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:2405.08449 · Physical Review A

Exploring dynamical phase transitions in the X Y chain through a linear quench: Early and long-term perspectives

Kaiyuan Cao, Hou Hong-Sheng, Peiqing Tong

Published Oct 9, 2024• 10 citations
arXivHTMLPDF

Abstract

We investigate the nonequilibrium dynamics induced by a finite-time linear quench in the $XY$ chain. Initially, we examine the dynamical quantum phase transition (DQPT-LE), characterized by the nonanalytic behavior of the Loschmidt amplitude. We find distinct behaviors of DQPT-LEs during and following the ramp. Following the ramp, the ramp crossing the critical point ${h}_{c}$ is the sufficient condition for the occurrence of DQPT-LE, but it is not during the ramp. Through adiabatic-impulse-adiabatic (AIA) approximation analysis, we establish that adequate distancing from the critical point is crucial for DQPT-LE manifestation during the ramp, elucidating the absence of DQPT-LE as the ramp gets faster. Additionally, we explore another type of dynamical phase transition (DQPT-OP), describing the long-term relaxation behavior of the order parameter. Our finding indicates that the asymptotic behavior of the time-dependent part induced by the linear quench is equivalent to that following a sudden quench, i.e., the time-dependent part exhibits power-law decays of $\ensuremath{\sim}{t}^{\ensuremath{-}3/2}$ and $\ensuremath{\sim}{t}^{\ensuremath{-}1/2}$ for the ramp to the commensurate and incommensurate phases, respectively. Moreover, we also delve into the steady part, which showcases nonanalytic singularities at the critical point.

DOI
10.1103/physreva.110.042209

Related papers

Dynamical relaxation of a long-range XY chain

Huang, Yu-Huang, Yin-Tao Zou, Chengxiang Ding

We study the universal real-time relaxation behaviors of a long-range quantum XY chain following a quench. Our research includes both the noncritical and critical quench. In the case of noncritical quench, i.e., neither the initial state nor the postquench Hamiltonian is at a critical point of equilibrium phase transition, a quench to the commensurate phase or incommensurate phase gives a scaling of $t^{-3/2}$ or $t^{-1/2}$, respectively, which is the same as the counterpart of the short-range XY model. However, for a quench to the boundary line between the commensurate and incommensurate phases, the scaling law $t^{-μ}$ may be different from the $t^{-3/4}$ law of the counterpart of the short-range model. More interestingly, the decaying exponent $μ$ may depend on the choice of the parameters of the postquench Hamiltonian because of the different asymptotic behaviors of the energy spectrum. Furthermore, in certain cases, the scaling behavior may be outside the range of predictions made by the stationary phase approximation, because an inflection point emerges in the energy spectrum. For the critical quench, i.e., the initial state or the postquench Hamiltonian is at a critical point of equilibrium phase transition, the aforementioned scaling law $t^{-μ}$ may be changed because of the gap-closing property of the energy spectrum of the critical point.

arXiv:2311.18293•Nov 30, 2023•0 citations

Dynamical relaxation of a long-range Kitaev chain

Yu-Huan Huang, Yin-Tao Zou, Chengxiang Ding

We study the universal real-time relaxation behaviors of a long-range Kitaev chain following a quench, and the results are compared to a short-range quantum XY chain. Our research includes both noncritical and critical quenches. In the case of noncritical quench, i.e., neither the initial state nor the postquench Hamiltonian is at a critical point of the equilibrium phase transition, a quench to the commensurate phase or incommensurate phase gives a scaling of ${t}^{\ensuremath{-}3/2}$ or ${t}^{\ensuremath{-}1/2}$, respectively, which is the same as the counterpart of the short-range quantum XY model. However, for a quench to the boundary line between the commensurate and incommensurate phases, the scaling law ${t}^{\ensuremath{-}\ensuremath{\mu}}$ may be different from the ${t}^{\ensuremath{-}3/4}$ law of the counterpart of the short-range quantum XY model. More interestingly, the decaying exponent $\ensuremath{\mu}$ may depend on the choice of the parameters of the postquench Hamiltonian because of the different asymptotic behaviors of the energy spectrum. Furthermore, in certain cases, the scaling behavior may be outside the range of predictions made by the stationary phase approximation, because an inflection point emerges in the energy spectrum. For the critical quench, i.e., the initial state or the postquench Hamiltonian is at a critical point of equilibrium phase transition, the aforementioned scaling law ${t}^{\ensuremath{-}\ensuremath{\mu}}$ may be changed because of the gap-closing property of the energy spectrum of the critical point.

