Session G50: Quantum Thermalization Dynamics

Local pairing of Feynman histories in many-body Floquet models

We study many-body quantum dynamics using Floquet quantum circuits in one space dimension as simple examples of systems with local interactions that support ergodic phases. Physical properties can be expressed in terms of multiple sums over Feynman histories, which for these models are paths or many-body orbits in Fock space. A natural simplification of such sums is the diagonal approximation, where the only terms that are retained are ones in which each path is paired with a partner that carries the complex conjugate weight. We identify the regime in which the diagonal approximation holds, and the nature of the leading corrections to it.  We show that properties are dominated at long times by contributions to orbit sums in which each orbit is paired locally with a conjugate, as in the diagonal approximation, but that in large systems these contributions consist of many spatial domains, with distinct local pairings in neighbouring domains. The existence of these domains is reflected in deviations of the SFF from RMT predictions, and of matrix element correlations from ETH predictions; deviations of both kinds diverge with system size. We also show that, within this picture, the transition from an ergodic to a many-body localised phase can be viewed as symmetry breaking.   

Thermalization of Randomly Coupled SYK Models

We investigate the thermalization of Sachdev-Ye-Kitaev (SYK) models coupled via random interactions following quenches from the perspective of entanglement. Previous studies have shown that when a system of two SYK models coupled by random two-body terms is quenched from the thermofield double state with sufficiently low effective temperature, the R\'enyi entropies do \emph{not} saturate to the expected thermal values in the large-$N$ limit. Using numerical large-$N$ methods, we first show that the R\'enyi entropies in a pair SYK models coupled by two-body terms can thermalize, if quenched from a state with sufficiently high effective temperature, and hence exhibit state-dependent thermalization. In contrast, SYK models coupled by single-body terms appear to always thermalize. We provide evidence that the subthermal behavior in the former system is likely a large-$N$ artifact by repeating the quench for finite $N$ and finding that the saturation value of the R\'enyi entropy extrapolates to the expected thermal value in the $N \to \infty$ limit. Finally, as a finer grained measure of thermalization, we compute the late-time spectral form factor of the reduced density matrix after the quench. While a single SYK dot exhibits perfect agreement with random matrix theory, both the quadratically and quartically coupled SYK models exhibit slight deviations.   

Tunable Local Thermalization of a Disordered, Floquet-Engineered Dipolar Ensemble

Ubiquity of local thermalization in ergodic quantum systems is a central paradigm of condensed matter physics. Unravelling the local relaxation dynamics in specific models however remains an open theoretical and experimental challenge. In this talk we report on recent advances in which dense ensembles of nitrogen vacancy centers in diamond are Floquet-engineered to modify the spin exchange anisotropy of their native dipolar interaction. In addition to standard Ramsey measurements as a global probe, we develop a novel technique to measure ensemble-averaged, infinite temperature autocorrelation functions of local operators. This technique, exploiting strong local disorder in the system, functions as an exquisite probe of local thermalization in the many-body system. As a function of the tunable exchange anisotropy, we show how both the timescale and shape of these measured spin autocorrelations are nontrivially modified, in striking contrast to the expectations set by the NMR literature. In particular, we show theoretically how the shape of the relaxation, as quantified by stretching exponents, encodes the correlation properties of the system's intrinsic spin bath that emerges to thermalize the system. In addition to providing novel physical heuristics to interpret the experimental results, our work establishes a general phenomenology for understanding out-of-equilibrium quench dynamics in disordered, long-range interacting systems.   

Strongly-interacting Integrable Floquet Hopping models from Rydberg blockade

In this paper, we introduce a class of strongly-interacting Floquet models, whose single-particle dynamics are equivalent to bosonic hopping models. We propose to experimentally realize strongly interacting versions of these using Rydberg atom arrays, where the interactions arise from the Rydberg blockade effect. We do this by developing a gate which conserves both particle number and respects the Rydberg blockade, dynamically conserving two U(1) charges. We study in detail the dynamics in a 1D chain, and identify a large class of integrable classical cellular automata. We prove integrability via a coordinate Bethe Ansatz solution, and harness the classical cellular automata description of the dynamics to gain insight into its origin. Remarkably, these automata remain integrable to natural quantum deformations, resulting in a large class of integrable quantum cellular automata. Using the thermodynamic Bethe Ansatz, we analyze the expected behavior of experimentally realizable quenches within the framework of generalized hydrodynamics, and find unusual diffusive behaviors in certain parameter regimes.   

