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:2112.02915 · Physical review. B./Physical review. B

Dynamical relaxation of correlators in periodically driven integrable quantum systems

Sreemayee Aditya, Sutapa Samanta, Arnab Sen, K. Sengupta, Diptiman Sen

Published Mar 3, 2022• 26 citations
arXivHTMLPDF

Abstract

We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as ${n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}$, where $n$ is the number of drive cycles and $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ denote positive integers. We find that, generically, $\ensuremath{\beta}=2$ within a dynamical phase characterized by a fixed $\ensuremath{\alpha}$; however, its value can change to $\ensuremath{\beta}=3$ or $\ensuremath{\beta}=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\ensuremath{\beta}$, and chart out the values of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale ${n}_{c}$ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, ${\ensuremath{\omega}}_{c}$, is approached from below $(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})$. We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.

DOI
10.1103/physrevb.105.104303

Related papers

Entanglement generation in periodically driven integrable systems: Dynamical phase transitions and steady state

Arnab Sen, Sourav Nandy, K. Sengupta

We study a class of periodically driven $d$-dimensional integrable models and show that after $n$ drive cycles with frequency $\ensuremath{\omega}$, pure states with non-area-law entanglement entropy ${S}_{n}(l)\ensuremath{\sim}{l}^{\ensuremath{\alpha}(n,\ensuremath{\omega})}$ are generated, where $l$ is the linear dimension of the subsystem, and $d\ensuremath{-}1\ensuremath{\le}\ensuremath{\alpha}(n,\ensuremath{\omega})\ensuremath{\le}d$. The exponent $\ensuremath{\alpha}(n,\ensuremath{\omega})$ eventually approaches $d$ (volume law) for large enough $l$ when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We identify and analyze the crossover phenomenon from an area $(S\ensuremath{\sim}{l}^{d\ensuremath{-}1}$ for $d\ensuremath{\ge}1$) to a volume $(S\ensuremath{\sim}{l}^{d})$ law and provide a criterion for their occurrence which constitutes a generalization of Hastings's theorem to driven integrable systems in one dimension. We also find that ${S}_{n}$ generically decays to ${S}_{\ensuremath{\infty}}$ as ${(\ensuremath{\omega}/n)}^{(d+2)/2}$ for fast and ${(\ensuremath{\omega}/n)}^{d/2}$ for slow periodic drives; these two dynamical phases are separated by a topological transition in the eigenspectrum of the Floquet Hamiltonian. This dynamical transition manifests itself in the temporal behavior of all local correlation functions and does not require a critical point crossing during the drive. We find that these dynamical phases show a rich re-entrant behavior as a function of $\ensuremath{\omega}$ for $d=1$ models and also discuss the dynamical transition for $d>1$ models. Finally, we study entanglement properties of the steady state and show that singular features (cusps and kinks in $d=1$) appear in ${S}_{\ensuremath{\infty}}$ as a function of $\ensuremath{\omega}$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.

arXiv:1511.03668•Physical review. B./Physical review. B•Dec 8, 2016•71 citations

Entanglement transitions in a periodically driven non-Hermitian Ising chain

Tista Banerjee, K. Sengupta

We study entanglement transitions in a periodically driven Ising chain in the presence of an imaginary transverse field $\ensuremath{\gamma}$ as a function of drive frequency ${\ensuremath{\omega}}_{D}$. In the high drive amplitude and frequency regime, we find a critical value $\ensuremath{\gamma}={\ensuremath{\gamma}}_{c}$ below which the steady-state half-chain entanglement entropy, ${S}_{L/2}$, scales with chain length $L$ as ${S}_{L/2}\ensuremath{\sim}lnL$; in contrast, for $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, it becomes independent of $L$. In the small $\ensuremath{\gamma}$ limit, we compute the coefficient, $\ensuremath{\alpha}$, of the $lnL$ term analytically using a Floquet perturbation theory and trace its origin to the presence of Fisher-Hartwig jump singularities in the correlation function of the driven chain. We also study the frequency dependence of ${\ensuremath{\gamma}}_{c}$ and show that ${\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0$ at special drive frequencies; at these frequencies, which we analytically compute, ${S}_{L/2}$ remain independent of $L$ for all $\ensuremath{\gamma}$. This behavior can be traced to an approximate emergent symmetry of the Floquet Hamiltonian at these drive frequencies which we identify. Finally, we discus the behavior of the driven system at low and intermediate drive frequencies. Our analysis shows the presence of volume law behavior of the entanglement in this regime ${S}_{\ensuremath{\ell}}\ensuremath{\sim}\ensuremath{\ell}$ for small subsystem length $\ensuremath{\ell}\ensuremath{\le}{\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$. We identify ${\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$ and tie its existence to the effective long-range nature of the Floquet Hamiltonian of the driven chain for small subsystem size. We discuss the applicability of our results to other integrable non-Hermitian models.