Physical review. B./Physical review. B•Mar 25, 2024•5 citations

Relaxation dynamics in the alternating XY chain following a quantum quench

Kaiyuan Cao, Yayun Hu, Peiqing Tong +2 more

Abstract We investigate the relaxation dynamics of the fermion two-point correlation function C mn ( t ) = 〈 ψ ( t ) ∣ c m † c n ∣ ψ ( t ) 〉 in the XY chain with alternating nearest-neighbor hopping interaction after a quench. We find that the deviation δ C mn (t) = C mn (t) − C mn (∞) decays with time following the power law behavior t −μ , where the exponent μ depends on whether the quench is to the commensurate phase (μ = 1) or incommensurate phase ( μ = 1 2 ). This decay of δ C mn (t) arises from the transient behavior of the double-excited quasiparticle occupations and the transitions between different excitation spectra. Furthermore, we find that the steady value C mn (∞) only involves the average fermion occupation numbers (i.e. the average excited single particle) over the time-evolved state, which are different from the ground state expectation values. We also observe nonanalytic singularities in the steady value C mn (∞) for the quench to the critical points of the quantum phase transitions (QPTs), suggesting its potential use as a signature of QPTs.

arXiv:2311.08025•Physica Scripta•Jul 8, 2024•1 citation

Exploring Dynamical Quantum Phase Transitions via the Dynamical Phase Accumulated at an Equilibrium Critical Point

Han-Chuan Kou, Peng Li, Zhi-Han Zhang +1 more

We investigate dynamical quantum phase transitions (DQPTs) generated by a two-step linear quench with a finite waiting time at the equilibrium critical point. The waiting stage leads to the accumulation of a dynamical phase that depends on the waiting time and directly reflects the critical spectral structure. In the transverse Ising chain, this accumulated phase modifies the critical structure of DQPTs from a single-critical-mode (SCM) regime to a multiple-critical-mode (MCM) regime. We analytically and numerically determine the boundary between the two regimes. Along this boundary, the waiting time approaches a constant in the sudden-quench limit, whereas in the slow-quench limit it follows Kibble--Zurek (KZ) scaling with the quench time. In the SCM regime, the rate function exhibits a conventional linear singularity, $r(t)-r(t^{*})\propto |t-t^{*}|^α$, with the critical exponent $α=1$ near the critical time. At the SCM--MCM boundary, this exponent changes to $α=1/2$, leading to a divergent first derivative of the rate function. Additional discrete parameter lines with $α=1/2$ also appear within the MCM regime in the parameter plane spanned by the waiting and quench times. Our findings establish a direct connection between the accumulated dynamical phase and the critical structure of DQPTs, further supported by an extended Su--Schrieffer--Heeger (SSH) model.

quant-pharXiv:2610.14591•Oct 13, 2026•0 citations

Dynamical Quantum Phase Transitions in Boundary Time Crystals

Sukrut Mondkar, Priya Ghosh, Ujjwal Sen

We demonstrate the existence of a dynamical quantum phase transition (DQPT) in a dissipative collective-spin model that exhibits the boundary time crystal (BTC) phase. We initialize the system in the ground state of the Hamiltonian in either the BTC or the non-BTC phase, and drive it across the BTC transition. The driving is done by an abrupt quench or by a finite-time linear ramp of a Hamiltonian control parameter under Markovian Lindblad dynamics. We diagnose DQPTs through zeros of the fidelity-based Loschmidt echo between the initial state and the evolving mixed state, which induce nonanalytic cusp-like features in the associated rate function. For quenches into the BTC phase, the Loschmidt echo exhibits repeated zeros due to the emergent time-periodic steady state, whereas for quenches into the non-BTC phase, the overlap vanishes and remains zero once the dynamics relaxes to a stationary state. We further show that the DQPT persists under the ramp protocol followed by unitary evolution with the final Hamiltonian. Finally, we analyze the finite-size scaling of the first critical time and find convergence to a constant in the thermodynamic limit, with distinct power-law approaches for the quench and the ramp protocols.

quant-pharXiv:2602.04792•Feb 5, 2026•0 citations

Signature of a nonequilibrium quantum phase transition in the long-time average of the Loschmidt echo

