$BKT$ transitions in classical and quantum long-range systems

TL;DR

This paper investigates the critical behavior and phase transitions in classical and quantum long-range interacting systems, focusing on the BKT universality class, using field theory, approximations, and applying findings to quantum spin chains.

Contribution

It provides a comprehensive field theoretical analysis of long-range 2D XY models, compares methods, and explores implications for quantum spin chains, revealing complex phase diagrams.

Findings

Rich phase diagram with power-law BKT scaling and symmetry breaking

Failure of Villain approximation in the long-range regime

Predictions for order parameter scaling near quantum transitions

Abstract

In the past decades considerable efforts have been made in order to understand the critical features of both classical and quantum long-range interacting models. The case of the Berezinskii-Kosterlitz-Thouless (BKT) universality class, as in the $2d$ classical $XY$ model, is considerably complicated by the presence, for short-range interactions, of a line of renormalization group fixed points. In this paper we discuss a field theoretical treatment of the $2d$ $XY$ model with long-range couplings and we compare it with results from the self-consistent harmonic approximation. These methods lead to a rich phase diagram, where both power-law BKT scaling and spontaneous symmetry breaking appear for the same (intermediate) decay rates of long-range interactions. We also discuss the Villain approximation for the $2d$ $XY$ model with power-law couplings, providing hints that, in the long-range regime, it fails to reproduce the correct critical behavior. The obtained results are then applied to the long-range quantum XXZ spin chain at zero temperature. We discuss the relation between the phase diagrams of the two models and we give predictions about the scaling of the order parameter of the quantum chain close to the transition.