arXiv:2309.07661•Physical review. B./Physical review. B•Mar 15, 2024•10 citations

Periodically driven integrable systems with long-range pair potentials

Sourav Nandy, K. Sengupta, Arnab Sen

Abstract We study periodically driven closed systems with a long-ranged Hamiltonian by considering a generalized Kitaev chain with pairing terms which decay with distance as a power law characterized by exponent α . Starting from an initial unentangled state, we show that all local quantities synchronize with the driving frequency ω at late times and relax to well-defined steady state values in the thermodynamic limit and after drive cycles for any α and ω . We introduce a distance measure, , that characterizes the approach of the reduced density matrix of a subsystem of l sites to that of the final steady state. We chart out the n dependence of and identify a critical value (which depends only on the time-averaged Hamiltonian) below which they generically decay to zero as . For , in contrast, for with at least one intermediate dynamical transition. An identical behavior is found for relaxation of all non-trivial correlation functions to their steady-state values. We also study the mutual information propagation to understand the nature of the entanglement spreading in space with increasing n for such periodically driven long-ranged systems. We point out existence of qualitatively new features (absent for ) in the space-time dependence of mutual information for , where is the largest critical frequency for the dynamical transition for a given α such as the presence of multiple light cone-like structures which persists even when α is large. We also show that the space-time dependence of the mutual information of long-ranged Hamiltonians with differs qualitatively from those with for any drive frequency and relate this to the behavior of the Floquet group velocity of such driven system.

arXiv:1709.08897•Journal of Physics A Mathematical and Theoretical•Jun 25, 2018•35 citations

Dynamical transition for a class of integrable models coupled to a bath

Madhumita Sarkar, K. Sengupta

We study the dynamics of correlation functions of a class of $d$-dimensional integrable models coupled linearly to a fermionic or bosonic bath in the presence of a periodic drive with a square pulse protocol. It is well known that in the absence of the bath, these models exhibit a dynamical phase transition; all correlators decay to their steady-state values as ${n}_{0}^{\ensuremath{-}(d+2)/2}[{n}_{0}^{\ensuremath{-}d/2}]$ above [below] a critical frequency ${\ensuremath{\omega}}_{c}$, where ${n}_{0}$ is the number of drive cycles. We find that the presence of a linearly coupled fermionic bath which maintains integrability of the system preserves this transition and leads to additional ones that have no analogs in closed quantum systems. We provide a semianalytic expression for the evolution operator for this system and use it to provide a phase diagram showing the different dynamical regimes as a function of the system-bath coupling strength and the bath parameters. In contrast, when such models are coupled to a bosonic bath which breaks integrability of the model, we find exponential decay of the correlators to their steady state. Our numerical analysis shows that this exponential decay sets in above a critical number of drive cycles ${n}_{c}$ which depends on the system-bath coupling strength and the amplitude of perturbation. Below ${n}_{c}$, the system retains the power-law behavior identical to that for the closed integrable models and the dynamical transition survives. We discuss the applicability of our results for interacting fermion systems and discuss experiments which can test our theory.

arXiv:2006.14640•Physical review. B./Physical review. B•Dec 24, 2020•17 citations

