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

Abstract

We consider two-dimensional $q$-state quantum clock models with quantum fluctuations connecting states with clock transitions with different choices for matrix elements. We study the quantum phase transitions in these models using quantum Monte Carlo simulations, with the aim of characterizing the cross-over from emergent U(1) symmetry at the transition (for $q\ge 4$) to $Z_q$ 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 $Z_q$ symmetry cross-over in all these systems is governed by a 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. We find the expected classical XY critical exponents and scaling dimensions $y_q$ of the clock fields. However, the initial weak violation of the U(1) symmetry in the ordered phase, characterized by a $Z_q$ symmetric order parameter ${\varphi }_q$, scales in an unexpected way. As a function of the system size $L$, close to the critical temperature ${\varphi }_q\propto L^p$, 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 ${\nu }^{\prime }$ governing the divergence of the U(1) to $Z_q$ cross-over length scale ${\xi }^{\prime }$ in the thermodynamic limit, according to the relationship ${\nu }^{\prime }=\nu (1+|y_q|/p)$, where $\nu$ is the conventional correlation length exponent. We present a phenomenological argument based on an anomalous renormalization of the clock field in the presence of anisotropy, possibly as a consequence of topological (vortex) line defects.

arXiv commentDue to small technical error pointed out by readers, we have replaced page 20, left column, last sentence "These operators instead ... is not necessary)."

References in corpus (72)

• "Deconfined" quantum critical points
• Emergent electrodynamics of skyrmions in a chiral magnet
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry
• Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions
• Computational Studies of Quantum Spin Systems
• Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State
• Critical behavior of the three-dimensional XY universality class
• Deconfined quantum critical points: symmetries and dualities
• Topological phases with parafermions: theory and blueprints
• The Loop Algorithm
• Quantum criticality with two length scales
• Walking, Weak first-order transitions, and Complex CFTs
• Antiferromagnetic to valence-bond-soild transitions in two-dimensional SU(N) Heisenberg models with multi-spin interactions
• Emergent $SO(5)$ Symmetry at the Néel to Valence-Bond-Solid Transition
• Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models
• Stochastic series expansion method for quantum Ising models with arbitrary interactions
• Topological defects as relics of emergent continuous symmetry and Higgs condensation of disorder in ferroelectrics
• Carving out OPE space and precise $O(2)$ model critical exponents
• Deconfined quantum criticality and Neel order via dimer disorder
• The sign problem in Monte Carlo simulations of frustrated quantum spin systems
• Anisotropic perturbations in three-dimensional O(N)-symmetric vector models
• From an Antiferromagnet to a Valence Bond Solid: Evidence for a First Order Phase Transition
• Monte Carlo study of an improved clock model in three dimensions
• Néel to valence-bond solid transition on the honeycomb lattice: Evidence for deconfined criticality
• Conformal Bootstrap Dashing Hopes of Emergent Symmetry
• Confinement transition of ${\mathbb{Z}}_2$ gauge theories coupled to massless fermions: emergent QCD${}_3$ and $SO(5)$ symmetry
• Conformal dimensions via large charge expansion
• Superconductivity from the Condensation of Topological Defects in a Quantum Spin-Hall Insulator
• Emergent $\mathrm{SU}(4)$ Symmetry in $\alpha$-ZrCl${}_3$ and Crystalline Spin-Orbital Liquids
• Quantum field theory for the chiral clock transition in one spatial dimension
• Critical Properties of Random Quantum Potts and Clock Models
• Dirac Fermions with Competing Orders: Non-Landau Transition with Emergent Symmetry
• Emergence of U(1) symmetry in the 3D XY model with Zq anisotropy
• Ordered phase and scaling in $Z_n$ models and the three-state antiferromagnetic Potts model in three dimensions
• Properties of Resonating-Valence-Bond Spin Liquids and Critical Dimer Models
• Emergence and spontaneous breaking of approximate O(4) symmetry at a weakly first-order deconfined phase transition
• Symmetry enhanced first-order phase transition in a two-dimensional quantum magnet
• Generic mixed columnar-plaquette phases in Rokhsar-Kivelson models
• Transitions to valence-bond solid order in a honeycomb lattice antiferromagnet
• Sweeping cluster algorithm for quantum spin systems with strong geometric restrictions
• Critical exponents can be different on the two sides of a transition: A generic mechanism
• Continuous easy-plane deconfined phase transition on the kagome lattice
• The plaquette phase of the square lattice quantum dimer model
• Critical Correlations for Short-Range Valence-Bond Wave Functions on the Square Lattice
• Anomalous quantum-critical scaling corrections in two-dimensional antiferromagnets
• Emergent SO(5) symmetry at the columnar ordering transition in the classical cubic dimer model
• Disorder induced cross-over effects at quantum critical points
• Phase-coherence and the boson-analogy of vortex liquids
• Critical Binder cumulant for isotropic Ising models on square and triangular lattices
• Criticality versus q in the 2+1-dimensional $Z_q$ clock model
• Berezinskii-Kosterlitz-Thouless transition on regular and Villain types of $q$-state clock models
• Level spectroscopy in a two-dimensional quantum magnet: Linearly dispersing spinons at the deconfined quantum critical point
• Nature of finite-temperature transition in anisotropic pyrochlore Er2Ti2O7
• Monte Carlo Renormalization Flows in the Space of Relevant and Irrelevant Operators: Application to Three-Dimensional Clock Models
• Continuous-time Diffusion Monte Carlo and the Quantum Dimer Model
• Emergent Symmetry and Dimensional Reduction at a Quantum Critical Point
• Widely existing mixed phase structure of quantum dimer model on square lattice
• Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet
• Finite-Volume Energy Spectrum, Fractionalized Strings, and Low-Energy Effective Field Theory for the Quantum Dimer Model on the Square Lattice
• Evidence of columnar order in the fully frustrated transverse field Ising model on the square lattice
• A Scaling Relation for Dangerously Irrelevant Symmetry-Breaking Fields
• Disordered bosons in one dimension: from weak to strong randomness criticality
• Finite-Scale Emergence of 2+1D Supersymmetry at First-Order Quantum Phase Transition
• Methods to determine the Hausdorff dimension of vortex loops in the three-dimensional XY model
• Correlated valence-bond states
• Three-dimensional vortex structures and dynamics in hexagonal manganites
• Emergent chiral symmetry in a three-dimensional interacting Dirac liquid
• Resonating valence-bond physics on the honeycomb lattice
• Emergent topological excitations in a two-dimensional quantum spin system
• Criticality versus q in the 2+1-dimensional $Z_q$ clock model
• Monte Carlo simulation of quantum Potts model

