Criticality of compact and noncompact quantum dissipative $Z_4$ models in $(1+1)$ dimensions

TL;DR

This study uses large-scale Monte Carlo simulations to compare the critical behavior of compact and noncompact quantum $Z_4$ models in (1+1) dimensions with Ohmic dissipation, revealing similar dynamical critical exponents.

Contribution

It provides the first detailed comparison of critical properties between compact and noncompact quantum $Z_4$ models with dissipation in (1+1)D, highlighting the irrelevance of variable compactness to critical scaling.

Findings

Both models exhibit a dynamical critical exponent z≈1.

The compact model has two phases separated by a critical line.

The noncompact model has three phases, including an intermediate quasi-long-range order phase.

Abstract

Using large-scale Monte Carlo computations, we study two versions of a $(1+1)D$ $Z_4$-symmetric model with Ohmic bond dissipation. In one of these versions, the variables are restricted to the interval $[0,2\pi>$, while the domain is unrestricted in the other version. The compact model features a completely ordered phase with a broken $Z_4$ symmetry and a disordered phase, separated by a critical line. The noncompact model features three phases. In addition to the two phases exhibited by the compact model, there is also an intermediate phase with isotropic quasi-long-range order. We calculate the dynamical critical exponent $z$ along the critical lines of both models to see if the compactness of the variable is relevant to the critical scaling between space and imaginary time. There appears to be no difference between the two models in that respect, and we find $z\approx1$ for the single phase transition in the compact model as well as for both transitions in the noncompact model.