Emergent conservation in the Floquet dynamics of integrable non-Hermitian models

Tista Banerjee, K. Sengupta

We study the dynamics of a class of integrable non-Hermitian free-fermionic models driven periodically using a continuous drive protocol characterized by an amplitude ${g}_{1}$ and frequency ${\ensuremath{\omega}}_{D}$. We derive an analytic, albeit perturbative, Floquet Hamiltonian for describing such systems using Floquet perturbation theory with ${g}_{1}^{\ensuremath{-}1}$ being the perturbation parameter. Our analysis indicates the existence of special drive frequencies at which an approximately conserved quantity emerges. The presence of such an almost conserved quantity is reflected in the dynamics of the fidelity, the correlation functions, and the half-chain entanglement entropy of the driven system. In addition, it also controls the nature of the steady state of the system. We show that one-dimensional transverse field Ising model, with an imaginary component of the transverse field, serves as an experimentally relevant example of this phenomenon. In this case, the transverse magnetization is approximately conserved; this conservation leads to complete suppression of oscillatory features in the transient dynamics of fidelity, magnetization, and entanglement of the driven chain at special drive frequencies. We discuss the nature of the steady state of the Ising chain near and away from these special frequencies, demonstrate the protocol independence of this phenomenon by showing its existence for discrete drive protocols, and suggest experiments which can test our theory.

arXiv:2209.12939•Physical review. B./Physical review. B•Apr 10, 2023•16 citations

Insights from the analytical solution of a periodically driven transverse-field Ising chain

Pritam Das, Anirban Dutta

We derive an exact analytical expression, at stroboscopic intervals, for the time-dependent wave function of a class of integrable quantum many-body systems, driven by the periodic delta-kick protocol. To investigate long-time dynamics, we use the wave function to obtain an exact analytical expression for the expectation values of the defect density, magnetization, residual energy, fidelity, and the correlation function after $n\mathrm{th}$ drive cycle. Periodically driven integrable closed quantum systems absorb energy, and the long-time universal dynamics are described by the periodic generalized Gibbs ensemble (GGE). We demonstrate that the expectation values of all observables are divided into two parts: one highly oscillatory term that depends on the drive cycle $n$, and the rest of the terms are independent of it. Typically, the $n$-independent part constitutes the saturation at large $n$ and periodic GGE. The contribution from the highly oscillatory term vanishes in large $n$. We also generalize our formalism to include square-pulse and sinusoidal driving protocols.

