On Anomalous Lieb-Robinson Bounds for the Fibonacci XY Chain

TL;DR

This paper establishes a new type of anomalous Lieb-Robinson bounds for the Fibonacci XY chain, showing decay in correlations is sub-ballistic and depends on the spectral properties of the associated Fibonacci Hamiltonian.

Contribution

It introduces a novel anomalous Lieb-Robinson bound with decay depending on a parameter related to the Fibonacci Hamiltonian's transport exponent, extending previous bounds to Fibonacci systems.

Findings

Lieb-Robinson bounds are sub-ballistic with decay in x-vt^α

Bounds hold for α exceeding the Fibonacci Hamiltonian's upper transport exponent

Method does not extend to power-law bounds in the random dimer model

Abstract

We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in $x-vt$ is replaced by exponential decay in $x-vt^\alpha$ with $0<\alpha<1$. In fact, we can characterize the values of $\alpha$ for which such a bound holds as those exceeding $\alpha_u^+$, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in \cite{DT07, DT08, D05, DGY} to our purposes. We also explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.