Feshbach-Schur RG for the Anderson Model

Transcription

1 Feshbach-Schur RG for the Anderson Model John Z. Imbrie University of Virginia Isaac Newton Institute October 26, 2018

2 Overview Consider the localization problem for the Anderson model of a quantum particle moving in a random potential. We develop a renormalization-group framework based on a sequence of Feshbach-Schur maps. Each map produces an effective Hamiltonian on a lower-dimensional space by localizing modes in space and in energy. Randomness in ever-larger neighborhoods produces nontrivial eigenvalue movement and separates eigenvalues, making the next step of the RG possible. We discuss a particularly challenging case where the disorder has a discrete distribution. Reference: arxiv:1705:01916

3 Schur complement (or Feshbach map) Let H be a symmetric matrix in block form, H = ( A B C D ), with C = B T. Assume that (D E) 1 ε 1, B γ, C γ. The Schur complement at energy λ is: F λ A B(D λ) 1 C. Let ε and γ/ε be small, and λ E ε/2. Then ( ) ϕ (i) If (F λ λ)ϕ = 0, then (H λ) (D λ) 1 Cϕ = 0, and all eigenvectors of H with eigenvalue λ are of this form. (ii) ( γ ) 2 λ F λ F E 2 E. ε (iii) The spectrum of H in [E ε/2, E + ε/2] is in close agreement with that of F E in the following sense. If λ 1 λ 2... λ m are the eigenvalues of H in [E ε/2, E + ε/2], then there are corresponding eigenvalues λ 1 λ 2... λ m of F E, and λ i λ i 2(γ/ε) 2 λ i E.

4 Utility of the Schur complement With the Schur complement, we are able to reduce the eigenvalue problem for a large matrix into an equivalent (and hopefully more tractable ) one for a smaller matrix. If one focuses on a window of energies in some neighborhood of E, the submatrix D is gapped, in the sense that (D λ) 1 2/ε for λ E ε/2. The degrees of freedom associated with D are eliminated, and one can work with a renormalized operator F λ for λ near E. The fact that F λ depends on λ means that we cannot find all of the eigenvalues in the interval by looking at one matrix. However, (ii) guarantees that the λ-dependence is mild, and then the closeness of the spectra of H and that of F E is guaranteed by (iii). In particular, if one finds a gap in the spectrum of F λ in the interval, then a similar gap occurs for H. This is a way to demonstrate non-degeneracy in the spectrum.

5 Iterated Schur complement: a natural for the renormalization group For example, let H =. Then invertibility is an issue near E = 0. Low-momentum modes lead to long-range correlations. One can try to handle the problem by introducing block-spin RG. Kinematics of block spin transformations a simple example: Consider models like φ 4 on the lattice, S(φ) = φ, Hφ + V (φ). One could project onto energies 2 k, say, in step k of the RG. But locality would not be preserved, because of the plane wave eigenvectors. Instead, introduce block fields, ψ x = average of φ on block B(x) centered at x. Change basis: φ y = ψ x 1 B(x) (y) + χ y, for y B(x). = constant + fluctuation dim 1 dim L d 1 (choose basis) H = ( ) ( A B C D using the basis ψχ ) block averages fluctuations

6 Kinematics of block spin transformations, cont. S(φ) = φ, Hφ + V (φ) H = ( A B C D ) using the basis ( ψχ ) block averages fluctuations D = fluctuation covariance, D 1 1 ε Terms B, C couple φ and χ, so one needs to shift the fluctuation field χ to obtain a centered Gaussian integral: χ χ D 1 Cψ (minimizer of quadratic φ, Hφ, given ψ) ( ψ χ ) ( A B C D ) ( ψχ ) = ψ, Aψ + χ, Dχ + 2 χ, Cψ ψ, Aψ + χ, Dχ 2 χ, Cψ + 2 χ, Cψ + ψ, BD 1 Cψ = ψ, F 0 ψ + χ, Dχ, with F 0 = A BD 1 C. We see that the Schur complement F 0 becomes the effective Laplacian for the block spins. The translation D 1 Cψ now appears in V. One is ready to integrate out the fluctuation χ to obtain the new effective action.