arXiv:2409.08830•Physical review. B./Physical review. B•Jan 27, 2025•3 citations
We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as n\ensuremath−(\ensuremathα+1)/\ensuremathβ{n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}n\ensuremath−(\ensuremathα+1)/\ensuremathβ, where nnn is the number of drive cycles and \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathβ\ensuremath{\beta}\ensuremathβ denote positive integers. We find that, generically, \ensuremathβ=2\ensuremath{\beta}=2\ensuremathβ=2 within a dynamical phase characterized by a fixed \ensuremathα\ensuremath{\alpha}\ensuremathα; however, its value can change to \ensuremathβ=3\ensuremath{\beta}=3\ensuremathβ=3 or \ensuremathβ=4\ensuremath{\beta}=4\ensuremathβ=4 either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining \ensuremathβ\ensuremath{\beta}\ensuremathβ, and chart out the values of \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathβ\ensuremath{\beta}\ensuremathβ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale nc{n}_{c}nc​ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, \ensuremathωc{\ensuremath{\omega}}_{c}\ensuremathωc​, is approached from below (\ensuremathω<\ensuremathωc)(\ensuremath{\omega}<{\ensuremath{\omega}}_{c})(\ensuremathω<\ensuremathωc​). We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.
We study a class of periodically driven ddd-dimensional integrable models and show that after nnn drive cycles with frequency $\ensuremath{\omega}$, pure states with non-area-law entanglement entropy ${S}_{n}(l)\ensuremath{\sim}{l}^{\ensuremath{\alpha}(n,\ensuremath{\omega})}$ are generated, where lll is the linear dimension of the subsystem, and $d\ensuremath{-}1\ensuremath{\le}\ensuremath{\alpha}(n,\ensuremath{\omega})\ensuremath{\le}d$. The exponent $\ensuremath{\alpha}(n,\ensuremath{\omega})$ eventually approaches ddd (volume law) for large enough lll when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. We identify and analyze the crossover phenomenon from an area $(S\ensuremath{\sim}{l}^{d\ensuremath{-}1}$ for $d\ensuremath{\ge}1$) to a volume $(S\ensuremath{\sim}{l}^{d})$ law and provide a criterion for their occurrence which constitutes a generalization of Hastings's theorem to driven integrable systems in one dimension. We also find that Sn{S}_{n}Sn​ generically decays to ${S}_{\ensuremath{\infty}}$ as ${(\ensuremath{\omega}/n)}^{(d+2)/2}$ for fast and ${(\ensuremath{\omega}/n)}^{d/2}$ for slow periodic drives; these two dynamical phases are separated by a topological transition in the eigenspectrum of the Floquet Hamiltonian. This dynamical transition manifests itself in the temporal behavior of all local correlation functions and does not require a critical point crossing during the drive. We find that these dynamical phases show a rich re-entrant behavior as a function of $\ensuremath{\omega}$ for d=1d=1d=1 models and also discuss the dynamical transition for d>1d>1d>1 models. Finally, we study entanglement properties of the steady state and show that singular features (cusps and kinks in d=1d=1d=1) appear in ${S}_{\ensuremath{\infty}}$ as a function of $\ensuremath{\omega}$ whenever there is a crossing of the Floquet bands. We discuss experiments which can test our theory.
We study entanglement transitions in a periodically driven Ising chain in the presence of an imaginary transverse field $\ensuremath{\gamma}$ as a function of drive frequency ${\ensuremath{\omega}}_{D}$. In the high drive amplitude and frequency regime, we find a critical value $\ensuremath{\gamma}={\ensuremath{\gamma}}_{c}$ below which the steady-state half-chain entanglement entropy, SL/2{S}_{L/2}SL/2​, scales with chain length LLL as ${S}_{L/2}\ensuremath{\sim}lnL$; in contrast, for $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}$, it becomes independent of LLL. In the small $\ensuremath{\gamma}$ limit, we compute the coefficient, $\ensuremath{\alpha}$, of the lnLlnLlnL term analytically using a Floquet perturbation theory and trace its origin to the presence of Fisher-Hartwig jump singularities in the correlation function of the driven chain. We also study the frequency dependence of ${\ensuremath{\gamma}}_{c}$ and show that ${\ensuremath{\gamma}}_{c}\ensuremath{\rightarrow}0$ at special drive frequencies; at these frequencies, which we analytically compute, SL/2{S}_{L/2}SL/2​ remain independent of LLL for all $\ensuremath{\gamma}$. This behavior can be traced to an approximate emergent symmetry of the Floquet Hamiltonian at these drive frequencies which we identify. Finally, we discus the behavior of the driven system at low and intermediate drive frequencies. Our analysis shows the presence of volume law behavior of the entanglement in this regime ${S}_{\ensuremath{\ell}}\ensuremath{\sim}\ensuremath{\ell}$ for small subsystem length $\ensuremath{\ell}\ensuremath{\le}{\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$. We identify ${\ensuremath{\ell}}^{*}({\ensuremath{\omega}}_{D})$ and tie its existence to the effective long-range nature of the Floquet Hamiltonian of the driven chain for small subsystem size. We discuss the applicability of our results to other integrable non-Hermitian models.
