Truncation effects in the charge representation of the O(2) model

TL;DR

This paper investigates how truncating the charge representation in the O(2) model affects its quantum phase transition, revealing that even minimal truncation preserves the transition's nature and critical properties.

Contribution

It introduces a dual charge representation with spin truncation, showing the persistence of the phase transition and analyzing the critical behavior for different truncation levels.

Findings

The phase transition persists for the smallest spin truncation S=1.

For S=1, the transition is an infinite-order Gaussian transition with BKT-like exponents.

True BKT transitions occur for S≥2, with different singularity behaviors.

Abstract

The O(2) model in Euclidean space-time is the zero-gauge-coupling limit of the compact scalar quantum electrodynamics. We obtain a dual representation of it called the charge representation. We study the quantum phase transition in the charge representation with a truncation to ``spin $S$," where the quantum numbers have an absolute value less than or equal to $S$. The charge representation preserves the gapless-to-gapped phase transition even for the smallest spin truncation $S = 1$. The phase transition for $S = 1$ is an infinite-order Gaussian transition with the same critical exponents $\delta$ and $\eta$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for $S \ge 2$. The essential singularity in the correlation length for $S = 1$ is different from that for $S \ge 2$. The exponential convergence of the phase-transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated $\hat{U}^{\pm} = \exp(\pm i \hat{\theta})$ operators by the spin ladder operators $\hat{S}^{\pm}$ in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for $S = 1$, which allows us to extract the $\eta$ exponent with high accuracy.