Multi-Scale Jacobi Method for Anderson Localization

A new KAM-style proof of Anderson localization is obtained. A sequence of local rotations is defined, such that off-diagonal matrix elements of the Hamiltonian are driven rapidly to zero. This leads to the first proof via multi-scale analysis of exponential decay of the eigenfunction correlator (this implies strong dynamical localization). The method has been used in recent work on many-body localization [arXiv:1403.7837].Comment: 34 pages, 8 figures, clarifications and corrections for published version; more detail in Section 4.

End-to-end Distance from the Green's Function for a Hierarchical Self-Avoiding Walk in Four Dimensions

In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Green's function for the process equals 1/x^2. If the process is modified so as to be weakly self-repelling, it was shown that at the critical killing rate (mass-squared) \beta^c, the Green's function behaves like the free one. - Now we analyze the end-to-end distance of the model and show that its expected value grows as a constant times \sqrt{T} log^{1/8}T (1+O((log log T)/log T)), which is the same law as has been conjectured for self-avoiding walks on the simple cubic lattice Z^4. The proof uses inverse Laplace transforms to obtain the end-to-end distance from the Green's function, and requires detailed properties of the Green's function throughout a sector of the complex \beta plane. These estimates are derived in a companion paper [math-ph/0205028].Comment: 29 pages, v2: reference

Dimensional Reduction for Directed Branched Polymers

Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D+1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: $\gamma(D+1)=\alpha(D)$ (the exponent describing the singularity of the pressure), and $\nu_{\perp}(D+1)=\nu(D)$ (the correlation length exponent of the repulsive gas). It also leads to the relation $\theta(D+1)=1+\sigma(D)$, where $\sigma(D)$ is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.Comment: 7 pages, 1 eps figure, ref 24 correcte

Many-body localization: stability and instability

Rare regions with weak disorder (Griffiths regions) have the potential to spoil localization. We describe a non-perturbative construction of local integrals of motion (LIOMs) for a weakly interacting spin chain in one dimension, under a physically reasonable assumption on the statistics of eigenvalues. We discuss ideas about the situation in higher dimensions, where one can no longer ensure that interactions involving the Griffiths regions are much smaller than the typical energy-level spacing for such regions. We argue that ergodicity is restored in dimension d>1, although equilibration should be extremely slow, similar to the dynamics of glasses.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.status: publishe