Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.07640 · Physical review. B./Physical review. B

Phases and phase transitions of the disordered quantum clock model

Pulloor Kuttanikkad Vishnu, Gaurav Khairnar, Rajesh Narayanan, Thomas Vojta

Published Mar 28, 2025• 3 citations
arXivHTMLPDF

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 multicritical 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 $q>4$.

DOI
10.1103/physrevb.111.094212

Related papers

True and quasi-long-range order in the generalized q -state clock model

Seung Ki Baek, Petter Minnhagen, Beom Jun Kim

From consideration of the order-parameter distribution, we propose an observable which makes a clear distinction between true and quasi-long-range orders in the two-dimensional generalized q-state clock model. Measuring this quantity by Monte Carlo simulations for q=8, we construct a phase diagram and identify critical properties across the phase-separation lines among the true long-range order, quasi-long-range order, and disorder. Our result supports the theoretical prediction that there appears a discontinuous order-disorder transition as soon as the two phase-separation lines merge.

arXiv:0912.2830•Physical Review E•Dec 4, 2009•29 citations

Two-Dimensional J 1 − J 2 Clock Model: Enhanced Symmetries, Emergent Orders, and Landau-Incompatible Transitions

Pulloor Kuttanikkad Vishnu, Abhishodh Prakash, Rajesh Narayanan +1 more

We present a comprehensive study on the frustrated J_{1}-J_{2} classical q-state clock model with even q>4 on a two-dimensional square lattice, revealing a rich ensemble of phases driven by competing interactions. In the unfrustrated regime (J_{1}>2J_{2}), the model reproduces the standard clock model phenomenology: a low-temperature Z_{q}-broken ferromagnet, an intermediate XY-like critical quasi-long-range-ordered phase with emergent U(1) symmetry, and a high-temperature paramagnet. For J_{1}<2J_{2}, frustration stabilizes five distinct regimes: the disordered paramagnet, a stripe-ordered phase breaking Z_{q}×Z_{2} symmetry, two Z_{2}-broken nematic phases (one with and one without quasi-long-range order), and an exotic stripe phase with emergent discrete Z_{q} spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden Z_{q} order emerges via a relevant operator in the infrared long-wavelength limit, rather than from an irrelevant perturbation, highlighting a nonstandard route to emergence. Using large-scale corner transfer matrix renormalization group calculations, complemented by classical Monte Carlo simulations, we map the complete phase diagram and identify Berezinskii-Kosterlitz-Thouless, Ising, first-order, and unconventional Landau-incompatible transitions between different phases. Finally, we propose an effective field-theoretic framework that encompasses these emergent orders and their interwoven transitions.

arXiv:2505.05194•Physical Review Letters•Nov 26, 2025•0 citations

Critical Properties of Random Quantum Potts and Clock Models

T. Senthil, Satya N. Majumdar

We study zero temperature phase transitions in two classes of random quantum systems---the $q$-state quantum Potts and clock models. For models with purely ferromagnetic interactions in one dimension, we show that for strong randomness there is a second order transition with critical properties that can be determined exactly by use of a renormalization group procedure. Somewhat surprisingly, the critical behavior is completely independent of $q$. For the $q>4$ clock model, we suggest the existence of a novel multicritical point at intermediate randomness. We also consider the $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ transition from a paramagnet to a spin glass in an infinite-range model, and find $q$ independent exponents.

arXiv:cond-mat/9511121•Physical Review Letters•Apr 15, 1996•83 citations

Monte Carlo simulations of the disordered three-color quantum Ashkin-Teller chain

Ahmed K. Ibrahim, Thomas Vojta

We investigate the zero-temperature quantum phase transitions of the disordered three-color quantum Ashkin-Teller spin chain by means of large-scale Monte Carlo simulations. We find that the first-order phase transitions of the clean system are rounded by the quenched disorder. For weak intercolor coupling, the resulting emergent quantum critical point between the paramagnetic phase and the magnetically ordered Baxter phase is of infinite-randomness type and belongs to the universality class of the random transverse-field Ising model, as predicted by recent strong-disorder renormalization group calculations. We also find evidence for unconventional critical behavior in the case of strong intercolor coupling, even though an unequivocal determination of the universality class is beyond our numerical capabilities. We compare our results to earlier simulations, and we discuss implications for the classification of phase transitions in the presence of disorder.

arXiv:1612.05617•Physical review. B./Physical review. B•Feb 1, 2017•4 citations

Phase transitions in the two-dimensional XY model with random phases: A Monte Carlo study

J. Maucourt, Daniel R. Grempel

We study the two-dimensional $\mathrm{XY}$ model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to $T=0$ in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.

arXiv:cond-mat/9703109•Physical review. B, Condensed matter•Aug 1, 1997•30 citations

