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:2607.25661

Discrete power-law decay of subsystem distance after a quantum quench

Bin Sui, Jiaju Zhang

quant-phcond-mat.stat-mechcond-mat.str-elhep-thSubmitted Jul 28, 2026• Announced Jul 29, 2026• 0 citations
arXivHTMLPDF

Abstract

We present a numerical study of subsystem distance decay following a global quantum quench in the infinite one-dimensional transverse-field Ising chain, using the mathematically rigorous Bures distance $B_A(t)$ to quantify the deviation of the time-evolved reduced density matrix from its stationary generalized Gibbs ensemble state. We show that the late-time decay follows a discrete power law $B_A(t) \sim t^{-λ}$, with the exponent $λ$ confined to discrete values: $1$, $5/4$, $3/2$, $7/4$, $2$, $5/2$, and potentially further values. The specific exponent is jointly determined by the pre- and post-quench transverse fields, as well as by properties of the symmetric excitation-fraction function $m_S(\varphi)$, defined on $\varphi\in[0,π]$ to characterize the pre-quench Hamiltonian eigenstates, including continuity, boundary values, and first-derivative boundary values, among others. The previously established $t^{-3/2}$ decay for the initial ground state of the pre-quench Hamiltonian is naturally recovered as a special case of this general classification. Our results reveal a universal discrete structure governing local equilibration dynamics in integrable quantum systems.

Comments
10 pages, 4 figures

Related papers

Reduced density matrix after a quantum quench

Maurizio Fagotti, Fabian H. L. Eßler

We consider the reduced density matrix (RDM) ${\ensuremath{\rho}}_{A}(t)$ for a finite subsystem $A$ after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of ${\ensuremath{\rho}}_{A}(t)$ is described by the RDM ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ of a generalized Gibbs ensemble. Here, we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question of how ${\ensuremath{\rho}}_{A}(t)$ approaches ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ as a function of time. To that end, we introduce a distance on the space of density matrices and show that it approaches zero as a universal power law ${t}^{\ensuremath{-}3/2}$ in time. As the RDM completely determines all local observables within $A$, this provides information on the relaxation of correlation functions of local operators. We then address the issue of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.

arXiv:1302.6944•Physical Review B•Jun 12, 2013•325 citations

Thermalization of long range Ising model in different dynamical regimes: A full counting statistics approach

Nishan Ranabhat, Mario Collura

We study the thermalization of the transverse field Ising chain with a power law decaying interaction \sim 1/r^{\alpha} ∼ 1 / r α following a global quantum quench of the transverse field in two different dynamical regimes. The thermalization behavior is quantified by comparing the full probability distribution function (PDF) of the evolving states with the corresponding thermal state given by the canonical Gibbs ensemble (CGE). To this end, we used the matrix product state (MPS)-based Time Dependent Variational Principle (TDVP) algorithm to simulate both real time evolution following a global quantum quench and the finite temperature density operator. We observe that thermalization is strongly suppressed in the region with strong confinement for all interaction strengths \alpha α , whereas thermalization occurs in the region with weak confinement.

arXiv:2212.00533•SciPost Physics Core•Apr 5, 2024•8 citations

Quantum quenches in the transverse field Ising chain: II. Stationary state properties

Pasquale Calabrese, Fabian H L Essler, Maurizio Fagotti

We consider the stationary state properties of the reduced density matrix as well as spin-spin correlation functions after a sudden quantum quench of the magnetic field in the transverse field Ising chain. We demonstrate that stationary state properties are described by a generalized Gibbs ensemble. We discuss the approach to the stationary state at late times.

arXiv:1205.2211•Journal of Statistical Mechanics Theory and Experiment•Jul 26, 2012•318 citations

