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:2302.12275 · Physical review. B./Physical review. B

Kohlrausch regime of slow relaxation and its phenomenology in isolated quantum many-body systems with strong disorder

Asmi Haldar

Published Feb 5, 2025• 4 citations
arXivHTMLPDF

Abstract

The Kohlrausch(-Williams-Watts) law of stretched exponential relaxation has been observed for over a century and a half in diverse complex classical systems. Here we show that this law describes relaxation quite generically in closed (executing Schrödinger dynamics), interacting disordered many-body systems, using interaction range and disorder strength as primary tuning parameters. This we observe for both time-independent and periodically driven (Floquet) systems. Finite-size analysis indicates the persistence of this nonthermal relaxation regime in the thermodynamic limit thus defining a distinct dynamical regime. This regime exhibits a peak in the time scale of the long-time relaxation (following a transient), upon crossing over from weak to strong disorder. We provide a simple picture of this behavior, which naturally accounts for its generic occurrence. Formation of spin-glass — one of the possible mechanisms for stretched relaxation appears incidental to the occurrence of Kohlrausch law in our context. Finally, we provide a simple non-Hermitian Hamiltonian formulation for the dynamics of a single spin embedded in the disordered chain. This provides an analytical formula that captures not only the Kohlrausch relaxation of the disorder averaged autocorrelation but also captures the largely diverse dynamics of an arbitrary target spin in the system. Our work hence also provides a concrete quantification of the “prethermal slowness” in many-body disordered system.

DOI
10.1103/physrevb.111.075113

Related papers

Glassy quantum dynamics of disordered Ising spins

Philipp Schultzen, Titus Franz, Sebastian Geier +6 more

We study the out-of-equilibrium dynamics of the quantum Ising model with power-law interactions and positional disorder. For arbitrary dimension $d$ and interaction range $\ensuremath{\alpha}\ensuremath{\ge}d$ we analytically find a stretched-exponential decay of the global magnetization and ensemble-averaged single-spin purity with a stretch power $\ensuremath{\beta}=d/\ensuremath{\alpha}$ in the thermodynamic limit. Numerically, we confirm that glassy behavior persists for finite system sizes and sufficiently strong disorder. We identify dephasing between disordered coherent pairs as the main mechanism leading to a relaxation of global magnetization, whereas genuine many-body interactions lead to a loss of single-spin purity which signifies the buildup of entanglement. The emergence of glassy dynamics in the quantum Ising model extends prior findings in classical and open quantum systems, where the stretched-exponential law is explained by a scale-invariant distribution of timescales, to both integrable and nonintegrable quantum systems.

arXiv:2104.00349•Physical review. B./Physical review. B•Jan 19, 2022•13 citations

Phenomenology of the Prethermal Many-Body Localized Regime

David M. Long, Philip J. D. Crowley, Vedika Khemani +1 more

The dynamical phase diagram of interacting disordered systems has seen substantial revision over the past few years. Theory must now account for a large prethermal many-body localized regime in which thermalization is extremely slow, but not completely arrested. We derive a quantitative description of these dynamics in short-ranged one-dimensional systems using a model of successive many-body resonances. The model explains the decay timescale of mean autocorrelators, the functional form of the decay-a stretched exponential-and relates the value of the stretch exponent to the broad distribution of resonance timescales. The Jacobi method of matrix diagonalization provides numerical access to this distribution, as well as a conceptual framework for our analysis. The resonance model correctly predicts the stretch exponents for several models in the literature. Successive resonances may also underlie slow thermalization in strongly disordered systems in higher dimensions, or with long-range interactions.

arXiv:2207.05761•Physical Review Letters•Sep 8, 2023•61 citations

Stretched-exponential melting of a dynamically frozen state under imprinted phase noise in the ising chain in a transverse field

Krishanu Roychowdhury, Arnab Das

Abstract The concept of dynamical freezing is a phenomenon where a suitable set of local observables freezes under a strong periodic drive in a quantum many-body system. This happens because of the emergence of approximate but perpetual conservation laws when the drive is strong enough. In this work, we probe the resilience of dynamical freezing to random perturbations added to the relative phases between the interfering states (elements of a natural basis) in the time-evolving wave function after each drive cycle. We study this in an integrable Ising chain in a time-periodic transverse field. Our key finding is, that the imprinted phase noise melts the dynamically frozen state, but the decay is “slow”: a stretched-exponential decay rather than an exponential one. Stretched-exponential decays (also known as Kohlrausch relaxation) are usually expected in complex systems with time-scale hierarchies due to strong disorders or other inhomogeneities resulting in jamming, glassiness, or localization. Here we observe this in a simple translationally invariant system dynamically frozen under a periodic drive. Moreover, the melting here does not obliterate the entire memory of the initial state but leaves behind a steady remnant that depends on the initial conditions. This underscores the stability of dynamically frozen states. Graphical abstract

