Universal behavior of the Shannon and R\'enyi mutual information of quantum critical chains

TL;DR

This paper investigates the universal scaling behavior of Shannon and Rényi mutual information in ground states of critical quantum spin chains, revealing basis-dependent relations to conformal field theory and central charge.

Contribution

It identifies a special conformal basis where mutual information scaling relates to the central charge, and explores basis dependence in various quantum spin models.

Findings

Mutual information in the conformal basis scales with the central charge.

Generic bases show no direct relation between scaling coefficients and central charge.

Numerical results support the basis-dependent scaling behavior across multiple models.

Abstract

We study the Shannon and R\'enyi mutual information (MI) in the ground state (GS) of different critical quantum spin chains. Despite the apparent basis dependence of these quantities we show the existence of some particular basis (we will call them conformal basis) whose finite-size scaling function is related to the central charge $c$ of the underlying conformal field theory of the model. In particular, we verified that for large index $n$, the MI of a subsystem of size $\ell$ in a periodic chain with $L$ sites behaves as $\frac{c}{4}\frac{n}{n-1}\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi \ell}{L})\Big{)}$, when the ground-state wavefunction is expressed in these special conformal basis. This is in agreement with recent predictions. For generic local basis we will show that, although in some cases $b_n\ln\Big{(}\frac{L}{\pi}\sin(\frac{\pi \ell}{L})\Big{)}$ is a good fit to our numerical data, in general there is no direct relation between $b_n$ and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, $Q=3,4$ quantum Potts chain, quantum Ashkin-Teller chain and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information ($n=1$), for the parafermionic $Z_Q$ quantum chains with $Q=5,6,7$ and $8$.