Phase transitions in the q -state clock model

Arpita Goswami, Ravi Kumar, Monikana Gope +1 more

The q-state clock model, sometimes called the discrete XY model, is known to show a second-order (symmetry-breaking) phase transition in two dimensions for q≤4 (q=2 corresponds to the Ising model). On the other hand, the q→∞ limit of the model corresponds to the XY model, which shows the infinite order (non-symmetry-breaking) Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions. Interestingly, the two-dimensional clock model with q≥5 is predicted to show three different phases and two associated phase transitions. There are varying opinions about the actual characters of phases and the associated transitions. In this work, we develop the basic and higher-order mean-field (MF) theories to study the q-state clock model systematically. Our MF calculations reaffirm that, for large q, there are three phases: (broken) Z_{q} symmetric ferromagnetic phase at the low temperature, emergent U(1) symmetric BKT phase at the intermediate temperature, and paramagnetic (disordered) phase at the high temperature. The phase transition at the higher temperature is found to be of the BKT type, and the other transition at the lower temperature is argued to be a large-order spontaneous symmetry-breaking type (the largeness of transition order yields the possibility of having some of the numerical characteristics of a BKT transition). The higher-order MF theory developed here better characterizes phases by estimating the spin-spin correlation between two neighbors.

arXiv:2407.17507•Physical review. E•May 19, 2025•5 citations
We study the phases and phase transitions of a disordered one-dimensional quantum qqq-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=6q=6q=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 multicritical 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=6q=6q=6, we expect the qualitative features of the behavior to hold for all q>4q>4q>4.
From consideration of the order-parameter distribution, we propose an observable which makes a clear distinction between true and quasi-long-range orders in the two-dimensional generalized q-state clock model. Measuring this quantity by Monte Carlo simulations for q=8, we construct a phase diagram and identify critical properties across the phase-separation lines among the true long-range order, quasi-long-range order, and disorder. Our result supports the theoretical prediction that there appears a discontinuous order-disorder transition as soon as the two phase-separation lines merge.
We present a comprehensive study on the frustrated J_{1}-J_{2} classical q-state clock model with even q>4 on a two-dimensional square lattice, revealing a rich ensemble of phases driven by competing interactions. In the unfrustrated regime (J_{1}>2J_{2}), the model reproduces the standard clock model phenomenology: a low-temperature Z_{q}-broken ferromagnet, an intermediate XY-like critical quasi-long-range-ordered phase with emergent U(1) symmetry, and a high-temperature paramagnet. For J_{1}<2J_{2}, frustration stabilizes five distinct regimes: the disordered paramagnet, a stripe-ordered phase breaking Z_{q}×Z_{2} symmetry, two Z_{2}-broken nematic phases (one with and one without quasi-long-range order), and an exotic stripe phase with emergent discrete Z_{q} spin degrees of freedom prohibited in the microscopic Hamiltonian. Remarkably, this seemingly forbidden Z_{q} order emerges via a relevant operator in the infrared long-wavelength limit, rather than from an irrelevant perturbation, highlighting a nonstandard route to emergence. Using large-scale corner transfer matrix renormalization group calculations, complemented by classical Monte Carlo simulations, we map the complete phase diagram and identify Berezinskii-Kosterlitz-Thouless, Ising, first-order, and unconventional Landau-incompatible transitions between different phases. Finally, we propose an effective field-theoretic framework that encompasses these emergent orders and their interwoven transitions.
We study zero temperature phase transitions in two classes of random quantum systems---the qqq-state quantum Potts and clock models. For models with purely ferromagnetic interactions in one dimension, we show that for strong randomness there is a second order transition with critical properties that can be determined exactly by use of a renormalization group procedure. Somewhat surprisingly, the critical behavior is completely independent of qqq. For the q>4q>4q>4 clock model, we suggest the existence of a novel multicritical point at intermediate randomness. We also consider the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 transition from a paramagnet to a spin glass in an infinite-range model, and find qqq independent exponents.
We investigate the zero-temperature quantum phase transitions of the disordered three-color quantum Ashkin-Teller spin chain by means of large-scale Monte Carlo simulations. We find that the first-order phase transitions of the clean system are rounded by the quenched disorder. For weak intercolor coupling, the resulting emergent quantum critical point between the paramagnetic phase and the magnetically ordered Baxter phase is of infinite-randomness type and belongs to the universality class of the random transverse-field Ising model, as predicted by recent strong-disorder renormalization group calculations. We also find evidence for unconventional critical behavior in the case of strong intercolor coupling, even though an unequivocal determination of the universality class is beyond our numerical capabilities. We compare our results to earlier simulations, and we discuss implications for the classification of phase transitions in the presence of disorder.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
The q-state clock model, sometimes called the discrete XY model, is known to show a second-order (symmetry-breaking) phase transition in two dimensions for q≤4 (q=2 corresponds to the Ising model). On the other hand, the q→∞ limit of the model corresponds to the XY model, which shows the infinite order (non-symmetry-breaking) Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two dimensions. Interestingly, the two-dimensional clock model with q≥5 is predicted to show three different phases and two associated phase transitions. There are varying opinions about the actual characters of phases and the associated transitions. In this work, we develop the basic and higher-order mean-field (MF) theories to study the q-state clock model systematically. Our MF calculations reaffirm that, for large q, there are three phases: (broken) Z_{q} symmetric ferromagnetic phase at the low temperature, emergent U(1) symmetric BKT phase at the intermediate temperature, and paramagnetic (disordered) phase at the high temperature. The phase transition at the higher temperature is found to be of the BKT type, and the other transition at the lower temperature is argued to be a large-order spontaneous symmetry-breaking type (the largeness of transition order yields the possibility of having some of the numerical characteristics of a BKT transition). The higher-order MF theory developed here better characterizes phases by estimating the spin-spin correlation between two neighbors.