Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems

TL;DR

This paper studies quantum dynamics governed by Jacobi and block Jacobi matrices with periodic or limit-periodic modulations, revealing different transport behaviors and applying findings to quantum many-body systems like the XY chain.

Contribution

It provides new insights into quantum transport phenomena in modulated Jacobi matrices and establishes explicit bounds for Lieb-Robinson velocity in quantum spin chains.

Findings

Ballistic transport in periodic cases

Quasi-ballistic transport and weak localization in limit-periodic cases

Explicit Lieb-Robinson velocity bounds for XY chain

Abstract

We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic sequence. In particular, we investigate the transport exponents. In the periodic case we demonstrate ballistic transport, while in the limit-periodic case we discuss various phenomena such as quasi-ballistic transport and weak dynamical localization. We also present applications to some quantum many body problems. In particular, we establish for the anisotropic XY chain on $\mathbb{Z}$ with periodic parameters an explicit strictly positive lower bound for the Lieb-Robinson velocity.