Non-equilibrium properties of quantum spin chains

Tatjana Puškarov

In this thesis, we study the properties of the transverse field Ising (TFI) chain in several non-equilibrium setups. One such setup is finite-time quantum quench—quench in which a Hamiltonian parameter is continuously changed during a finite-time, as opposed to the usual instantaneous change in the sudden quenches. A different type of quantum quench, which does not involve changing a Hamiltonian parameter, can be realised by the so-called partitioning protocol. In this protocol, two independent chains are prepared at different temperatures and then coupled into a single system and evolved as such. Another possibility to take the system out of equilibrium is to introduce a driving—for example, by suddenly and continuously changing a Hamiltonian parameter over long times. We discuss the general treatment of each of these situations, and apply them to several quench and driving protocols in the TFI chain. We analyse the behaviour of several observables, such as the total energy, spin correlation functions and Loschmidt echo, during and after quenches and during driving. While we choose the specific protocols to display the dependence of observables on the protocol features, we also investigate the universal behaviours of the observables in the late-time limit. Isolated quenched TFI chains do not relax at late times, however, local observables in these systems do—they relax to a special stationary state described by the generalised Gibbs ensemble (GGE). A similar situation happens in the case of periodic driving of the TFI chain—the late time state of the driven TFI chain, at least as far as the local observables are concerned, it is a periodic steady state described by the periodic Gibbs ensemble (PGE). In this thesis, we observe the relaxation behaviour for various quench and driving protocols and observables. In addition to studying statistical ensembles in the context of relaxation of out-of-equilibrium systems, we describe a machine learning algorithm based on the generalised Gibbs ensemble. We first describe a traditional machine learning algorithm, the Boltzmann machine, and its quantum version. Based on our knowledge of the relaxation of the out-of-equilibrium TFI chain, we modify the existing Boltzmann machine algorithm to include the many GGE charges and test it on the commonly used MNIST dataset. Rather than learning a complicated energy function and using the Boltzmann distribution, we then use the relatively simple TFI Hamiltonian and learn the effective temperatures of the GGE and find reasonably low error rates, while learning a comparatively low number of parameters.

•Dec 10, 2018•0 citations

Quantum Quench in the Transverse-Field Ising Chain

Pasquale Calabrese, Fabian H. L. Eßler, Maurizio Fagotti

We consider the time evolution of observables in the transverse-field Ising chain after a sudden quench of the magnetic field. We provide exact analytical results for the asymptotic time and distance dependence of one- and two-point correlation functions of the order parameter. We employ two complementary approaches based on asymptotic evaluations of determinants and form-factor sums. We prove that the stationary value of the two-point correlation function is not thermal, but can be described by a generalized Gibbs ensemble (GGE). The approach to the stationary state can also be understood in terms of a GGE. We present a conjecture on how these results generalize to particular quenches in other integrable models.

arXiv:1104.0154•Physical Review Letters•Jun 2, 2011•685 citations

Relaxation and thermalization after a quantum quench: Why localization is important

Simone Ziraldo, Giuseppe E. Santoro

We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/$XY$ spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.

