Since perturbations are omnipresent in physics, understanding their impact on the dynamics of quantum many-body systems is a vitally important but notoriously difficult question. On the one hand, random-matrix and typicality arguments suggest a rather simple damping in the overwhelming majority of cases, e.g., exponential damping according to Fermi's golden rule. On the other hand, counterexamples are known to exist, and it remains unclear how frequent and under which conditions such counterexamples appear. In our work, we consider the spin-1/2 XXZ chain as a paradigmatic example of a quantum many-body system and study the dynamics of the magnetization current in the easy-axis regime. Using numerical simulations based on dynamical quantum typicality, we show that the standard autocorrelation function is damped in a nontrivial way and only a modified version of this function is damped in a simple manner. Employing projection-operator techniques in addition, we demonstrate that both the nontrivial and simple damping relation can be understood on perturbative grounds. Our results are in agreement with earlier findings for the particle current in the Hubbard chain.
Since perturbations are omnipresent in physics, understanding their impact on the dynamics of quantum many-body systems is a vitally important but notoriously difficult question. On the one hand, random-matrix and typicality arguments suggest a rather simple damping in the overwhelming majority of cases, e.g., exponential damping according to Fermi's golden rule. On the other hand, counterexamples are known to exist, and it remains unclear how frequent and under which conditions such counterexamples appear. In our work, we consider the spin-1/2 XXZ chain as a paradigmatic example of a quantum many-body system and study the dynamics of the magnetization current in the easy-axis regime. Using numerical simulations based on dynamical quantum typicality, we show that the standard autocorrelation function is damped in a nontrivial way and only a modified version of this function is damped in a simple manner. Employing projection-operator techniques in addition, we demonstrate that both the nontrivial and simple damping relation can be understood on perturbative grounds. Our results are in agreement with earlier findings for the particle current in the Hubbard chain.
Understanding how the dynamics of a given quantum system with many degrees of freedom is altered by the presence of a generic perturbation is a notoriously difficult question. Recent works predict that, in the overwhelming majority of cases, the unperturbed dynamics is just damped by a simple function, e.g., exponentially as expected from Fermi's golden rule. While these predictions rely on random-matrix arguments and typicality, they can only be verified for a specific physical situation by comparing to the actual solution or measurement. Crucially, it also remains unclear how frequent and under which conditions counterexamples to the typical behavior occur. In this work, we discuss this question from the perspective of projection-operator techniques, where exponential damping of a density matrix occurs in the interaction picture but not necessarily in the Schrödinger picture. We show that a nontrivial damping in the Schrödinger picture can emerge if the dynamics in the unperturbed system possesses rich features, for instance due to the presence of strong interactions. This suggestion has consequences for the time dependence of correlation functions. We substantiate our theoretical arguments by large-scale numerical simulations of charge transport in the extended Fermi-Hubbard chain, where the nearest-neighbor interactions are treated as a perturbation to the integrable reference system.
Given a quantum many-body system and the expectation-value dynamics of some operator, we study how this reference dynamics is altered due to a perturbation of the system's Hamiltonian. Based on projection operator techniques, we unveil that if the perturbation exhibits a random-matrix structure in the eigenbasis of the unperturbed Hamiltonian, then this perturbation effectively leads to an exponential damping of the original dynamics. Employing a combination of dynamical quantum typicality and numerical linked cluster expansions, we demonstrate that our theoretical findings for random matrices can, in some cases, be relevant for the dynamics of realistic quantum many-body models as well. Specifically, we study the decay of current autocorrelation functions in spin-1/2 ladder systems, where the rungs of the ladder are treated as a perturbation to the otherwise uncoupled legs. We find a convincing agreement between the exact dynamics and the lowest-order prediction over a wide range of interchain couplings.
