Identification of topologically different valence bond states in spin ladders

TL;DR

This paper compares three methods for identifying topologically distinct valence-bond ground states in spin ladders, revealing their equivalence and connection to symmetry breaking phenomena.

Contribution

It introduces a proper bosonization of string order parameters and demonstrates the equivalence of three different methods for characterizing topological states in spin ladders.

Findings

The three methods are shown to be equivalent.

The methods reflect the $Z_2\times Z_2$ symmetry breaking.

A proper bosonization of string order parameters is proposed.

Abstract

We discuss relation between three different methods to identify topologically distinct short-range valence-bond ground states in spin-$\frac12$ two-leg ladders: the string order parameters, the level-crossing method, and the ground-state expectation value of the twist operator. For this purpose we reexamine a phase transition in the frustrated spin ladder. We propose a proper bosonization of the string order parameters. Then, these three methods are shown to be equivalent reflecting the $Z_2\times Z_2$ symmetry breaking of the order and the disorder fields of the two-dimensional Ising model.