Ballistic Transport for Limit-Periodic Jacobi Matrices with Applications to Quantum Many-Body Problems

TL;DR

This paper demonstrates that limit-periodic Jacobi matrices with rapid approximation exhibit ballistic quantum transport, with applications to quantum many-body systems like XY spin chains, establishing bounds on Lieb-Robinson velocities.

Contribution

It proves ballistic transport for limit-periodic Jacobi matrices under rapid approximation and applies this to bound velocities in quantum spin chains.

Findings

Quantum dynamics are strongly ballistic under certain conditions.

Transport exponents for localized states are equal to one.

Lower bounds on Lieb-Robinson velocity are established for specific spin chains.

Abstract

We study Jacobi matrices that are uniformly approximated by periodic operators. We show that if the rate of approximation is sufficiently rapid, then the associated quantum dynamics are ballistic in a rather strong sense; namely, the (normalized) Heisenberg evolution of the position operator converges strongly to a self-adjoint operator that is injective on the space of absolutely summable sequences. In particular, this means that all transport exponents corresponding to well-localized initial states are equal to one. Our result may be applied to a class of quantum many-body problems. Specifically, we establish a lower bound on the Lieb--Robinson velocity for an isotropic XY spin chain on the integers with limit-periodic couplings.