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• Widely existing mixed phase structure of quantum dimer model on square lattice
• Emergent O(4) symmetry at the phase transition from plaquette-singlet to antiferromagnetic order in quasi-two-dimensional quantum magnets
• Nonequilibrium dynamics in deconfined quantum critical point revealed by imaginary-time evolution
• Three-dimensional lattice SU($N_c$) gauge theories with multiflavor scalar fields in the adjoint representation
• Valley-polarized nematic order in twisted moiré systems: In-plane orbital magnetism and non-Fermi liquid to Fermi liquid crossover
• Classification of electronic nematicity in three-dimensional crystals and quasicrystals
• Equilibration of Topological Defects Near the Deconfined Quantum Multicritical Point
• Dual dynamic scaling in deconfined quantum criticality
• Multicomponent gauge-Higgs models with discrete Abelian gauge groups
• A critical nematic phase with pseudogap-like behavior in twisted bilayers
• Strong-coupling expansion of multi-band interacting models: mapping onto the transverse-field $J_1$-$J_2$ Ising model
• Field digitization scaling in a ${\mathbb{Z}}_N\subset U(1)$ symmetric model
• Walking behavior induced by $\mathcal{P}\mathcal{T}$ symmetry breaking in a non-Hermitian $\mathrm{XY}$ model with clock anisotropy
• Relaxation Critical Dynamics with Emergent Symmetry
• Finite-time scaling with two characteristic time scales: Driven critical dynamics with emergent symmetry
• Scalar gauge-Higgs models with discrete Abelian symmetry groups
• Qubit encodings for lattices of dipolar planar rotors