We study spin transport in the one-dimensional anisotropic S=1 Heisenberg model. Particular emphasis is given to dynamics at infinite temperature, where current autocorrelations and spatiotemporal correlation functions are obtained by means of an efficient pure-state approach based on the concept of typicality. Our comprehensive numerical analysis unveils that high-temperature spin transport is diffusive in the easy-axis regime for strong exchange anisotropies. This finding is based on the combination of numerous signatures, such as (i) Gaussian spreading of correlations, (ii) a time-independent diffusion coefficient, (iii) power-law decay of equal-site correlations, (iv) exponentially decaying long-wavelength modes, and (v) Lorentzian line shapes of the dynamical structure factor. Moreover, we provide evidence that some of these signatures are not exclusively restricted to the infinite-temperature limit but can persist at lower temperatures as well, where we complement our results by additional quantum Monte Carlo simulations of large systems. In contrast to the easy-axis regime, we show that in the case of an isotropic chain, the signatures (i)--(v) are much less pronounced or even entirely absent, suggesting the existence of anomalous spin transport despite the nonintegrability of the model. Eventually, upon introducing a random on-site magnetic field, we observe a breakdown of diffusion and distinctly slower dynamics. In particular, our results exhibit qualitative similarities to disordered spin-1/2 chains and might be consistent with the onset of many-body localization in the S=1 model for sufficiently strong disorder.
We use the concept of typicality to study the real-time dynamics of spin and energy currents in spin-1/2 models in one dimension and at nonzero temperatures. These chains are the integrable XXZ chain and a nonintegrable modification due to the presence of a staggered magnetic field oriented in the z direction. In the framework of linear response theory, we numerically calculate autocorrelation functions by propagating a single pure state, drawn at random as a typical representative of the full statistical ensemble. By comparing to small-system data from exact diagonalization (ED) and existing short-time data from time-dependent density matrix renormalization group, we show that typicality is satisfied in finite systems over a wide range of temperature and is fulfilled in both integrable and nonintegrable systems. For the integrable case, we calculate the long-time dynamics of the spin current and extract the spin Drude weight for large systems outside the range of ED. We particularly provide strong evidence that the high-temperature Drude weight vanishes at the isotropic point. For the nonintegrable case, we obtain the full relaxation curve of the energy current and determine the heat conductivity as a function of magnetic field, exchange anisotropy, and temperature.
We numerically study the relaxation of correlation functions in weakly perturbed integrable XXZ chain. The decay of the spin-current and the energy-current correlations at zero magnetization are well described by single, but quite distinct, relaxation rates governed by the square of the perturbation strength g g . However, at finite magnetization a single correlation function reveals multiple relaxation rates. The result can be understood in terms of multi-scale relaxation scenario, where various relaxation times are linked with various quantities which are conserved in the reference integrable system. On the other hand, the correlations of non-commuting quantities, being conserved at particular anisotropies \Delta Δ , decay non-exponentially with characteristic time scale linear in g g .
Based on the references [P1–P6], we investigate in this cumulative PhD thesis the dynamics and transport in closed and open interacting spin chain models. We focus on (i) the connection between the open and the closed system dynamics [P1, P2], (ii) the influence of various system parameters on the transport behavior in closed and open systems [P3, P5], and in addtion (iii) the dynamics and transport in closed models [P4, P6]. As for (i), we explore the connection of the dynamics in open systems, modeled by the Lindblad equation, and the correlation functions in the closed system [P1]. Building on this connection for magnetization transport, we study whether the time evolution of the open quantum system can be described in terms of classical correlation functions generated by the Hamiltonian equations of motion for real vectors [P2]. Second, we study in (ii) the influence of different parameters on the diffusion constants in closed and open systems [P3, P5]. In Ref. [P3], we study the impact of integrability-breaking perturbations in both closed and open spin chain systems. While we find a continuous change of the diffusion constant with perturbation strength in the closed system, in the open system, in the weak-perturbation limit, we indicate the possibility of a diverging diffusion constant. Also, we investigate the influence of the system-bath coupling on the transport behavior in boundary-driven open systems [P5]. We observe that the diffusion constants depend on the system-bath coupling and that in the nonequilibrium steady state, they inevitably retain finite-size effects, limiting the computation of long-time transport coefficients. Additional for (iii), in Ref. [P4], we first examine the impact of perturbations on the dynamics in closed spin chain models. We employ projector-operator techniques to understand, on perturbative grounds, the nontrivial damping of the standard correlation function. Second, we also discuss a dynamical quantum typicality approach for studying the energy transport in bipartite systems at low temperatures [P6].