Abstract We study periodically driven closed systems with a long-ranged Hamiltonian by considering a generalized Kitaev chain with pairing terms which decay with distance as a power law characterized by exponent α . Starting from an initial unentangled state, we show that all local quantities synchronize with the driving frequency ω at late times and relax to well-defined steady state values in the thermodynamic limit and after drive cycles for any α and ω . We introduce a distance measure, , that characterizes the approach of the reduced density matrix of a subsystem of l sites to that of the final steady state. We chart out the n dependence of and identify a critical value (which depends only on the time-averaged Hamiltonian) below which they generically decay to zero as . For , in contrast, for with at least one intermediate dynamical transition. An identical behavior is found for relaxation of all non-trivial correlation functions to their steady-state values. We also study the mutual information propagation to understand the nature of the entanglement spreading in space with increasing n for such periodically driven long-ranged systems. We point out existence of qualitatively new features (absent for ) in the space-time dependence of mutual information for , where is the largest critical frequency for the dynamical transition for a given α such as the presence of multiple light cone-like structures which persists even when α is large. We also show that the space-time dependence of the mutual information of long-ranged Hamiltonians with differs qualitatively from those with for any drive frequency and relate this to the behavior of the Floquet group velocity of such driven system.
We study the dynamics of correlation functions of a class of ddd-dimensional integrable models coupled linearly to a fermionic or bosonic bath in the presence of a periodic drive with a square pulse protocol. It is well known that in the absence of the bath, these models exhibit a dynamical phase transition; all correlators decay to their steady-state values as ${n}_{0}^{\ensuremath{-}(d+2)/2}[{n}_{0}^{\ensuremath{-}d/2}]$ above [below] a critical frequency ${\ensuremath{\omega}}_{c}$, where n0{n}_{0}n0​ is the number of drive cycles. We find that the presence of a linearly coupled fermionic bath which maintains integrability of the system preserves this transition and leads to additional ones that have no analogs in closed quantum systems. We provide a semianalytic expression for the evolution operator for this system and use it to provide a phase diagram showing the different dynamical regimes as a function of the system-bath coupling strength and the bath parameters. In contrast, when such models are coupled to a bosonic bath which breaks integrability of the model, we find exponential decay of the correlators to their steady state. Our numerical analysis shows that this exponential decay sets in above a critical number of drive cycles nc{n}_{c}nc​ which depends on the system-bath coupling strength and the amplitude of perturbation. Below nc{n}_{c}nc​, the system retains the power-law behavior identical to that for the closed integrable models and the dynamical transition survives. We discuss the applicability of our results for interacting fermion systems and discuss experiments which can test our theory.
We study the dynamics of a class of integrable non-Hermitian free-fermionic models driven periodically using a continuous drive protocol characterized by an amplitude g1{g}_{1}g1​ and frequency ${\ensuremath{\omega}}_{D}$. We derive an analytic, albeit perturbative, Floquet Hamiltonian for describing such systems using Floquet perturbation theory with ${g}_{1}^{\ensuremath{-}1}$ being the perturbation parameter. Our analysis indicates the existence of special drive frequencies at which an approximately conserved quantity emerges. The presence of such an almost conserved quantity is reflected in the dynamics of the fidelity, the correlation functions, and the half-chain entanglement entropy of the driven system. In addition, it also controls the nature of the steady state of the system. We show that one-dimensional transverse field Ising model, with an imaginary component of the transverse field, serves as an experimentally relevant example of this phenomenon. In this case, the transverse magnetization is approximately conserved; this conservation leads to complete suppression of oscillatory features in the transient dynamics of fidelity, magnetization, and entanglement of the driven chain at special drive frequencies. We discuss the nature of the steady state of the Ising chain near and away from these special frequencies, demonstrate the protocol independence of this phenomenon by showing its existence for discrete drive protocols, and suggest experiments which can test our theory.
We derive an exact analytical expression, at stroboscopic intervals, for the time-dependent wave function of a class of integrable quantum many-body systems, driven by the periodic delta-kick protocol. To investigate long-time dynamics, we use the wave function to obtain an exact analytical expression for the expectation values of the defect density, magnetization, residual energy, fidelity, and the correlation function after nthn\mathrm{th}nth drive cycle. Periodically driven integrable closed quantum systems absorb energy, and the long-time universal dynamics are described by the periodic generalized Gibbs ensemble (GGE). We demonstrate that the expectation values of all observables are divided into two parts: one highly oscillatory term that depends on the drive cycle nnn, and the rest of the terms are independent of it. Typically, the nnn-independent part constitutes the saturation at large nnn and periodic GGE. The contribution from the highly oscillatory term vanishes in large nnn. We also generalize our formalism to include square-pulse and sinusoidal driving protocols.