arXiv:1211.4465•Physical Review B•Feb 4, 2013•51 citations
We present a numerical study of subsystem distance decay following a global quantum quench in the infinite one-dimensional transverse-field Ising chain, using the mathematically rigorous Bures distance BA(t)B_A(t)BA​(t) to quantify the deviation of the time-evolved reduced density matrix from its stationary generalized Gibbs ensemble state. We show that the late-time decay follows a discrete power law BA(t)∼t−λB_A(t) \sim t^{-λ}BA​(t)∼t−λ, with the exponent λλλ confined to discrete values: 111, 5/45/45/4, 3/23/23/2, 7/47/47/4, 222, 5/25/25/2, and potentially further values. The specific exponent is jointly determined by the pre- and post-quench transverse fields, as well as by properties of the symmetric excitation-fraction function mS(φ)m_S(\varphi)mS​(φ), defined on φ∈[0,π]\varphi\in[0,π]φ∈[0,π] to characterize the pre-quench Hamiltonian eigenstates, including continuity, boundary values, and first-derivative boundary values, among others. The previously established t−3/2t^{-3/2}t−3/2 decay for the initial ground state of the pre-quench Hamiltonian is naturally recovered as a special case of this general classification. Our results reveal a universal discrete structure governing local equilibration dynamics in integrable quantum systems.
We consider the reduced density matrix (RDM) ${\ensuremath{\rho}}_{A}(t)$ for a finite subsystem AAA after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of ${\ensuremath{\rho}}_{A}(t)$ is described by the RDM ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ of a generalized Gibbs ensemble. Here, we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question of how ${\ensuremath{\rho}}_{A}(t)$ approaches ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ as a function of time. To that end, we introduce a distance on the space of density matrices and show that it approaches zero as a universal power law ${t}^{\ensuremath{-}3/2}$ in time. As the RDM completely determines all local observables within AAA, this provides information on the relaxation of correlation functions of local operators. We then address the issue of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.
We study the thermalization of the transverse field Ising chain with a power law decaying interaction \sim 1/r^{\alpha} ∼ 1 / r α following a global quantum quench of the transverse field in two different dynamical regimes. The thermalization behavior is quantified by comparing the full probability distribution function (PDF) of the evolving states with the corresponding thermal state given by the canonical Gibbs ensemble (CGE). To this end, we used the matrix product state (MPS)-based Time Dependent Variational Principle (TDVP) algorithm to simulate both real time evolution following a global quantum quench and the finite temperature density operator. We observe that thermalization is strongly suppressed in the region with strong confinement for all interaction strengths \alpha α , whereas thermalization occurs in the region with weak confinement.
In this thesis, we study the properties of the transverse field Ising (TFI) chain in several non-equilibrium setups. One such setup is finite-time quantum quench—quench in which a Hamiltonian parameter is continuously changed during a finite-time, as opposed to the usual instantaneous change in the sudden quenches. A different type of quantum quench, which does not involve changing a Hamiltonian parameter, can be realised by the so-called partitioning protocol. In this protocol, two independent chains are prepared at different temperatures and then coupled into a single system and evolved as such. Another possibility to take the system out of equilibrium is to introduce a driving—for example, by suddenly and continuously changing a Hamiltonian parameter over long times. We discuss the general treatment of each of these situations, and apply them to several quench and driving protocols in the TFI chain. We analyse the behaviour of several observables, such as the total energy, spin correlation functions and Loschmidt echo, during and after quenches and during driving. While we choose the specific protocols to display the dependence of observables on the protocol features, we also investigate the universal behaviours of the observables in the late-time limit. Isolated quenched TFI chains do not relax at late times, however, local observables in these systems do—they relax to a special stationary state described by the generalised Gibbs ensemble (GGE). A similar situation happens in the case of periodic driving of the TFI chain—the late time state of the driven TFI chain, at least as far as the local observables are concerned, it is a periodic steady state described by the periodic Gibbs ensemble (PGE). In this thesis, we observe the relaxation behaviour for various quench and driving protocols and observables. In addition to studying statistical ensembles in the context of relaxation of out-of-equilibrium systems, we describe a machine learning algorithm based on the generalised Gibbs ensemble. We first describe a traditional machine learning algorithm, the Boltzmann machine, and its quantum version. Based on our knowledge of the relaxation of the out-of-equilibrium TFI chain, we modify the existing Boltzmann machine algorithm to include the many GGE charges and test it on the commonly used MNIST dataset. Rather than learning a complicated energy function and using the Boltzmann distribution, we then use the relatively simple TFI Hamiltonian and learn the effective temperatures of the GGE and find reasonably low error rates, while learning a comparatively low number of parameters.
We consider the time evolution of observables in the transverse-field Ising chain after a sudden quench of the magnetic field. We provide exact analytical results for the asymptotic time and distance dependence of one- and two-point correlation functions of the order parameter. We employ two complementary approaches based on asymptotic evaluations of determinants and form-factor sums. We prove that the stationary value of the two-point correlation function is not thermal, but can be described by a generalized Gibbs ensemble (GGE). The approach to the stationary state can also be understood in terms of a GGE. We present a conjecture on how these results generalize to particular quenches in other integrable models.
We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/XYXYXY spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.