W . If d � 3 then ℓ = N (delocalization). Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 3
Assumptions • Let d � 1 and N ∈ N . Consider random matrices H = ( H xy ) whose entries are indexed by x, y ∈ Λ N := {− N, . . . , N } d . • Assume that the entries H xy are independent (up to H = H ∗ ) with variances given by � x − y � 1 E | H xy | 2 = W d f . W Here f is a probability density of zero mean on R d (the “band shape”). • Assume that H xy is symmetric and exhibits subexponential decay. Note that � y E | H xy | 2 = 1 . Let { λ α } be the family of eigenvalues of H . Then � � 1 1 1 | Λ N | E Tr H 2 = E | H xy | 2 = 1 , E λ 2 α = | Λ N | | Λ N | α x,y i.e. the eigenvalues of H are of order 1. In fact, Sp( H ) → [ − 2 , 2] . Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 4
The diffusive scaling Define the quantum transition probability from 0 to x in time t through � �� �� � 2 . δ x , e − i tH/ 2 δ 0 ̺ ( t, x ) := E Note that ̺ ( t, · ) is a probability on Λ N for all t . Consider the diffusive regime x = η 1 / 2 WX , t = ηT , for η → ∞ . Here X and T are of order one. For d = 1 , diffusion cannot hold for x ≫ W 2 = ⇒ choose η = W κ for 0 < κ < 2 . Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 5
Quantum diffusion Theorem(Quantum diffusion) [Erd˝ os, K] Fix 0 < κ < 1 / 3 and pick a test function ϕ ∈ C b ( R d ) . Then � � � � � � x W dκ T, x lim ̺ ϕ = R d d X L ( T, X ) ϕ ( X ) , W 1+ dκ/ 2 W →∞ x ∈ Λ N uniformly in N � W 1+ d/ 6 and T � 0 in compacts. Here � 1 λ 2 d λ 4 L ( T, X ) := √ 1 − λ 2 G ( λT, X ) π 0 is a superposition of heat kernels 1 2 T X · Σ − 1 X , e − 1 G ( T, X ) := (2 πT ) d/ 2 √ det Σ where Σ = (Σ ij ) is the covariance matrix of the probability density f : � Σ ij := R d d X X i X j f ( X ) . Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 6
Interpretation of λ The quantum particle spends a macroscopic time λT moving according to a random walk, with jump rate O (1) in time t and transition kernel p ( y ← x ) = E | H xy | 2 . The remaining fraction (1 − λ ) T is the time the particle “wastes” in backtracking. Probability density of λ : λ 2 4 √ π 1 − λ 2 0 0.2 0.4 0.6 0.8 1 Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 7
Corollary: delocalization Informally: fraction of eigenvectors localized on scales ℓ � W 1+ dκ/ 2 converges to 0 . Let { ψ α } | Λ N | α =1 be an orthonormal family of eigenvectors of H . Fix K > 0 and γ > 0 and define the random subset of eigenvectors � � | x − u | � γ � � α ( x ) | 2 exp B ω | ψ ω ℓ := α ∈ A : ∃ u ∈ Λ N � K . ℓ x Theorem(Delocalization) [Erd˝ os, K] For any κ < 1 / 3 we have W →∞ E | B ℓ | lim | Λ N | = 0 , where ℓ = W 1+ dκ/ 2 . Proof. Expand e − i tH/ 2 δ 0 = � α ψ α (0) e − i tλ α / 2 ψ α . Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 8
Naive (and doomed) attempt: power series expansion of e − i tH/ 2 The moment method (Wigner 1955: semicircle law, . . . ) involves computing � E Tr H n = E H x 1 x 2 H x 2 x 3 · · · H x n x 1 x 1 ,...,x n for large n . Because of E H xy = 0 , nonzero terms have a complete pairing (or a higher-order lumping). Graphical representation: path x 1 , x 2 , . . . , x n , x 1 . • The path is nonbacktracking if x i � = x i +2 for all i . • The path is fully backtracking if it can be obtained from x 1 by successive replacements of the form a �→ aba . (This generates “double-edged trees”.) Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 9
