Unconventional U(1) to $\mathbf{Z_q}$ cross-over in quantum and classical ${\bf q}$-state clock models

Abstract

We consider two-dimensional q-state quantum clock models with quantum fluctuations connecting states with all-to-all clock transitions with different choices for the matrix elements. We study the quantum phase transitions in these models using quantum Monte Carlo simulations and finite-size scaling, with the aim of characterizing the cross-over from emergent U(1) symmetry at the transition (for q ≥ 4) to Zq symmetry of the ordered state. We also study classical three-dimensional clock models with spatial anisotropy corresponding to the space-time anisotropy of the quantum systems. The U(1) to Zq symmetry cross-over in all these systems is governed by a so-called dangerously irrelevant operator. We specifically study q = 5 and q = 6 models with different forms of the quantum fluctuations and different anisotropies in the classical models. In all cases, we find the expected classical XY critical exponents and scaling dimensions yq of the clock fields. However, the initial weak violation of the U(1) symmetry in the ordered phase, characterized by a Zq symmetric order parameter φq, scales in an unexpected way. As a function of the system size (length) L, close to the critical temperature φq ∝ L p , where the known value of the exponent is p = 2 in the classical isotropic clock model. In contrast, for strongly anisotropic classical models and the quantum models we find p = 3. For weakly anisotropic classical models we observe a cross-over from p = 2 to p = 3 scaling. The exponent p directly impacts the exponent ν governing the divergence of the U(1) to Zq cross-over length scale ξ in the thermodynamic limit, according to the relationship ν = ν(1 + |yq|/p), where ν is the conventional correlation length exponent. We present a phenomenological argument for p = 3 based on an anomalous renormalization of the clock field in the presence of anisotropy, possibly as a consequence of topological (vortex) line defects. Thus, our study points to an intriguing interplay between conventional and dangerously irrelevant perturbations, which may affect also other quantum systems with emergent symmetries.

$\mathbb{Z}_N$ lattice gauge theory in a ladder geometry

“…This causes indeed the appearance of a crossover between a U(1) symmetric and a Z N symmetric regime in the ordered phase. This analysis has been numerically well-verified for the classical 3D model (see, for example, [77,79]), and it has been recently confirmed also for the quantum 2D case [80], thus leading to the conclusion that no gapless phase exists for the clock limit g → 0 and the dual pure LGT λ → 0 [77].…”

“…In three dimensions, it is indeed known that the classical clock model does not display an extended gapless phase due to the P background terms being "dangerously irrelevant" perturbations and the system displays only a single phase transition in the 3D XY model universality class. This is due to the fact that the irrelevant P operators become relevant when approaching the U (1) symmetric gapless Nambu-Goldstone fixed point [78], thus causing a second step in the RG flow towards the ordered ferromagnetic phase of the clock model (see the phase diagrams evaluated in [79,80]). As a consequence, the system is characterized by two different length scales ξ < ξ , both diverging at the critical point, which are associated with the onset of U (1) and Z N symmetric features respectively.…”

Higher-charge three-dimensional compact lattice Abelian-Higgs models

“…For larger values of q, we expect a continuous transition. It belongs to the Ising universality class for q = 4 [64], and to the 3D XY universality class for q ≥ 5 [64][65][66].…”