Extraction of topological information in Tomonaga-Luttinger liquids

TL;DR

This paper introduces a method to extract topological information from Tomonaga-Luttinger liquids using the expectation values of a twist operator, revealing universal values at fixed points and phase transitions.

Contribution

It demonstrates that the expectation value of the twist operator uniquely identifies topological phases and phase transitions in one-dimensional correlated systems, providing a new topological characterization tool.

Findings

Universal values of ±1/2 at TL fixed points

Discontinuous changes at phase transitions

Topological phase identification in various systems

Abstract

We discuss expectation values of the twist operator $U$ appearing in the Lieb-Schultz-Mattis theorem (or the polarization operator for periodic systems) in excited states of the one-dimensional correlated systems $z_L^{(q,\pm)}\equiv\braket{\Psi_{q/2}^{\pm}|U^q|\Psi_{q/2}^{\pm}}$, where $\ket{\Psi_{p}^{\pm}}$ denotes the excited states given by linear combinations of momentum $2pk_{\rm F}$ with parity $\pm 1$. We found that $z_L^{(q,\pm)}$ gives universal values $\pm 1/2$ on the Tomonaga-Luttinger (TL) fixed point, and its signs identify the topology of the dominant phases. Therefore, this expectation value changes between $\pm 1/2$ discontinuously at a phase transition point with the U(1) or SU(2) symmetric Gaussian universality class. This means that $z_L^{(q,\pm)}$ extracts the topological information of TL liquids. We explain these results based on the free-fermion picture and the bosonization theory, and also demonstrate them in several physical systems.