Energy relaxation in Dirac semimetals

Electron-phonon interactions mediate BCS-superconductivity and charge density wave phase transitions but also are the main pathway to thermalization in electronic systems. We compute the temperature as a function of time after a sudden excitation, i.e., laser pulse, in simple models of Weyl/Dirac semimetals, nodal-line semimetals and graphene. These materials have zero-energy nodal points that constrain the electron-phonon dynamics in particular ways. Above the Bloch-Gruneisen temperature, we find inverse log, inverse, power law, exponential, and linear relaxation behaviors in various limits. The most common relaxation behavior at high and low temperature is linear due to optical and (longitudinal) acoustic phonons respectively. The nodal line constrains the electron momenta in scattering processes to be near the nodal line and hence the relaxation is a power law in graphene and Weyl/Dirac semimetals but exponential in nodal-line semimetals at low densities and temperatures.   

Conserved Quantities from Entanglement Hamiltonian

We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities. By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigen-operators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly in power law if the system is integrable, with the exponent depending on if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigen-operators of which correspond to the identity and the subregion Hamiltonian.   

Mechanisms for the emergence of Gaussian correlations

We comprehensively investigate two distinct mechanisms leading to memory loss of non-Gaussian correlations after switching off the interactions in an isolated quantum system undergoing out-of-equilibrium dynamics. The first mechanism is based on spatial scrambling and results in the emergence of locally Gaussian steady states in large systems evolving over long times. The second mechanism, characterized as `canonical transmutation', is based on the mixing of a pair of canonically conjugate fields, one of which initially exhibits non-Gaussian fluctuations while the other is Gaussian and dominates the dynamics, resulting in the emergence of relative Gaussianity even at finite system sizes and times. We evaluate signatures of the occurrence of the two candidate mechanisms in a recent experiment that has observed Gaussification in an atom-chip controlled ultracold gas and elucidate evidence that it is canonical transmutation rather than spatial scrambling that is responsible for Gaussification in the experiment. Both mechanisms are shown to share the common feature that the Gaussian correlations revealed dynamically by the quench are already present though practically inaccessible at the initial time. On the way, we present novel observations based on the experimental data, demonstrating clustering of equilibrium correlations, analyzing the dynamics of full counting statistics, and utilizing tomographic reconstructions of quantum field states. Our work aims at providing an accessible presentation of the potential of atom-chip experiments to explore fundamental aspects of quantum field theories in quantum simulations.   

Dynamics of a time-dependent spin chain

We study the dynamics of a time-dependent quantum spin chain. We analytically determine the spin dynamics and find exotic discrete resonant peaks surfing above continuum spectrum from low to high energies. Those results are further confirmed by large-scale infinite time-evolving block decimation (iTEBD) calculation. Our results provide a practicable approach to go beyond integrability.   

Relative entropy of fermionic states from Wigner function characteristics

The study of entanglement and its various measures has ubiquitous interest in quantum physics. Particularly in quantum condensed matter, novel states both at the bottom and the middle of the spectrum have been shown to exhibit universal scaling of entanglement entropy (EE) both in static and dynamical situations. Using recent Wigner-characteristic based techniques to calculate EE, we give a microscopic understanding of the universal scaling of EE with subsystem fraction from the structure of eigenstates. We will also show how to adapt the technique to compute the relative entropy between two fermionic density matrices. Of particular interest is the relative entropy between the reduced density matrix of a many body quantum state and a thermal ensemble defined on the same degrees of freedom. This provides an information theoretic measure of how close reduced density matrices obtained by partial tracing an arbitrary Fock state are close to a thermal state. We contrast this with the usual Eigenstate Thermalisation Hypothesis prediction based on expectation values of local observables to adjudge thermalisation, or the lack thereof.   

Influence functional of quantum many-body systems

Feynman-Vernon influence functional (IF) was originally introduced to describe the effect of a quantum environment on the dynamics of an open quantum system. We apply the IF approach to describe quantum many-body dynamics in isolated spin systems, viewing the system as an environment for its local subsystems. While the IF can be computed exactly only in certain many-body models, it generally satisfies a self-consistency equation, provided the system, or an ensemble of systems, are translationally invariant. We view the IF as a fictitious wavefunction in the temporal domain, and approximate it using matrix-product states (MPS). This approach is efficient provided the temporal entanglement of the IF is sufficiently low. We illustrate the versatility of the IF approach by analyzing several models that exhibit a range of dynamical behaviors, from thermalizing to many-body localized, in both Floquet and Hamiltonian settings. The IF approach offers a new lens on many-body non-equilibrium phenomena, both in ergodic and non-ergodic regimes, connecting the theory of open quantum systems theory to quantum statistical physics.   

