Conformal properties of 1D quantum systems with long-range interactions

TL;DR

This paper explores the conformal properties of one-dimensional quantum systems with long-range Coulomb-like interactions, revealing how the conformal anomaly varies with interaction decay and how correlations are affected by system deformations.

Contribution

It provides a detailed analysis of the conformal anomaly as a function of the decay parameter and studies the impact of deformations on correlation length in long-range interacting systems.

Findings

Conformal anomaly c=0 for σ<1, indicating a Wigner crystal phase.

Conformal anomaly c=1 for σ>1, indicating a Luttinger liquid phase.

Correlation length is modified by interactions for σ>1, requiring redefinition to maintain physical consistency.

Abstract

We investigate conformal properties of a one-dimensional quantum system with a long-range, Coulomb-like potential of the form $\frac{1}{|x|^{\sigma}}$, with $\sigma >0$. We compute the conformal anomaly $c$ as function of $\sigma$. We obtain $c=0$ for $\sigma <1$ (Wigner crystal) and $c=1$ for $\sigma >1$ (Luttinger liquid). By studying the scaling regime of the system when it is deformed by the inclusion of a term $t \rho(x)$, (with $\rho(x)$ a scaling operator), we show that, for $\sigma >1$, the correlation length $\xi$ gets an additional interaction dependent factor. This leads to a necessary redefinition of $\xi$ in order to avoid an unphysical dependence of the central charge on the coupling constant.