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:1901.04818 · Physical review. B./Physical review. B

Nonequilibrium critical dynamics in the quantum chiral clock model

Rui-Zhen Huang, Shuai Yin

Published May 14, 2019• 11 citations
arXivHTMLPDF

Abstract

In this paper we study the driven critical dynamics in the three-state quantum chiral clock model. This is motivated by a recent experiment [Keesling et al., Nature 568, 207 (2019)], which verified the Kibble-Zurek mechanism and the finite-time scaling in a reconfigurable one-dimensional array of $^{87}\mathrm{Rb}$ atoms with programmable interactions. This experimental model has the same universality class as the quantum chiral clock model and has been shown to possess a nontrivial noninteger dynamic exponent $z$. Besides the case of changing the transverse field as realized in the experiment, we also consider the driven dynamics under changing longitudinal field. For both cases, we verify the finite-time scaling for a noninteger dynamic exponent $z$. Furthermore, we determine the critical exponents $\ensuremath{\beta}$ and $\ensuremath{\delta}$ numerically. We also investigate the dynamic scaling behavior including the thermal effects, which are inevitably involved in experiments. From a nonequilibrium dynamic point of view, our results strongly support a direct continuous phase transition between the ordered phase and the disordered phase. Also, we show that the method based on the finite-time scaling theory provides a promising approach to determine the critical point and critical properties.

DOI
10.1103/physrevb.99.184104

Related papers

Probing quantum critical dynamics on a programmable Rydberg simulator

Alexander Keesling, Ahmed Omran, Harry Levine +12 more

Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. The resulting quantum criticality has been explored by probing linear response for systems near thermal equilibrium. While time dependent phenomena associated with classical phase transitions have been studied in various scientific fields, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is of fundamental importance both for exploring novel approaches to quantum information processing and realizing exotic new phases of matter. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT at variable speeds, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) for an Ising-type QPT, explore scaling universality, and observe corrections beyond simple QKZM predictions. This approach is subsequently used to investigate novel QPTs associated with chiral clock model providing new insights into exotic systems, and opening the door for precision studies of critical phenomena and applications to quantum optimization.

•Sep 14, 2018•3 citations

Numerical study of the chiral Z 3 quantum phase transition in one spatial dimension

Rhine Samajdar, Soonwon Choi, Hannes Pichler +2 more

Recent experiments on a one-dimensional chain of trapped alkali-metal atoms [Bernien et al., Nature (London) 551, 579 (2017)] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous ${\mathbb{Z}}_{3}$ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley et al. [Phys. Rev. B 69, 075106 (2004)]. By symmetry arguments, the transition is expected to be in the universality class of the ${\mathbb{Z}}_{3}$ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order--disorder transition in these models and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization-group techniques. We use finite-size scaling to determine the dynamical critical exponent $z$ and the correlation length exponent $\ensuremath{\nu}$. Our analysis presents the only known instance of a strongly coupled generic transition between gapped states with $z\ensuremath{\ne}1$, implying an underlying nonconformal critical-field theory.

arXiv:1806.01867•Physical Review A•Aug 13, 2018•107 citations

Resolving chiral transitions in one-dimensional Rydberg arrays with quantum Kibble-Zurek mechanism and finite-time scaling

Jose Soto Garcia, Natalia Chepiga

The experimental realization of the quantum Kibble-Zurek mechanism in arrays of trapped Rydberg atoms has brought the problem of commensurate-incommensurate transition back into the focus of active research. Relying on equilibrium simulations of finite intervals, direct chiral transitions at the boundary of the period-3 and period-4 phases have been predicted. Here, we study how these chiral transitions can be diagnosed experimentally with critical dynamics. We demonstrate that chiral transitions can be distinguished from the floating phases by comparing Kibble-Zurek dynamics on arrays with different numbers of atoms. Furthermore, by sweeping in the opposite direction and keeping track of the order parameter, we identify the location of conformal points. Finally, combining forward and backward sweeps, we extract all critical exponents characterizing the transition.