Computing quench dynamics for thermalizing quantum-many body systems in the influence matrix approach

Efficient simulation of quantum many-body systems is the central challenge in the field of computational quantum physics. Recently, an approach based on Feynman and Vernon's influence functional was proposed which allows the computation of local dynamics in interacting systems. In contrast to conventional methods its efficiency relies on the low amount of temporal entanglement instead of spatial entanglement. In this talk we will discuss different types of systems and dynamical regimes where this approach can be sucessfully employed. In particular we analyze thermalizing and many-body localized dynamics in one dimensional systems, as well as transport in interacting heisenberg chains. Finally we will show how to tackle 2D systems using these tools.   

Classification of Non-Interacting Quantum Baths: Reconstruction of the Influence Matrix from Keldysh Correlation Functions

The recently developed influence matrix (IM) approach has opened new opportunities for the efficient simulation of quantum many-body dynamics in the thermalising and localised regimes. This approach is based on the Feynman-Vernon influence functional, which encodes the dynamical influence of a many-body bath on its local subsystems.

While a self-consistent approach to the computation of the IM using tensor-network techniques has proven to be versatile and efficient, a deeper analytical insight into the structure of IM is desirable to describe and classify quantum many body baths. In this talk, I will present a complementary perspective in which the IM is viewed as generating functional for Keldysh correlation functions. By computing these correlation functions explicitly, we reconstruct the IM for generic non-interacting systems of fermions. We will discuss the entanglement scaling of IM, as well as the dynamical properties of the resulting bath depending on the fermions spectral properties.   

Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes

We study a mechanism whereby quantum information present in the initial state of a quantum many-body system can be protected for arbitrary times due to a combination of symmetry and spatial locality. Remarkably, the mechanism is sufficiently generic that the dynamics can be fully ergodic upon resolving the protecting symmetry and fixing the encoded quantum state, resulting in an infinite-temperature quantum memory. After exemplifying the mechanism in a strongly nonintegrable two-dimensional (2D) spin model inspired by the surface code, we find it has a natural interpretation in the language of noiseless subsystems and stabilizer subsystem codes. This interpretation yields a number of further examples, including a nonintegrable Hamiltonian with quantum memory based on the Bacon-Shor code. The lifetime of the encoded quantum information in these models is infinite provided the dynamics respect the stabilizer symmetry of the underlying subsystem code. In the presence of symmetry-violating perturbations, we make contact with previous work leveraging the concept of prethermalization to show that the encoded quantum information retains a parametrically long lifetime under dynamics with an enlarged continuous symmetry group. We identify conditions on the underlying subsystem code that enable such a prethermal enhancement of the memory lifetime.   

Charge transport and chaos in the complex Brownian SYK chain

Extended chaotic quantum systems with global symmetries allow for transport of localized charges and display quantum information scrambling, both of which depend on the nature of interactions in the system. The properties of transport and scrambling are therefore implicitly related, and to study the interplay between the two, we utilize the Complex Brownian SYK model defined on a one-dimensional lattice. We investigate the charge-transport through the two-point correlator of charge density in both the free (quadratic inter-site couplings) and interacting (quartic inter-site couplings) model, which post-disorder averaging maps to the imaginary time dynamics of an effective spin-chain with an SU(2) x U(1) symmetry. The quantum information scrambling is probed via the out-of-time-ordered correlator, the computation of which is also mapped to an effective spin chain that evolves in imaginary time and displays an SU(4) x U(1) symmetry. This mapping to spin models with SU(n) symmetries enables us to analytically solve the free model and numerically investigate the interacting model for large system size using tensor network techniques, which we then utilize to outline the relation between charge transport and scrambling in the Complex Brownian SYK chain.   

Simulating the real time diffusive dynamics of the 2D Fermi-Hubbard on a large lattice

I apply the Liouvillean graph effective model of [1] to the Fermi-Hubbard model.  I first consider the regimes in which the effective model is well-justified, and find that it requires interaction strength $U \sim t$ the hopping amplitude.  I then compute the finite-wavelength effective charge diffusion coefficents and momentum relaxation rate, and compare these results to the cold-atom experiment of [2].  Finally, I compute the true long-wavelength diffusion coefficent and THING; illuminating finite-size effects in the experiment [2], and compare to the Mott-Ioffe-Regel limit.

[1] Christopher David White, "Effective dissipation rate in a Liouvillean graph picture of high-temperature quantum hydrodynamics", arXiv:2108.00019

[2] Brown et al, "Bad metallic transport in a cold atom Fermi-Hubbard system", Science 363,379, 2019