Phases and Phase Transitions of the Disordered Quantum Clock Model

TL;DR

This study investigates the phase diagram and critical behavior of a disordered quantum clock model, revealing how disorder influences phase transitions and identifying an infinite-randomness critical point at strong disorder.

Contribution

The paper provides the first large-scale Monte Carlo analysis of the disordered quantum clock model, detailing how disorder affects phase transitions and critical exponents.

Findings

Disorder shrinks the quasi-long-range ordered phase.

Weak disorder does not alter Berezinskii-Kosterlitz-Thouless transitions.

Strong disorder leads to an infinite-randomness critical point.

Abstract

We study the phases and phase transitions of a disordered one-dimensional quantum $q$-state clock Hamiltonian using large-scale Monte Carlo simulations. Making contact with earlier results, we confirm that the clean, translational invariant version of the model, for $q=6$, hosts an intermediate emergent quasi-long-range ordered (QLRO) phase between the symmetry-broken true long-range ordered (TLRO) phase and the disordered (paramagnetic) phase. With increasing disorder strength, the quasi-long-range ordered phase shrinks and finally vanishes at a multi-critical point, beyond which there is a direct transition from the TLRO phase to the paramagnetic phase. After establishing the phase diagram, we characterize the critical behaviors of the various quantum phase transitions in the model. We find that weak disorder is an irrelevant perturbation of the Berezinskii-Kosterlitz-Thouless transitions that separate the QLRO phase from the TLRO and paramagnetic phases. For stronger disorder, some of the critical exponents become disorder-dependent already before the system reaches the multicritical point. We also show that beyond the multicritical point, the direct transition from the TLRO phase to the paramagnetic phase is governed by an infinite-randomness critical point in line with strong-disorder renormalization group predictions. While our numerical results are for $q=6$, we expect the qualitative features of the behavior to hold for all $q>4$.