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arXiv:2409.09128 · The European Physical Journal B

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

Krishanu Roychowdhury, Arnab Das

Published Sep 1, 2024• 2 citations
arXivHTMLPDF

Abstract

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

DOI
10.1140/epjb/s10051-024-00776-3

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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
Dynamical localization is one of the most startling manifestations of quantum interference, where the evolution of a simple system is frozen out under a suitably tuned coherent periodic drive. Here we show that, although any randomness in the interactions of a many-body system kills dynamical localization eventually, spectacular remnants survive even when the disorder is strong. We consider a disordered quantum Ising chain where the transverse magnetization relaxes exponentially with time with a decay time-scale $\ensuremath{\tau}$ due to random longitudinal interactions between the spins. We show that, under external periodic drive, this relaxation slows down ($\ensuremath{\tau}$ shoots up) by orders of magnitude as the ratio of the drive frequency $\ensuremath{\omega}$ and amplitude h0{h}_{0}h0​ tends to certain specific values (the freezing condition). If $\ensuremath{\omega}$ is increased while maintaining the ratio ${h}_{0}/\ensuremath{\omega}$ at a fixed freezing value, then $\ensuremath{\tau}$ diverges exponentially with $\ensuremath{\omega}.$ The results can be easily extended for a larger family of disordered fermionic and bosonic systems.
We study the real-time dynamics of a quantum Ising chain driven periodically by instantaneous quenches of the transverse field between $+{\ensuremath{\Gamma}}_{0}$ and ${\ensuremath{\Gamma}}_{0}$ back and forth in equal intervals of time. Two interesting phenomena are reported and analyzed. (i) We observe dynamical many-body freezing (DMF), i.e., strongly nonmonotonic freezing of the response with respect to the driving parameters (pulse width and height) resulting from coherent suppression of dynamics of all quasiparticle modes. For certain combinations of the pulse height and the period, maximal freezing (DMF peaks) is observed, where a massive collapse of the entire Floquet spectrum occurs and the many-body system remains frozen extremely close to the initial state for all time. (ii) Second, away from the freezing peak, we observe the emergence of a distinct oscillation with a single nontrivial frequency, which can be much lower than the driving frequency. This remarkable slow oscillation involving many high-energy modes dominates the response in the limit of long observation time. We identify this slow oscillation as the unique survivor of destructive quantum interference between the many-body modes. The oscillation tends to decay algebraically with time to a constant value. All the key features are demonstrated analytically with numerical evaluations for specific results.
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 discuss quantum dynamics in the transverse field Ising model in two spatial dimensions. We show that, up to a prethermal timescale, which we quantify, the Hilbert space ``shatters'' into dynamically disconnected subsectors. We identify this shattering as originating from the interplay of a U(1)\text{U}(1)U(1) conservation law and a one-form Z2{\mathbb{Z}}_{2}Z2​ constraint. We show that the number of dynamically disconnected sectors is exponential in system volume, and includes a subspace exponential in system volume within which the dynamics is exactly localized, even in the absence of quenched disorder. Depending on the emergent sector in which we work, the shattering can be weak (such that typical initial conditions thermalize with respect to their emergent symmetry sector), or strong (such that typical initial conditions exhibit localized dynamics). We present analytical and numerical evidence that a first-order-like ``freezing transition'' between weak and strong shattering occurs as a function of the symmetry sector, in a nonstandard thermodynamic limit. We further numerically show that on the ``weak'' (melted) side of the transition domain wall dynamics follows ordinary diffusion.
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.
We discuss quantum dynamics in the transverse field Ising model in two spatial dimensions. We show that, up to a prethermal timescale, which we quantify, the Hilbert space 'shatters' into dynamically disconnected subsectors. We identify this shattering as originating from the interplay of a U(1)\mathrm{U}(1)U(1) conservation law and a one-form Z2\mathbb{Z}_2Z2​ constraint. We show that the number of dynamically disconnected sectors is exponential in system volume, and includes a subspace exponential in system volume within which the dynamics is exactly localized, even in the absence of quenched disorder. Depending on the emergent sector in which we work, the shattering can be weak (such that typical initial conditions thermalize with respect to their emergent symmetry sector), or strong (such that typical initial conditions exhibit localized dynamics). We present analytical and numerical evidence that a first order-like 'freezing transition' between weak and strong shattering occurs as a function of the symmetry sector, in a non-standard thermodynamic limit. We further numerically show that on the 'weak' (melted) side of the transition domain wall dynamics follows ordinary diffusion.