arXiv:2403.03081•Physical review. B./Physical review. B•Sep 9, 2024•12 citations

Finite-time scaling with two characteristic time scales: Driven critical dynamics with emergent symmetry

Yu-Rong Shu, Liying Yang, Shuai Yin

Critical points with emergent symmetry exhibit intriguing scaling properties induced by two divergent length scales, attracting extensive investigations recently. We study the driven critical dynamics in a three-dimensional $q$-state clock model, in which the ordered phase breaks the $Z_q$ discrete symmetry, while an emergent $U(1)$ symmetry appears at the critical point. By increasing the temperature at a finite velocity $v$ to traverse the critical point from the ordered phase, we uncover rich dynamic scaling properties beyond the celebrated Kibble-Zurek mechanism. Our findings reveal the existence of two finite-time scaling (FTS) regions, characterized by two driving-induced time scales $ζ_d\propto v^{-z/r}$ and $ζ_d'\propto v^{-z/r'}$, respectively. Here $z$ is the dynamic exponent, $r$ is the usual critical exponent of $v$, and $r'$ represents an additional critical exponent of $v$ associated with the dangerously irrelevant scaling variable. While the square of the order parameter $M^2$ obeys the usual FTS form, the angular order parameter $ϕ_q$ shows remarkably distinct scaling behaviors controlled by both FTS regions. For small $v$, $ϕ_q$ is dominated by the time scale $ζ_d$, whereas for large $v$, $ϕ_q$ is governed by the second time scale $ζ_d'$. We verify the universality of these scaling properties in models with both isotropic and anisotropic couplings. Our theoretical insights provide a promising foundation for further experimental investigations in the hexagonal RMnO$_3$ (R=rare earth) materials.

arXiv:2503.16796•Physical review. B./Physical review. B•Mar 20, 2026•2 citations

Universal Driven Critical Dynamics of Entanglement Entropy

Chang-Yu Shen, Shuai Yin, Zi-Xiang Li

The Kibble-Zurek mechanism (KZM) and finite-time scaling (FTS) provide a foundational framework for driven critical dynamics, yet their predictive power has been largely confined to local observables. Here, we establish a universal finite-time scaling theory for the nonequilibrium dynamics of quantum entanglement. Using unbiased quantum Monte Carlo simulations, we investigate the corner entanglement entropy of (2+1)-dimensional interacting Dirac fermions driven from ordered phases toward a quantum critical point. We find that the corner entanglement accurately obeys a universal driven scaling governed by the driving rate and system size, persisting whether the initial ordered state is fully gapped or hosts gapless Goldstone modes. Crucially, this dynamical entanglement exhibits a logarithmic dependence on the driving rate, from which the universal corner coefficient of the underlying conformal field theory can be robustly extracted far from equilibrium. These results generalize the KZM from local observables to the intrinsic nonlocal quantum information measures, offering a practical blueprint for characterizing quantum criticality and entanglement on programmable quantum simulators.

cond-mat.str-elCross-listedarXiv:2609.03854•Sep 4, 2026•0 citations

Quantum field theory for the chiral clock transition in one spatial dimension

Seth Whitsitt, Rhine Samajdar, Subir Sachdev A

We describe the quantum phase transition in the $N$-state chiral clock model in spatial dimension $d=1$. With couplings chosen to preserve time-reversal and spatial inversion symmetries, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For such couplings and $N=3$, the clock model is expected to have a direct phase transition from a gapped phase with a broken global ${\mathbb{Z}}_{N}$ symmetry, to a gapped phase with the ${\mathbb{Z}}_{N}$ symmetry restored. The transition has dynamical critical exponent $z\ensuremath{\ne}1$, and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in $d=1$, involving the onset of a single-boson condensate in the background of a higher-dimensional $N$-boson condensate. We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in $2\ensuremath{-}d$, with $4\ensuremath{-}N$ chosen to be of order $2\ensuremath{-}d$. At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for $N=3$, finding good evidence for a direct phase transition, and obtain estimates for $z$ and the correlation length exponent $\ensuremath{\nu}$.

