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arXiv:2409.17238 · Physical review. B./Physical review. B

Spectral gaps of local quantum channels in the weak-dissipation limit

J. Alexander Jacoby, David A. Huse, Sarang Gopalakrishnan

Published Mar 10, 2025• 19 citations
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Abstract

We consider the dynamics of generic chaotic quantum many-body systems with no conservation laws, subject to weak bulk dissipation. It was recently observed [T. Mori, Phys. Rev. B 109, 064311 (2024)] that the generator of these dissipative dynamics, a quantum channel $\mathcal{E}$, retains a nonzero gap as the dissipation strength $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ if the thermodynamic limit is taken first. We use a hydrodynamic description of operator spreading in the presence of dissipation to estimate the gap of $\mathcal{E}$ as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, to calculate the operator-size distribution of the slowest eigenmodes of $\mathcal{E}$, and to relate the gap to the long-time decay rates of autocorrelation functions under unitary Floquet dynamics. We provide a microscopic derivation of this hydrodynamic perspective for random unitary circuits. We argue that the gap in the $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ limit can change nonanalytically as one tunes the parameters of the unitary dynamics.

DOI
10.1103/physrevb.111.104303

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We consider the dynamics of generic chaotic quantum many-body systems with no conservation laws, subject to weak bulk dissipation. It was recently observed [T. Mori, Phys. Rev. B 109, 064311 (2024)] that the generator of these dissipative dynamics, a quantum channel E\mathcal{E}E, retains a nonzero gap as the dissipation strength \ensuremathγ\ensuremath→0\ensuremath{\gamma}\ensuremath{\rightarrow}0\ensuremathγ\ensuremath→0 if the thermodynamic limit is taken first. We use a hydrodynamic description of operator spreading in the presence of dissipation to estimate the gap of E\mathcal{E}E as \ensuremathγ\ensuremath→0\ensuremath{\gamma}\ensuremath{\rightarrow}0\ensuremathγ\ensuremath→0, to calculate the operator-size distribution of the slowest eigenmodes of E\mathcal{E}E, and to relate the gap to the long-time decay rates of autocorrelation functions under unitary Floquet dynamics. We provide a microscopic derivation of this hydrodynamic perspective for random unitary circuits. We argue that the gap in the \ensuremathγ\ensuremath→0\ensuremath{\gamma}\ensuremath{\rightarrow}0\ensuremathγ\ensuremath→0 limit can change nonanalytically as one tunes the parameters of the unitary dynamics.
In thermalizing many-body quantum systems without conservation laws, such as ergodic Floquet and random unitary circuits, local expectation values are predicted to decay to their equilibrium values exponentially quickly. In this work, we derive a relationship between said exponential decay rate g ¯ and the operator spreading properties of a local unitary evolution. A hydrodynamical picture for operator spreading allows us to argue that, for random unitary circuits, g ¯ is encoded by the leading eigenvalue of a dynamical map obtained by enriching unitary dynamics with dissipation, in the limit of weak dissipation. We argue that the size of the eigenvalue does not depend on the details of this weak dissipation (given mild assumptions on properties of the ergodic dynamics), so long as it only suppresses large operators significantly. Our calculations are based on analytical results for random unitary circuits, but we argue that similar results hold for ergodic Floquet systems. These conjectures are in accordance with existing results that numerically obtain quantum many-body analogs of classical Ruelle-Pollicott resonances [Prosen, ; Mori, ] by studying unitary evolutions subject to weak dissipation.
Markovian open many-body quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate towards the stationary state, but it has recently been pointed out that the spectral gap does not necessarily determine the overall relaxation time. Our understanding on the relaxation process before the asymptotically long-time regime is still limited. We here present a collective relaxation dynamics of autocorrelation functions in the stationary state. As a key quantity in the analysis, we introduce the instantaneous decay rate, which characterizes the transient relaxation and converges to the conventional asymptotic decay rate in the long-time limit. Our theory predicts that a bulk-dissipated system generically shows an accelerated decay before the asymptotic regime due to the scrambling of quantum information associated with the operator spreading.
We study relaxation times, also called mixing times, of quantum many-body systems described by a Lindblad master equation. We in particular study the scaling of the spectral gap with the system length, the so-called dynamical exponent, identifying a number of transitions in the scaling. For systems with bulk dissipation we generically observe different scaling for small and for strong dissipation strength, with a critical transition strength going to zero in the thermodynamic limit. We also study a related phase transition in the largest decay mode. For systems with only boundary dissipation we show a generic bound that the gap cannot be larger than ∼1/L. In integrable systems with boundary dissipation one typically observes scaling of ∼1/L(3), while in chaotic ones one can have faster relaxation with the gap scaling as ∼1/L and thus saturating the generic bound. We also observe transition from exponential to algebraic gap in systems with localized modes.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
Recent experiments have reported that novel physics emerge in open quantum many-body systems due to an interplay of interactions and dissipation, which stimulate theoretical studies of the many-body Lindblad equation. Although the strong dissipation regime receives considerable interest in this context, this work focuses on the weak bulk dissipation. By examining the spectral property of the many-body Lindblad generator for specific models, we find that its spectral gap shows singularity in the weak dissipation limit when the thermodynamic limit is taken first. Based on analytical arguments and numerical calculations, we conjecture that such a singularity is generic in bulk-dissipated quantum many-body systems and is related to the concept of the Ruelle-Pollicott resonance in chaos theory, which determines the timescale of thermalization of an isolated system. This conjecture suggests that the many-body Lindblad equation in the weak dissipation regime contains nontrivial information on intrinsic properties of a quantum many-body system.
Dissipation is a ubiquitous phenomenon that affects the fate of chaotic quantum many-body dynamics. Here, we show that chaos can be robust against dissipation but can also assist and anomalously enhance relaxation. We compute exactly the dissipative form factor of a generic Floquet quantum circuit with arbitrary on-site dissipation modeled by quantum channels and find that, for long enough times, the system always relaxes with two distinctive regimes characterized by the presence or absence of gap-closing. While the system can sustain a robust ramp for a long (but finite) time interval in the gap-closing regime, relaxation is "assisted" by quantum chaos in the regime where the gap remains nonzero. In the latter regime, we prove that, if the thermodynamic limit is taken first, the gap does not close even in the dissipationless limit. We complement our analytical findings with numerical results for quantum qubit circuits.