Klein bottle entropy of compactified boson conformal field theory

TL;DR

This paper extends the concept of Klein bottle entropy to the compactified boson conformal field theory, providing a new method to determine the Luttinger parameter from lattice models with numerical validation.

Contribution

It generalizes Klein bottle entropy to the compactified boson CFT and establishes a direct relation to the compactification radius, enabling efficient extraction of the Luttinger parameter.

Findings

Numerical results agree with CFT predictions for the XXZ chain.

The relation $ ln g = ln R$ simplifies Luttinger parameter extraction.

Provides accurate Luttinger parameter estimates for the S=1 XXZ chain.

Abstract

We extend the scope of the Klein bottle entropy, originally introduced by [Tu, Phys. Rev. Lett. 119, 261603 (2017)] in the rational conformal field theory (CFT), to the compactified boson CFT, which are relevant to the studies of Luttinger liquids. We first review the Klein bottle entropy in rational CFT and discuss details of how to extract the Klein bottle entropy from lattice models using the example of the transverse field Ising model. We then go beyond the scope of rational CFT and study the Klein bottle entropy $\ln g$ in the compactified boson CFT, which turns out to have a straightforward relation to the compactification radius $R$, $\ln g = \ln R$. This relation indicates a convenient and efficient method to extract the Luttinger parameter from lattice model calculations. Our numerical results on the Klein bottle entropy in the spin-$1/2$ XXZ chain show excellent agreement with the CFT predictions, up to some small deviations near the isotropic point, which we attribute to the marginally irrelevant terms. For the $S = 1$ XXZ chain that cannot be exactly solved, our numerical results provide an accurate numerical determination of the Luttinger parameter in this model.