We study a generic but simple nonintegrable quantum many-body system of locally interacting particles, namely, a kicked-parameter (t,V) model of spinless fermions on a one-dimensional lattice (equivalent to a kicked Heisenberg XX-Z chain of 1/2 spins). The statistical properties of the dynamics (quantum ergodicity and quantum mixing) and the nature of quantum transport in the thermodynamic limit are considered as the kick parameters (which control the degree of nonintegrability) are varied. We find and demonstrate ballistic transport and nonergodic, nonmixing dynamics (implying infinite conductivity at all temperatures) in the integrable regime of zero or very small kick parameters, and more generally and importantly, also in the nonintegrable regime of intermediate values of kicked parameters, whereas only for sufficiently large kick parameters do we recover quantum ergodicity and mixing implying normal (diffusive) transport. We propose an order parameter (charge stiffness D) which controls the phase transition from nonmixing and nonergodic dynamics (ordered phase, D>0) to mixing and ergodic dynamics (disordered phase, D=0) in the thermodynamic limit. Furthermore, we find exponential decay of time correlation functions in the regime of mixing dynamics. The results are obtained consistently within three different numerical and analytical approaches: (i) time evolution of a finite system and direct computation of time correlation functions, (ii) full diagonalization of finite systems and statistical analysis of stationary data, and (iii) algebraic construction of quantum invariants of motion of an infinite system, in particular the time-averaged observables.
We study a generic but simple nonintegrable quantum many-body system of locally interacting particles, namely, a kicked-parameter (t,V) model of spinless fermions on a one-dimensional lattice (equivalent to a kicked Heisenberg XX-Z chain of 1/2 spins). The statistical properties of the dynamics (quantum ergodicity and quantum mixing) and the nature of quantum transport in the thermodynamic limit are considered as the kick parameters (which control the degree of nonintegrability) are varied. We find and demonstrate ballistic transport and nonergodic, nonmixing dynamics (implying infinite conductivity at all temperatures) in the integrable regime of zero or very small kick parameters, and more generally and importantly, also in the nonintegrable regime of intermediate values of kicked parameters, whereas only for sufficiently large kick parameters do we recover quantum ergodicity and mixing implying normal (diffusive) transport. We propose an order parameter (charge stiffness D) which controls the phase transition from nonmixing and nonergodic dynamics (ordered phase, D>0) to mixing and ergodic dynamics (disordered phase, D=0) in the thermodynamic limit. Furthermore, we find exponential decay of time correlation functions in the regime of mixing dynamics. The results are obtained consistently within three different numerical and analytical approaches: (i) time evolution of a finite system and direct computation of time correlation functions, (ii) full diagonalization of finite systems and statistical analysis of stationary data, and (iii) algebraic construction of quantum invariants of motion of an infinite system, in particular the time-averaged observables.
Numerical evidence is given for nonergodic (nonmixing) behavior, exhibiting ideal transport, of a simple nonintegrable many-body quantum system in the thermodynamic limit, namely, the kicked t- V model of spinless fermions on a ring. However, for sufficiently large kick parameters t and V we recover quantum ergodicity, and normal transport, which can be described by random matrix theory.
A dynamical Lie algebraic method for the construction of quantum invariants of motion in non-integrable many-body systems of infinite size is proposed and applied to a simple but generic toy model, namely an infinite kicked t - V chain of interacting spinless fermions. The transition from an integrable via quasi-integrable ( intermediate ) to a quantum ergodic (quantum mixing) regime in parameter space is investigated. A dynamical phase transition between an ergodic and intermediate (neither ergodic nor completely integrable) regime in thermodynamic limit is proposed. The existence or non-existence of local conservation laws corresponds to the intermediate or ergodic regime, respectively. The computation of time-correlation functions of typical observables by means of local conservation laws is found to be fully consistent with direct calculations on finite systems.
We study the infinite temperature dynamics of a prototypical one-dimensional system expected to exhibit many-body localization. Using numerically exact methods, we establish the dynamical phase diagram of this system based on the statistics of its eigenvalues and its dynamical behavior. We show that the nonergodic phase is reentrant as a function of the interaction strength, illustrating that localization can be reinforced by sufficiently strong interactions even at infinite temperature. Surprisingly, within the accessible time range, the ergodic phase shows subdiffusive behavior, suggesting that the diffusion coefficient vanishes throughout much of the phase diagram in the thermodynamic limit. Our findings strongly suggest that Wigner-Dyson statistics of eigenvalue spacings may appear in a class of ergodic but subdiffusive systems.
We analytically study the time evolution of the expectation values of observables in periodically kicked many-body quantum systems. Starting from an initial state, we compute both the transient and the long-time properties of the observables. Our derivation explains the criteria and the mechanism that lead to the infinite-temperature statistical average of observables at long times, irrespective of the initial state. When the criteria are violated, the observables oscillate with time. These oscillations are subharmonic and robust to small perturbations, suggesting the emergence of a discrete time crystal phase. We demonstrate these features explicitly in periodically kicked nonintegrable spin chains. For a spin chain with two kicks per cycle, we show that the kicked chain can exhibit an ergodic or a discrete-time crystal phase for the same kicking strengths, depending on the initial state preparation. We complement our time-evolution study of observables with the spectral form factor of these kicked models.
Integrable and nonintegrable systems have very different transport properties. In this paper, we highlight these differences for specific one-dimensional models of interacting lattice fermions. The technique used is a finite temperature numerical calculation of the adiabatic stiffness (also called the Drude weight or charge stiffness) and isothermal stiffness (also called the ``Meissner'' stiffness) in electrical and energy transport, and the momentum dependent dynamical conductivities $\ensuremath{\sigma}(q,\ensuremath{\omega})$ and $\ensuremath{\kappa}(q,\ensuremath{\omega})$. We apply a flux twist to break the Kramers degeneracy; thus, allowing us to focus on the effect of the dynamical degeneracies in the integrable system. In this situation, we show that the isothermal stiffness goes to zero rapidly with the system size for both types of systems even at high temperatures; while the adiabatic stiffness appears to go to zero in the nonintegrable system and to a finite value in the integrable one. We analyze this difference in terms of the statistics of the current matrix elements and the degeneracies of the systems, and show that in the integrable system, despite the presence of degeneracies, the dominant contribution to the adiabatic stiffness comes from large-current-carrying nondegenerate states. We also show that energy transport at nonzero $\ensuremath{\omega}$ and q occurs within a banded continuum in the integrable system indicative of ballistic transport while the nonintegrable system shows diffusion but with the existence of overdamped excitations at large values of momentum.