Bozhen Zhou, Chao Yang, Shu Chen

We unveil the role of the long-time average of the Loschmidt echo in the characterization of nonequilibrium quantum phase transitions by studying sudden quench processes across quantum phase transitions in various quantum systems. While the dynamical quantum phase transitions are characterized by the emergence of a series of zero points at critical times during time evolution, we demonstrate that nonequilibrium quantum phase transitions can be identified by nonanalyticities in the long-time average of the Loschmidt echo. The nonanalytic behaviors are illustrated by a sharp change in the long-time average of the Loschmidt echo or the corresponding rate function or the emergence of divergence in the second derivative of the rate function when the driving quench parameter crosses the phase transition points. The connection between the second derivative of the rate function and fidelity susceptibility is also discussed.

arXiv:1908.11572•Physical review. B./Physical review. B•Nov 27, 2019•30 citations
We investigate the nonequilibrium dynamics induced by a finite-time linear quench in the XYXYXY chain. Initially, we examine the dynamical quantum phase transition (DQPT-LE), characterized by the nonanalytic behavior of the Loschmidt amplitude. We find distinct behaviors of DQPT-LEs during and following the ramp. Following the ramp, the ramp crossing the critical point hc{h}_{c}hc​ is the sufficient condition for the occurrence of DQPT-LE, but it is not during the ramp. Through adiabatic-impulse-adiabatic (AIA) approximation analysis, we establish that adequate distancing from the critical point is crucial for DQPT-LE manifestation during the ramp, elucidating the absence of DQPT-LE as the ramp gets faster. Additionally, we explore another type of dynamical phase transition (DQPT-OP), describing the long-term relaxation behavior of the order parameter. Our finding indicates that the asymptotic behavior of the time-dependent part induced by the linear quench is equivalent to that following a sudden quench, i.e., the time-dependent part exhibits power-law decays of \ensuremath∼t\ensuremath−3/2\ensuremath{\sim}{t}^{\ensuremath{-}3/2}\ensuremath∼t\ensuremath−3/2 and \ensuremath∼t\ensuremath−1/2\ensuremath{\sim}{t}^{\ensuremath{-}1/2}\ensuremath∼t\ensuremath−1/2 for the ramp to the commensurate and incommensurate phases, respectively. Moreover, we also delve into the steady part, which showcases nonanalytic singularities at the critical point.
We study the universal real-time relaxation behaviors of a long-range quantum XY chain following a quench. Our research includes both the noncritical and critical quench. In the case of noncritical quench, i.e., neither the initial state nor the postquench Hamiltonian is at a critical point of equilibrium phase transition, a quench to the commensurate phase or incommensurate phase gives a scaling of t−3/2t^{-3/2}t−3/2 or t−1/2t^{-1/2}t−1/2, respectively, which is the same as the counterpart of the short-range XY model. However, for a quench to the boundary line between the commensurate and incommensurate phases, the scaling law t−μt^{-μ}t−μ may be different from the t−3/4t^{-3/4}t−3/4 law of the counterpart of the short-range model. More interestingly, the decaying exponent μμμ may depend on the choice of the parameters of the postquench Hamiltonian because of the different asymptotic behaviors of the energy spectrum. Furthermore, in certain cases, the scaling behavior may be outside the range of predictions made by the stationary phase approximation, because an inflection point emerges in the energy spectrum. For the critical quench, i.e., the initial state or the postquench Hamiltonian is at a critical point of equilibrium phase transition, the aforementioned scaling law t−μt^{-μ}t−μ may be changed because of the gap-closing property of the energy spectrum of the critical point.
We study the universal real-time relaxation behaviors of a long-range Kitaev chain following a quench, and the results are compared to a short-range quantum XY chain. Our research includes both noncritical and critical quenches. In the case of noncritical quench, i.e., neither the initial state nor the postquench Hamiltonian is at a critical point of the equilibrium phase transition, a quench to the commensurate phase or incommensurate phase gives a scaling of ${t}^{\ensuremath{-}3/2}$ or ${t}^{\ensuremath{-}1/2}$, respectively, which is the same as the counterpart of the short-range quantum XY model. However, for a quench to the boundary line between the commensurate and incommensurate phases, the scaling law ${t}^{\ensuremath{-}\ensuremath{\mu}}$ may be different from the ${t}^{\ensuremath{-}3/4}$ law of the counterpart of the short-range quantum XY model. More interestingly, the decaying exponent $\ensuremath{\mu}$ may depend on the choice of the parameters of the postquench Hamiltonian because of the different asymptotic behaviors of the energy spectrum. Furthermore, in certain cases, the scaling behavior may be outside the range of predictions made by the stationary phase approximation, because an inflection point emerges in the energy spectrum. For the critical quench, i.e., the initial state or the postquench Hamiltonian is at a critical point of equilibrium phase transition, the aforementioned scaling law ${t}^{\ensuremath{-}\ensuremath{\mu}}$ may be changed because of the gap-closing property of the energy spectrum of the critical point.