arXiv:2409.09128•The European Physical Journal B•Sep 1, 2024•2 citations

Emergent pair localization in a many-body quantum spin system

Titus Franz, Sebastian Geier, A. Braemer +8 more

Understanding how closed quantum systems dynamically approach thermal equilibrium presents a major unresolved problem in statistical physics. Generically, non-integrable quantum systems are expected to thermalize as they comply with the Eigenstate Thermalization Hypothesis. However, in the presence of strong disorder, the dynamics can possibly slow down to a degree that systems fail to thermalize on experimentally accessible timescales, as in spin glasses or many-body localized systems. In general, particularly in long-range interacting quantum systems, the specific nature of the disorder necessary for the emergence of a prethermal, metastable state--distinctly separating the timescales of initial relaxation and subsequent slow thermalization--remains an open question. We study an ensemble of Heisenberg spins with a tunable distribution of random coupling strengths realized by a Rydberg quantum simulator. We observe a drastic change in the late-time magnetization when increasing disorder strength. The data is well described by models based on pairs of strongly interacting spins, which are treated as thermal for weak disorder and isolated for strong disorder. Our results indicate a crossover into a pair-localized prethermal regime in a closed quantum system of thousands of spins in the critical case where the exponent of the power law interaction matches the spatial dimension.

arXiv:2207.14216•Jul 28, 2022•0 citations

Slow Nonthermalizing Dynamics in a Quantum Spin Glass

Louk Rademaker, Dmitry A. Abanin

Spin glasses and many-body localization (MBL) are prime examples of ergodicity breaking, yet their physical origin is quite different: the former phase arises due to rugged classical energy landscape, while the latter is a quantum-interference effect. Here, we study quantum dynamics of an isolated 1D spin glass under application of a transverse field. At high energy densities, the system is ergodic, relaxing via a resonance avalanche mechanism, that is also responsible for the destruction of MBL in nonglassy systems with power-law interactions. At low energy densities, the interaction-induced fields obtain a power-law soft gap, making the resonance avalanche mechanism inefficient. This leads to the persistence of the spin-glass order, as demonstrated by resonance analysis and by numerical studies. A small fraction of resonant spins forms a thermalizing system with long-range entanglement, making this regime distinct from the conventional MBL. The model considered can be realized in systems of trapped ions, opening the door to investigating slow quantum dynamics induced by glassiness.

arXiv:1910.04421•Physical Review Letters•Dec 28, 2020•19 citations

Dynamical crossover behavior in the relaxation of quenched quantum many-body systems

Aamir A. Makki, Souvik Kumar Bandyopadhyay, Somnath Maity +1 more

A crossover between different power-law relaxation behaviors of many-body periodically driven integrable systems has come to light in recent years. We demonstrate, using integrable quantum systems, that similar kinds of dynamical transitions may also occur in the relaxation of such systems following a sudden quench. Particularly, we observe two distinct power-law relaxation behaviors following a sudden quench in the integrable XY model, depending upon whether the quenched Hamiltonian lies in the commensurate or the incommensurate phase. The relaxation behavior for quenches at and near the boundary line, called the disorder line (DL), separating these phases is also characterized. The relaxation at the DL shows a scaling exponent different from that for quenches to either of the commensurate or incommensurate phases. The transitions occur through a crossover from the commensurate or incommensurate scaling behavior to the DL scaling behavior. The crossover time diverges as a power law as the parameters of the final quenched Hamiltonian approach the DL. The transitions are also observed to be robust under weak integrability-breaking perturbations but disappear following strongly chaotic quenches.

