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:1505.02924 · Journal of Statistical Mechanics Theory and Experiment

Asymptotic work statistics of periodically driven Ising chains

Angelo Russomanno, Shraddha Sharma, Amit Dutta, Giuseppe E. Santoro

Published Aug 26, 2015• 38 citations
arXivHTMLPDF

Abstract

We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically time-varying transverse field of frequency , we arrive at the characteristic cumulant generating function G ( u ), which is then used to calculate the work distribution function P ( W ). By applying the Floquet theory we show that, in the infinite time limit, P ( W ) converges, starting from the initial ground state, towards an asymptotic steady state value whose small- W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h 0 , the initial transverse field, , and the equilibrium quantum critical point , which we find to generate a sequence of non-equilibrium critical points , with l integer. When , we find a ‘universal’ edge singularity in P ( W ) at a threshold value of which is entirely determined by . The form of that singularity—Dirac delta derivative or square root—depends on h 0 being or not at a non-equilibrium critical point h * l . On the contrary, when , G ( u ) decays as a power-law for large u , leading to different types of edge singularity at . Generalizing our calculations to the case in which we initialize the system in a finite temperature density matrix, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.

DOI
10.1088/1742-5468/2015/08/p08030

Related papers

Entanglement entropy in a periodically driven Ising chain

Angelo Russomanno, Giuseppe E. Santoro, Rosario Fazio

Abstract In this work we study the entanglement entropy of a uniform quantum Ising chain in transverse field undergoing a periodic driving of period τ . By means of Floquet theory we show that, for any subchain, the entanglement entropy tends asymptotically to a value τ -periodic in time. We provide a semi-analytical formula for the leading term of this asymptotic regime: It is constant in time and obeys a volume law. The entropy in the asymptotic regime is always smaller than the thermal one: because of integrability the system locally relaxes to a generalized Gibbs ensemble (GGE) density matrix. The leading term of the asymptotic entanglement entropy is completely determined by this GGE density matrix. Remarkably, the asymptotic entropy shows marked features in correspondence to some non-equilibrium quantum phase transitions undergone by a Floquet state analog of the ground state.

arXiv:1603.03663•Journal of Statistical Mechanics Theory and Experiment•Jul 1, 2016•34 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

Dynamical relaxation of correlators in periodically driven integrable quantum systems

Sreemayee Aditya, Sutapa Samanta, Arnab Sen +2 more

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.

arXiv:2112.02915•Physical review. B./Physical review. B•Mar 3, 2022•26 citations

Periodic Steady Regime and Interference in a Periodically Driven Quantum System

Angelo Russomanno, Alessandro Silva, Giuseppe E. Santoro

We study the coherent dynamics of a quantum many-body system subject to a time-periodic driving. We argue that in many cases, destructive interference in time makes most of the quantum averages time periodic, after an initial transient. We discuss in detail the case of a quantum Ising chain periodically driven across the critical point, finding that, as a result of quantum coherence, the system never reaches an infinite temperature state. Floquet resonance effects are moreover observed in the frequency dependence of the various observables, which display a sequence of well-defined peaks or dips. Extensions to nonintegrable systems are discussed.

arXiv:1204.5084•Physical Review Letters•Dec 17, 2012•191 citations

Loschmidt echo and dynamical fidelity in periodically driven quantum systems

Shraddha Sharma, Angelo Russomanno, Giuseppe E. Santoro +1 more

We study the dynamical fidelity $\mathcal{F} (t)$ and the Loschmidt echo $\mathcal{L} (t)$, following a periodic driving of the transverse magnetic field of a quantum Ising chain (back and forth across the quantum critical point) by calculating the overlap between the initial ground state and the state reached after $n$ periods $τ$. We show that $\log{\mathcal{F}}(nτ)/L$ (the logarithm of the fidelity per-site) reaches a steady value in the asymptotic limit $n\to \infty$, and we derive an exact analytical expression for this quantity. Remarkably, the steady state value of $[\log{\mathcal{F}}(nτ\to \infty)]/L$ shows memory of non-trivial phase information which is instead hidden in the case of thermodynamic quantities; this conclusion, moreover, is not restricted to 1-dimensional models.

arXiv:1402.0758•Europhysics Letters (EPL)•Jun 1, 2014•63 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
We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically time-varying transverse field of frequency , we arrive at the characteristic cumulant generating function G ( u ), which is then used to calculate the work distribution function P ( W ). By applying the Floquet theory we show that, in the infinite time limit, P ( W ) converges, starting from the initial ground state, towards an asymptotic steady state value whose small- W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h 0 , the initial transverse field, , and the equilibrium quantum critical point , which we find to generate a sequence of non-equilibrium critical points , with l integer. When , we find a ‘universal’ edge singularity in P ( W ) at a threshold value of which is entirely determined by . The form of that singularity—Dirac delta derivative or square root—depends on h 0 being or not at a non-equilibrium critical point h * l . On the contrary, when , G ( u ) decays as a power-law for large u , leading to different types of edge singularity at . Generalizing our calculations to the case in which we initialize the system in a finite temperature density matrix, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.
Abstract In this work we study the entanglement entropy of a uniform quantum Ising chain in transverse field undergoing a periodic driving of period τ . By means of Floquet theory we show that, for any subchain, the entanglement entropy tends asymptotically to a value τ -periodic in time. We provide a semi-analytical formula for the leading term of this asymptotic regime: It is constant in time and obeys a volume law. The entropy in the asymptotic regime is always smaller than the thermal one: because of integrability the system locally relaxes to a generalized Gibbs ensemble (GGE) density matrix. The leading term of the asymptotic entanglement entropy is completely determined by this GGE density matrix. Remarkably, the asymptotic entropy shows marked features in correspondence to some non-equilibrium quantum phase transitions undergone by a Floquet state analog of the ground state.
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.
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 nnn 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 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, ${\ensuremath{\omega}}_{c}$, 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.
We study the coherent dynamics of a quantum many-body system subject to a time-periodic driving. We argue that in many cases, destructive interference in time makes most of the quantum averages time periodic, after an initial transient. We discuss in detail the case of a quantum Ising chain periodically driven across the critical point, finding that, as a result of quantum coherence, the system never reaches an infinite temperature state. Floquet resonance effects are moreover observed in the frequency dependence of the various observables, which display a sequence of well-defined peaks or dips. Extensions to nonintegrable systems are discussed.
We study the dynamical fidelity F(t)\mathcal{F} (t)F(t) and the Loschmidt echo L(t)\mathcal{L} (t)L(t), following a periodic driving of the transverse magnetic field of a quantum Ising chain (back and forth across the quantum critical point) by calculating the overlap between the initial ground state and the state reached after nnn periods τττ. We show that log⁡F(nτ)/L\log{\mathcal{F}}(nτ)/LlogF(nτ)/L (the logarithm of the fidelity per-site) reaches a steady value in the asymptotic limit n→∞n\to \inftyn→∞, and we derive an exact analytical expression for this quantity. Remarkably, the steady state value of [log⁡F(nτ→∞)]/L[\log{\mathcal{F}}(nτ\to \infty)]/L[logF(nτ→∞)]/L shows memory of non-trivial phase information which is instead hidden in the case of thermodynamic quantities; this conclusion, moreover, is not restricted to 1-dimensional 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.