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.
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.
We address the longstanding challenge in quantum many-body theory of reconciling unitary dynamics with irreversible relaxation. In classical chaos, the unitary evolution operator develops Ruelle-Pollicott (RP) resonances inside the unit circle in the continuum limit, leading to mixing. In the semiclassical limit, chaotic single-particle quantum systems relax with the same RP resonances. In contrast, the theory of quantum many-body RP resonances and their link to irreversibility remain underdeveloped. Here, we relate the spectral form factor to the sum of autocorrelation functions and, in generic many-body lattice systems without conservation laws, we argue that all quantum many-body RP resonances converge inside the unit disk, highlighting the role of nonunitary and the thermodynamic limit. While we conjecture this picture to be general, we analytically prove the emergence of irreversibility in the random phase model (RPM), a paradigmatic Floquet quantum circuit model, in the limit of large local Hilbert space dimension. To this end, we couple it to local environments and compute the exact time evolution of autocorrelation functions, the dissipative form factor, and out-of-time-order correlation functions (OTOCs). Although valid for any dissipation strength, we then focus on weak dissipation to clarify the origin of irreversibility in unitary systems. When the dissipationless limit is taken after the thermodynamic limit, the unitary quantum map develops an infinite tower of decaying RP resonances-chaotic systems display so-called anomalous relaxation. We identify the exact RP resonances in the RPM and prove that the same RP resonances are obtained from operator truncation. We also show that the OTOC in the RPM can undergo a two-stage relaxation, and that during the second stage, the approach to the stationary value is again controlled by the leading RP resonance. Finally, we demonstrate how conservation laws, many-body localization, and nonlocal interactions merge the leading RP resonance into the unit circle, thereby suppressing anomalous relaxation. Our findings clarify the microscopic origin of macroscopic relaxation in quantum systems and emphasize the role of RP resonances as a largely unexplored signature of many-body quantum chaos.
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.
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, 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 E as $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, to calculate the operator-size distribution of the slowest eigenmodes of 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.
We systematically investigate whether classical hydrodynamic field theories can predict the long-time dynamics of many-particle quantum systems. We study both numerically and analytically the time evolution of a chain of spins (or qubits) subjected to stroboscopic dynamics. The time evolution is implemented by a sequence of local and nearest-neighbor gates that conserve the total magnetization. The long-time dynamics of such a system is believed to be describable by a hydrodynamic field theory, which, importantly, includes the effect of noise. Based on a field theoretical analysis and symmetry arguments, we map each operator in the spin model to the corresponding fields in hydrodynamics. This allows us to predict which expectation values decay exponentially and which decay with a hydrodynamic long-time tail. We illustrate these findings by studying the time evolution of all 255 Hermitian operators that can be defined on four neighboring sites. All operators not protected by hydrodynamics decay exponentially, while the others show a slow hydrodynamic decay. While most hydrodynamic power laws seem to follow the analytical predictions, we also discuss cases where there is an apparent discrepancy between analytics and the finite-size numerical data.
We study the dynamics of Loschmidt echoes in noisy quantum many-body systems without conservation laws. We first show that the operator Loschmidt echo in noisy unitary dynamics is equivalent to the operator norm of the corresponding dissipative dynamics upon noise averaging. We then analyze this quantity in two complementary ways, revealing universal dynamical behavior. First, we develop a heuristic picture for generic Floquet systems that connects Loschmidt echoes, out-of-time-order correlators, and operator growth, which is valid at any dissipation strength. We assert that the Loschmidt echo has two dynamical regimes depending on the time t and the strength of the noise p : Gaussian decay for p t ≪ 1 and exponential decay (with a noise-independent decay rate) for p t ≫ 1 . Lastly, we rigorously prove all our results for a solvable chaotic many-body quantum circuit, the dissipative random phase model—thus providing exact insight into dissipative quantum chaos.
We study Ruelle-Pollicott resonances of translationally invariant magnetization-conserving qubit circuits via the spectrum of the quasi-momentum-resolved truncated propagator of extensive observables. Diffusive transport of the conserved magnetization is reflected in the Gaussian quasi-momentum k k dependence of the leading eigenvalue (Ruelle-Pollicott resonance) of the truncated propagator for small k k . This, in particular, allows us to extract the diffusion constant. For large k k , the leading Ruelle-Pollicott resonance is not related to transport and governs the exponential decay of correlation functions. Additionally, we conjecture the existence of a continuum of eigenvalues below the leading diffusive resonance, which governs non-exponential decay, for instance, power-law hydrodynamic tails. We expect our conclusions to hold for generic systems with exactly one U(1) conserved quantity.