Abstract We investigate the relaxation dynamics of the fermion two-point correlation function C mn ( t ) = 〈 ψ ( t ) ∣ c m † c n ∣ ψ ( t ) 〉 in the XY chain with alternating nearest-neighbor hopping interaction after a quench. We find that the deviation δ C mn (t) = C mn (t) − C mn (∞) decays with time following the power law behavior t −μ , where the exponent μ depends on whether the quench is to the commensurate phase (μ = 1) or incommensurate phase ( μ = 1 2 ). This decay of δ C mn (t) arises from the transient behavior of the double-excited quasiparticle occupations and the transitions between different excitation spectra. Furthermore, we find that the steady value C mn (∞) only involves the average fermion occupation numbers (i.e. the average excited single particle) over the time-evolved state, which are different from the ground state expectation values. We also observe nonanalytic singularities in the steady value C mn (∞) for the quench to the critical points of the quantum phase transitions (QPTs), suggesting its potential use as a signature of QPTs.
We investigate dynamical quantum phase transitions (DQPTs) generated by a two-step linear quench with a finite waiting time at the equilibrium critical point. The waiting stage leads to the accumulation of a dynamical phase that depends on the waiting time and directly reflects the critical spectral structure. In the transverse Ising chain, this accumulated phase modifies the critical structure of DQPTs from a single-critical-mode (SCM) regime to a multiple-critical-mode (MCM) regime. We analytically and numerically determine the boundary between the two regimes. Along this boundary, the waiting time approaches a constant in the sudden-quench limit, whereas in the slow-quench limit it follows Kibble--Zurek (KZ) scaling with the quench time. In the SCM regime, the rate function exhibits a conventional linear singularity, r(t)−r(t∗)∝∣t−t∗∣αr(t)-r(t^{*})\propto |t-t^{*}|^αr(t)−r(t∗)∝∣t−t∗∣α, with the critical exponent α=1α=1α=1 near the critical time. At the SCM--MCM boundary, this exponent changes to α=1/2α=1/2α=1/2, leading to a divergent first derivative of the rate function. Additional discrete parameter lines with α=1/2α=1/2α=1/2 also appear within the MCM regime in the parameter plane spanned by the waiting and quench times. Our findings establish a direct connection between the accumulated dynamical phase and the critical structure of DQPTs, further supported by an extended Su--Schrieffer--Heeger (SSH) model.
We demonstrate the existence of a dynamical quantum phase transition (DQPT) in a dissipative collective-spin model that exhibits the boundary time crystal (BTC) phase. We initialize the system in the ground state of the Hamiltonian in either the BTC or the non-BTC phase, and drive it across the BTC transition. The driving is done by an abrupt quench or by a finite-time linear ramp of a Hamiltonian control parameter under Markovian Lindblad dynamics. We diagnose DQPTs through zeros of the fidelity-based Loschmidt echo between the initial state and the evolving mixed state, which induce nonanalytic cusp-like features in the associated rate function. For quenches into the BTC phase, the Loschmidt echo exhibits repeated zeros due to the emergent time-periodic steady state, whereas for quenches into the non-BTC phase, the overlap vanishes and remains zero once the dynamics relaxes to a stationary state. We further show that the DQPT persists under the ramp protocol followed by unitary evolution with the final Hamiltonian. Finally, we analyze the finite-size scaling of the first critical time and find convergence to a constant in the thermodynamic limit, with distinct power-law approaches for the quench and the ramp protocols.
We unveil the role of the long-time average of the Loschmidt echo in the characterization of nonequilibrium quantum phase transitions by studying sudden quench processes across quantum phase transitions in various quantum systems. While the dynamical quantum phase transitions are characterized by the emergence of a series of zero points at critical times during time evolution, we demonstrate that nonequilibrium quantum phase transitions can be identified by nonanalyticities in the long-time average of the Loschmidt echo. The nonanalytic behaviors are illustrated by a sharp change in the long-time average of the Loschmidt echo or the corresponding rate function or the emergence of divergence in the second derivative of the rate function when the driving quench parameter crosses the phase transition points. The connection between the second derivative of the rate function and fidelity susceptibility is also discussed.