7 Application of iterated Schur complements to random Schrödinger Consider the Anderson model in Z d : The potentials {v x } x Z d H = γ + v with γ 1. are iid random variables. We are particularly interested in the case of a discrete distribution; specifically we may give each v x a uniform distribution 1 on {0, N 1, 2 N 1,..., 1}, with N 1. For N = 2, this is the Anderson-Bernoulli model, the Ising model of random Schrödinger operators. We are interested in showing exponential decay of eigenfunctions (localization). We would also like to control the probability of degeneracies or near-degeneracies in the spectrum of H. Related work: Bourgain, Kenig, Invent. Math. 2005: Localization for the continuum version of Anderson-Bernoulli (N = 2, Z d R d ) for energies near the bottom of the spectrum, E = 0.

8 Large N result Let I δ (E) denote the interval [E δ, E + δ], and let N (I ) denote the number of eigenvalues of H in I. Theorem. Choose a sufficiently large p. Then for N sufficiently large (depending on p) and γ sufficiently small (depending on N), E N (I δ (E)) Λ (log γ δ) p, and ( ) P min E α E β < δ Λ 2 (log γ δ) p. α β for any rectangle Λ and any δ [γ diam(λ)/2, 1]. Also, α ψ α(x)ψ α (y) decays exponentially, and its disorder average is bounded by C x y p.

9 Resonant set and block decomposition of H Given an energy E, we expect eigenfunctions whose eigenvalues are close to E will be localized near sites where v x E is small. Eigenfunctions should decay rapidly away from such sites. Resonant Set: Let ε = 1/N and γ ε 20. The resonant step for the first step is R (1) = {i Λ : v i E ε}. The nonresonant set is R (1)c = Λ \ R (1). Block decomposition: H = ( ) A B C D where A, D are H restricted to R (1), R (1)c, respectively, and B, C contain the nearest-neighbor terms from the Laplacian that connect the two regions. As D operates on the subspace associated with nonresonant sites, where v i E ε, one has that (D λ) 1 3/ε for λ E ε/2. Consequence: If ψ is an eigenvector of H with eigenvalue λ I ε/2 (E) then ( ) ψ = with ϕ an eigenvector of F λ = A B(D λ) 1 C. ϕ (D λ) 1 ϕ Also, (D λ) 1 decays exponentially, so ψ(x) decays rapidly with dist(x, R (1) ).

10 Iteration preliminaries Connected components of the resonant set 1. Decompose the resonant set R (1) into connected blocks or supersites B using proximity conditions.

11 Iteration preliminaries Accurate diagonalization as in quasidegenerate perturbation theory 2. Obtain a very accurate approximation to the energy levels associated with each block B. We need to do much better than the values of v i for i B because in the next step, the energy window will be shrunk considerably. To do this, let B be a neighborhood of B of width L 1. Here we introduce a sequence of length scales L k = L 0 2 k with L 0 a moderately large number. Then replace: F λ F (1) λ = A B(D λ) 1 C where the matrix D is now restricted to B \ B, instead of Λ \ B. In terms of a random-walk expansion for (D λ) 1 (x, y), take only walks that remain inside B. Then we have that F λ = α F (B α ) + O(γ 2L 1 ). This is analogous to the statement H = diag(v 1,..., v Λ ) + O(γ). Weyl s inequality: spectrum moves no more than the norm of the perturbation.

12 Iterated Schur Complement Recall the length scales L k = L 0 2 k. Energy window width in step k: ε k = γ L k. Resonant set R (k) consists of blocks still resonant to E within ε k. New proximity conditions: k th -step blocks separated by a distance L 3/2 k. k th -step block B k has a neighborhood B k of width L k (to approximate its spectrum and determine if it remains resonant). Block decomposition of the (k 1) st Schur complement leads to the k th Schur complement: ( F (k 1) A (k) B λ = (k) ), C (k) D (k) F (k) λ = A (k) B (k) (D (k) λ) 1 C (k).