arXiv:1808.07056•Physical review. B./Physical review. B•Nov 9, 2018•81 citations
In this paper we study the driven critical dynamics in the three-state quantum chiral clock model. This is motivated by a recent experiment [Keesling et al., Nature 568, 207 (2019)], which verified the Kibble-Zurek mechanism and the finite-time scaling in a reconfigurable one-dimensional array of 87Rb^{87}\mathrm{Rb}87Rb atoms with programmable interactions. This experimental model has the same universality class as the quantum chiral clock model and has been shown to possess a nontrivial noninteger dynamic exponent zzz. Besides the case of changing the transverse field as realized in the experiment, we also consider the driven dynamics under changing longitudinal field. For both cases, we verify the finite-time scaling for a noninteger dynamic exponent zzz. Furthermore, we determine the critical exponents \ensuremathβ\ensuremath{\beta}\ensuremathβ and \ensuremathδ\ensuremath{\delta}\ensuremathδ numerically. We also investigate the dynamic scaling behavior including the thermal effects, which are inevitably involved in experiments. From a nonequilibrium dynamic point of view, our results strongly support a direct continuous phase transition between the ordered phase and the disordered phase. Also, we show that the method based on the finite-time scaling theory provides a promising approach to determine the critical point and critical properties.
Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. The resulting quantum criticality has been explored by probing linear response for systems near thermal equilibrium. While time dependent phenomena associated with classical phase transitions have been studied in various scientific fields, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is of fundamental importance both for exploring novel approaches to quantum information processing and realizing exotic new phases of matter. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT at variable speeds, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) for an Ising-type QPT, explore scaling universality, and observe corrections beyond simple QKZM predictions. This approach is subsequently used to investigate novel QPTs associated with chiral clock model providing new insights into exotic systems, and opening the door for precision studies of critical phenomena and applications to quantum optimization.
Recent experiments on a one-dimensional chain of trapped alkali-metal atoms [Bernien et al., Nature (London) 551, 579 (2017)] have observed a quantum transition associated with the onset of period-3 ordering of pumped Rydberg states. This spontaneous Z3{\mathbb{Z}}_{3}Z3​ symmetry breaking is described by a constrained model of hard-core bosons proposed by Fendley et al. [Phys. Rev. B 69, 075106 (2004)]. By symmetry arguments, the transition is expected to be in the universality class of the Z3{\mathbb{Z}}_{3}Z3​ chiral clock model with parameters preserving both time-reversal and spatial-inversion symmetries. We study the nature of the order--disorder transition in these models and numerically calculate its critical exponents with exact diagonalization and density-matrix renormalization-group techniques. We use finite-size scaling to determine the dynamical critical exponent zzz and the correlation length exponent $\ensuremath{\nu}$. Our analysis presents the only known instance of a strongly coupled generic transition between gapped states with $z\ensuremath{\ne}1$, implying an underlying nonconformal critical-field theory.
The experimental realization of the quantum Kibble-Zurek mechanism in arrays of trapped Rydberg atoms has brought the problem of commensurate-incommensurate transition back into the focus of active research. Relying on equilibrium simulations of finite intervals, direct chiral transitions at the boundary of the period-3 and period-4 phases have been predicted. Here, we study how these chiral transitions can be diagnosed experimentally with critical dynamics. We demonstrate that chiral transitions can be distinguished from the floating phases by comparing Kibble-Zurek dynamics on arrays with different numbers of atoms. Furthermore, by sweeping in the opposite direction and keeping track of the order parameter, we identify the location of conformal points. Finally, combining forward and backward sweeps, we extract all critical exponents characterizing the transition.
