Entropy growth of shift-invariant states on a quantum spin chain

TL;DR

This paper investigates the asymptotic behavior of entropy in pure shift-invariant states on quantum spin chains, revealing sublinear growth patterns and proposing models with various growth rates.

Contribution

It introduces a detailed analysis of entropy growth in shift-invariant quantum states, including classes derived from quasi-free states with complex interval structures.

Findings

Entropy $S_N$ grows slower than $(\log N)^2$ for finite unions of intervals.

Numerical evidence suggests $S_N$ behaves like $\log N$.

For infinitely many intervals, $S_N$ can grow as $N^\alpha$ with $\alpha$ in (0,1).

Abstract

We study the entropy of pure shift-invariant states on a quantum spin chain. Unlike the classical case, the local restrictions to intervals of length $N$ are typically mixed and have therefore a non-zero entropy $S_N$ which is, moreover, monotonically increasing in $N$. We are interested in the asymptotics of the total entropy. We investigate in detail a class of states derived from quasi-free states on a CAR algebra. These are characterised by a measurable subset of the unit interval. As the entropy density is known to vanishes, $S_N$ is sublinear in $N$. For states corresponding to unions of finitely many intervals, $S_N$ is shown to grow slower than $(\log N)^2$. Numerical calculations suggest a $\log N$ behaviour. For the case with infinitely many intervals, we present a class of states for which the entropy $S_N$ increases as $N^\alpha$ where $\alpha$ can take any value in $(0,1)$.