arXiv:2112.02930•Physical review. B./Physical review. B•Feb 1, 2022•24 citations
The Kohlrausch(-Williams-Watts) law of stretched exponential relaxation has been observed for over a century and a half in diverse complex classical systems. Here we show that this law describes relaxation quite generically in closed (executing Schrödinger dynamics), interacting disordered many-body systems, using interaction range and disorder strength as primary tuning parameters. This we observe for both time-independent and periodically driven (Floquet) systems. Finite-size analysis indicates the persistence of this nonthermal relaxation regime in the thermodynamic limit thus defining a distinct dynamical regime. This regime exhibits a peak in the time scale of the long-time relaxation (following a transient), upon crossing over from weak to strong disorder. We provide a simple picture of this behavior, which naturally accounts for its generic occurrence. Formation of spin-glass — one of the possible mechanisms for stretched relaxation appears incidental to the occurrence of Kohlrausch law in our context. Finally, we provide a simple non-Hermitian Hamiltonian formulation for the dynamics of a single spin embedded in the disordered chain. This provides an analytical formula that captures not only the Kohlrausch relaxation of the disorder averaged autocorrelation but also captures the largely diverse dynamics of an arbitrary target spin in the system. Our work hence also provides a concrete quantification of the “prethermal slowness” in many-body disordered system.
We study the out-of-equilibrium dynamics of the quantum Ising model with power-law interactions and positional disorder. For arbitrary dimension ddd and interaction range $\ensuremath{\alpha}\ensuremath{\ge}d$ we analytically find a stretched-exponential decay of the global magnetization and ensemble-averaged single-spin purity with a stretch power $\ensuremath{\beta}=d/\ensuremath{\alpha}$ in the thermodynamic limit. Numerically, we confirm that glassy behavior persists for finite system sizes and sufficiently strong disorder. We identify dephasing between disordered coherent pairs as the main mechanism leading to a relaxation of global magnetization, whereas genuine many-body interactions lead to a loss of single-spin purity which signifies the buildup of entanglement. The emergence of glassy dynamics in the quantum Ising model extends prior findings in classical and open quantum systems, where the stretched-exponential law is explained by a scale-invariant distribution of timescales, to both integrable and nonintegrable quantum systems.
The dynamical phase diagram of interacting disordered systems has seen substantial revision over the past few years. Theory must now account for a large prethermal many-body localized regime in which thermalization is extremely slow, but not completely arrested. We derive a quantitative description of these dynamics in short-ranged one-dimensional systems using a model of successive many-body resonances. The model explains the decay timescale of mean autocorrelators, the functional form of the decay-a stretched exponential-and relates the value of the stretch exponent to the broad distribution of resonance timescales. The Jacobi method of matrix diagonalization provides numerical access to this distribution, as well as a conceptual framework for our analysis. The resonance model correctly predicts the stretch exponents for several models in the literature. Successive resonances may also underlie slow thermalization in strongly disordered systems in higher dimensions, or with long-range interactions.
Abstract The concept of dynamical freezing is a phenomenon where a suitable set of local observables freezes under a strong periodic drive in a quantum many-body system. This happens because of the emergence of approximate but perpetual conservation laws when the drive is strong enough. In this work, we probe the resilience of dynamical freezing to random perturbations added to the relative phases between the interfering states (elements of a natural basis) in the time-evolving wave function after each drive cycle. We study this in an integrable Ising chain in a time-periodic transverse field. Our key finding is, that the imprinted phase noise melts the dynamically frozen state, but the decay is “slow”: a stretched-exponential decay rather than an exponential one. Stretched-exponential decays (also known as Kohlrausch relaxation) are usually expected in complex systems with time-scale hierarchies due to strong disorders or other inhomogeneities resulting in jamming, glassiness, or localization. Here we observe this in a simple translationally invariant system dynamically frozen under a periodic drive. Moreover, the melting here does not obliterate the entire memory of the initial state but leaves behind a steady remnant that depends on the initial conditions. This underscores the stability of dynamically frozen states. Graphical abstract
Understanding how closed quantum systems dynamically approach thermal equilibrium presents a major unresolved problem in statistical physics. Generically, non-integrable quantum systems are expected to thermalize as they comply with the Eigenstate Thermalization Hypothesis. However, in the presence of strong disorder, the dynamics can possibly slow down to a degree that systems fail to thermalize on experimentally accessible timescales, as in spin glasses or many-body localized systems. In general, particularly in long-range interacting quantum systems, the specific nature of the disorder necessary for the emergence of a prethermal, metastable state--distinctly separating the timescales of initial relaxation and subsequent slow thermalization--remains an open question. We study an ensemble of Heisenberg spins with a tunable distribution of random coupling strengths realized by a Rydberg quantum simulator. We observe a drastic change in the late-time magnetization when increasing disorder strength. The data is well described by models based on pairs of strongly interacting spins, which are treated as thermal for weak disorder and isolated for strong disorder. Our results indicate a crossover into a pair-localized prethermal regime in a closed quantum system of thousands of spins in the critical case where the exponent of the power law interaction matches the spatial dimension.
Spin glasses and many-body localization (MBL) are prime examples of ergodicity breaking, yet their physical origin is quite different: the former phase arises due to rugged classical energy landscape, while the latter is a quantum-interference effect. Here, we study quantum dynamics of an isolated 1D spin glass under application of a transverse field. At high energy densities, the system is ergodic, relaxing via a resonance avalanche mechanism, that is also responsible for the destruction of MBL in nonglassy systems with power-law interactions. At low energy densities, the interaction-induced fields obtain a power-law soft gap, making the resonance avalanche mechanism inefficient. This leads to the persistence of the spin-glass order, as demonstrated by resonance analysis and by numerical studies. A small fraction of resonant spins forms a thermalizing system with long-range entanglement, making this regime distinct from the conventional MBL. The model considered can be realized in systems of trapped ions, opening the door to investigating slow quantum dynamics induced by glassiness.
A crossover between different power-law relaxation behaviors of many-body periodically driven integrable systems has come to light in recent years. We demonstrate, using integrable quantum systems, that similar kinds of dynamical transitions may also occur in the relaxation of such systems following a sudden quench. Particularly, we observe two distinct power-law relaxation behaviors following a sudden quench in the integrable XY model, depending upon whether the quenched Hamiltonian lies in the commensurate or the incommensurate phase. The relaxation behavior for quenches at and near the boundary line, called the disorder line (DL), separating these phases is also characterized. The relaxation at the DL shows a scaling exponent different from that for quenches to either of the commensurate or incommensurate phases. The transitions occur through a crossover from the commensurate or incommensurate scaling behavior to the DL scaling behavior. The crossover time diverges as a power law as the parameters of the final quenched Hamiltonian approach the DL. The transitions are also observed to be robust under weak integrability-breaking perturbations but disappear following strongly chaotic quenches.