A fully backtracking path is paired by construction; its contribution is 1. Proof. Sum over all vertices, starting from the leaves. Each summation yields a factor � y E | H xy | 2 = 1 . Fully backtracking paths give the leading order contribution as W → ∞ . Wigner’s original derivation of the semicircle law involved counting the number of fully backtracking paths. Applying this strategy to ̺ leads to trouble: the expansion � i n − n ′ t n + n ′ 0 x H n ′ 2 n + n ′ n ! n ′ ! E H n ̺ ( t, x ) = x 0 n,n ′ � 0 is unstable as t → ∞ . The main contribution comes from fully backtracking graphs, whose number is of the order 4 n + n ′ . The main contribution to the sum over n, n ′ comes from n + n ′ ∼ t (Poisson), diverges like e 4 t as t → ∞ . Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 10
Getting rid of the trees: perturbative renormalization Simple example: Let z = E + i η with η > 0 and compute E G ii ( z ) = E ( H − z ) − 1 ii . Assuming that the semicircle law holds, we know what to expect: � � E G ii ( z ) = E 1 G jj ( z ) = E 1 1 λ α − z = E m N ( z ) ≈ m ( z ) N N j α where m N ( z ) is the Stieltjes transform of the empirical eigenvalue density N − 1 � α δ λ α , and √ � 1 4 − x 2 m ( z ) := d x x − z 2 π is the Stieltjes transform of the semicircle law. Note: m ( z ) is uniquely characterized by 1 z + m ( z ) + m ( z ) = 0 , | m ( z ) | < 1 ( z / ∈ [ − 2 , 2]) . Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 11
Choose µ ≡ µ ( z ) ∈ C and expand G ( z ) around µ − 1 : � µ + z � n ∞ � 1 µ + H − µ − z = 1 1 µ + 1 − H H − z = . µ µ µ n =1 Thus we get � � E G ii = 1 µ + 1 µ 2 ( µ + z ) + 1 + 1 ( µ + z ) 2 + E H 2 µ 4 ( · · · ) + · · · . ii µ 3 Choose µ so that red terms cancel: µ 2 ( µ + z ) = − 1 1 ii = − 1 z + µ + 1 µ 3 E H 2 ⇐ ⇒ µ = 0 . µ 3 We need | µ | > 1 for convergence: choose µ = m − 1 . Using a graphical expansion, one can check that this choice of µ leads to a systematic cancellation of leading-order pairings (trees) up to all orders. What remains are higher-order corrections of size O ( W − d ) . In particular, E G ii = m + o (1) . This is essentially two-legged subdiagram renormalization in perturbative field theory. Works also for more complicated objects like E | G ij | 2 . Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 12
Renormalization using Chebyshev polynomials A more systematic and powerful renormalization: use a beautiful algebraic identity due to Bai, Yin, Feldheim, Sodin, . . . . Define the n -th nonbacktracking power of H through n − 2 � � H ( n ) x 0 x n := H x 0 x 1 · · · H x n − 1 x n 1 ( x i � = x i +2 ) . x 1 ,...,x n − 1 i =0 Assume from now on that H xy = 1 (1 � | x − y | � W ) Unif( S 1 ) . √ W d We shall see later how to relax this condition. Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 13
Lemma[Bai, Yin] H ( n ) = HH ( n − 1) − H ( n − 2) Proof. Introduce 1 = 1 ( x 0 � = x 2 ) + 1 ( x 0 = x 2 ) into ( HH ( n − 1) ) x 0 x n . Feldheim and Sodin inferred that H ( n ) = � U n ( H ) , where � U n ( ξ ) = U n ( ξ/ 2) and U n is the standard Chebyshev polynomial of the second kind. Indeed, we have U n ( ξ ) = ξ � � U n − 1 ( ξ ) − � U n − 2 ( ξ ) . Thus, we expand the propagator e − i tH/ 2 in terms of Chebyshev polynomials: � e − i tξ = α n ( t ) U n ( ξ ) . n � 0 We can compute the coefficients � 1 � α n ( t ) = 2 1 − ξ 2 d ξ = 2( − i) n n + 1 e − i tξ U n ( ξ ) J n +1 ( t ) , π t − 1 where J n is the n -th Bessel function of the first kind. Antti Knowles Quantum Diffusion and Delocalization for Random Band Matrices 14
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