13 Estimates for the multi-scale percolation problem To obtain localization, we need to demonstrate that the resonant blocks (= percolation clusters) do not percolate. Connections between blocks are made at a distance L 3/2 k, but a compensating effect is the fact that blocks become less and less likely to remain resonant. Roughly speaking, the expected size of percolation clusters will stay bounded uniformly in k if the probability of a block remaining resonant is bounded by L p k with p > d. For a continuous distribution of potentials, the probability would go like ε k = γ L k, by the Wegner estimate. For a discrete distribution, narrowing the spectral window does not automatically lead to a reduction in probability. Therefore, we need to demonstrate eigenvalue movement and separation (degeneracy breaking) based on the effect of potentials in the annuli Bk \ B k 1.

14 Eigenfunctions of γ + v cannot grow faster than exponentially on Z d Kagome lattice: Compactly supported e-fns Z d : normalized e-fn γ r somewhere on B r

15 This implies the existence of influential sites where an eigenfunction has a lower bound γ L k Use randomness at a single site in each annulus of radius L k : the most influential one. It splits an isolated eigenvalue into N values with spacing γ L k. The probability that the eigenvalue sits in an interval of size γ L k is N k = L p k, where p = log 2 N. This is summable decay for p > d, which means resonances don t percolate. If not isolated: the potential drives a rank 1 perturbation, which splits an n-fold (near) degeneracy to an (n 1)-fold degeneracy, with probability 1 1/N.

16 Result: Cantor-like splitting of eigenvalue distribution on a logarithmic length scale This leads to log-hölder continuity of the density of states: E N (I δ (E)) Λ (log γ δ) p. Large p, in particular p > d = separation of δ resonances by a distance L (log γ δ) p/d, so γ L δ. Thus the interaction strength is typically much smaller than δ, which implies localization.

17 Unique continuation in R d Usual unique continuation principle: If ψ(x) is a harmonic function that vanishes on an open set, then ψ(x) 0 everywhere. Quantitative unique continuation (Bourgain-Kenig 2005): If ψ(x) is an eigenfunction of + v then ψ(x) 2 dx exp( y 4/3 log y ). B 1 (y) The possibility of faster-than-exponential growth is realized for complex v examples (Meshkov 1992). Longstanding conjecture by Landis (from the 60 s) that this does not happen for real v.

18 Weak form of unique continuation in Z d for solutions to ( + v)ψ = 0 Although eigenfunctions cannot decay uniformly faster than exponentially on the lattice, they can vanish on large sets. Consider, for example, this solution to ( 4)ψ = 0: In a box X, ψ X (x) exp( c z ) for at least one x X in a 45 light cone of R (where ψ is known to be nonzero).

19 Some recent results in dimension 2 Landis conjecture in dimension 2: Kenig, Sylvestre, Wang, Comm. PDE A form of unique continuation in Z 2 : Buhovsky, Logunov, Malinnikova, Sodin, arxiv: Localization near the edge for Anderson-Bernoulli in Z 2 : Ding, Smart, arxiv:

20 The Anderson-Bernoulli model (work in progress with Svitlana Mayboroda) H = + v with v {0, 1} on Z d. A plethora of degeneracies have the potential for spoiling localization. To combat the problem, take E > 0 small (Lifshitz tail regime). To get spectrum near E one needs a region of size l 0 E 1/2 with mostly 0 s. Large deviation arguments = probability exp( E d/2 ) = separation of resonant spots as in the large N case exp(e d/2 )

21 Issues with N = 2 1. Recall that in the large N case, we used the potential at one site per annulus (because of the Cantor condition: we need the upper bound for next-generation effects γ.85l k+1 = γ 1.7L k to fit inside the lower bound on the current effect γ L k.) We reach length scale L k after k eigenvalue splittings (using the potential at one site per annulus), and then each atom has probability N k = 2 kp = L p k, where p = log 2 N. To prevent percolation of resonances we need p > d. But L p k is no longer integrable when N = 2. So we need to find a way to use many sites per annulus. 2. Decay length is many lattice steps = it is difficult to pin down the location of a site where the eigenfunction is guaranteed to be big (the location should not depend on the randomness in the new annulus).

22 To be continued...