Critical points with emergent symmetry exhibit intriguing scaling properties induced by two divergent length scales, attracting extensive investigations recently. We study the driven critical dynamics in a three-dimensional qqq-state clock model, in which the ordered phase breaks the ZqZ_qZq​ discrete symmetry, while an emergent U(1)U(1)U(1) symmetry appears at the critical point. By increasing the temperature at a finite velocity vvv to traverse the critical point from the ordered phase, we uncover rich dynamic scaling properties beyond the celebrated Kibble-Zurek mechanism. Our findings reveal the existence of two finite-time scaling (FTS) regions, characterized by two driving-induced time scales ζd∝v−z/rζ_d\propto v^{-z/r}ζd​∝v−z/r and ζd′∝v−z/r′ζ_d'\propto v^{-z/r'}ζd′​∝v−z/r′, respectively. Here zzz is the dynamic exponent, rrr is the usual critical exponent of vvv, and r′r'r′ represents an additional critical exponent of vvv associated with the dangerously irrelevant scaling variable. While the square of the order parameter M2M^2M2 obeys the usual FTS form, the angular order parameter ϕqϕ_qϕq​ shows remarkably distinct scaling behaviors controlled by both FTS regions. For small vvv, ϕqϕ_qϕq​ is dominated by the time scale ζdζ_dζd​, whereas for large vvv, ϕqϕ_qϕq​ is governed by the second time scale ζd′ζ_d'ζd′​. We verify the universality of these scaling properties in models with both isotropic and anisotropic couplings. Our theoretical insights provide a promising foundation for further experimental investigations in the hexagonal RMnO3_33​ (R=rare earth) materials.
The Kibble-Zurek mechanism (KZM) and finite-time scaling (FTS) provide a foundational framework for driven critical dynamics, yet their predictive power has been largely confined to local observables. Here, we establish a universal finite-time scaling theory for the nonequilibrium dynamics of quantum entanglement. Using unbiased quantum Monte Carlo simulations, we investigate the corner entanglement entropy of (2+1)-dimensional interacting Dirac fermions driven from ordered phases toward a quantum critical point. We find that the corner entanglement accurately obeys a universal driven scaling governed by the driving rate and system size, persisting whether the initial ordered state is fully gapped or hosts gapless Goldstone modes. Crucially, this dynamical entanglement exhibits a logarithmic dependence on the driving rate, from which the universal corner coefficient of the underlying conformal field theory can be robustly extracted far from equilibrium. These results generalize the KZM from local observables to the intrinsic nonlocal quantum information measures, offering a practical blueprint for characterizing quantum criticality and entanglement on programmable quantum simulators.
We describe the quantum phase transition in the NNN-state chiral clock model in spatial dimension d=1d=1d=1. With couplings chosen to preserve time-reversal and spatial inversion symmetries, such a model is in the universality class of recent experimental studies of the ordering of pumped Rydberg states in a one-dimensional chain of trapped ultracold alkali atoms. For such couplings and N=3N=3N=3, the clock model is expected to have a direct phase transition from a gapped phase with a broken global ZN{\mathbb{Z}}_{N}ZN​ symmetry, to a gapped phase with the ZN{\mathbb{Z}}_{N}ZN​ symmetry restored. The transition has dynamical critical exponent $z\ensuremath{\ne}1$, and so cannot be described by a relativistic quantum field theory. We use a lattice duality transformation to map the transition onto that of a Bose gas in d=1d=1d=1, involving the onset of a single-boson condensate in the background of a higher-dimensional NNN-boson condensate. We present a renormalization group analysis of the strongly coupled field theory for the Bose gas transition in an expansion in $2\ensuremath{-}d$, with $4\ensuremath{-}N$ chosen to be of order $2\ensuremath{-}d$. At two-loop order, we find a regime of parameters with a renormalization group fixed point which can describe a direct phase transition. We also present numerical density-matrix renormalization group studies of lattice chiral clock and Bose gas models for N=3N=3N=3, finding good evidence for a direct phase transition, and obtain estimates for zzz and the correlation length exponent $\ensuremath{\nu}$.