Valence Bond Entanglement and Fluctuations in Random Singlet Phases

TL;DR

This paper investigates valence bond fluctuations in disordered quantum spin chains, revealing a crossover from uniform-like behavior to random singlet states, and analyzes how disorder influences this transition.

Contribution

It introduces a numerical method to study valence bond fluctuations in weakly disordered systems and characterizes the crossover to random singlet phases through scaling analysis.

Findings

Fluctuations saturate at large block sizes in the random singlet phase.

A crossover length scale depends on disorder strength.

Results apply to various models including Heisenberg and Ising chains.

Abstract

The ground state of the uniform antiferromagnetic spin-1/2 Heisenberg chain can be viewed as a strongly fluctuating liquid of valence bonds, while in disordered chains these bonds lock into random singlet states on long length scales. We show that this phenomenon can be studied numerically, even in the case of weak disorder, by calculating the mean value of the number of valence bonds leaving a block of $L$ contiguous spins (the valence-bond entanglement entropy) as well as the fluctuations in this number. These fluctuations show a clear crossover from a small $L$ regime, in which they behave similar to those of the uniform model, to a large $L$ regime in which they saturate in a way consistent with the formation of a random singlet state on long length scales. A scaling analysis of these fluctuations is used to study the dependence on disorder strength of the length scale characterizing the crossover between these two regimes. Results are obtained for a class of models which include, in addition to the spin-1/2 Heisenberg chain, the uniform and disordered critical 1D transverse-field Ising model and chains of interacting non-Abelian anyons.