Describing open quantum systems in terms of effective non-Hermitian Hamiltonians gives rise to nonunitary time evolution. In this paper we study the impact of nonunitary dynamics on the emergent hydrodynamics in quantum systems with a global conservation law. To this end we demonstrate how linear-response correlation functions can be generalized and interpreted in the case of non-Hermitian systems. Moreover we show that dynamical quantum typicality provides an efficient numerical approach to evaluate such correlation functions, even though the nonunitary dynamics leads to subtleties that are absent in the Hermitian case. As a point of reference for our analysis, we consider the Hermitian spin-$1/2$ XXZ chain, whose high-temperature transport properties have been characterized extensively in recent years. Here we explore the resulting hydrodynamics for different non-Hermitian perturbations of the XXZ chain. We also discuss the role of integrability by studying the complex energy-level statistics of the non-Hermitian quantum models.
Describing open quantum systems in terms of effective non-Hermitian Hamiltonians gives rise to nonunitary time evolution. In this paper we study the impact of nonunitary dynamics on the emergent hydrodynamics in quantum systems with a global conservation law. To this end we demonstrate how linear-response correlation functions can be generalized and interpreted in the case of non-Hermitian systems. Moreover we show that dynamical quantum typicality provides an efficient numerical approach to evaluate such correlation functions, even though the nonunitary dynamics leads to subtleties that are absent in the Hermitian case. As a point of reference for our analysis, we consider the Hermitian spin-1/2 XXZ chain, whose high-temperature transport properties have been characterized extensively in recent years. Here we explore the resulting hydrodynamics for different non-Hermitian perturbations of the XXZ chain. We also discuss the role of integrability by studying the complex energy-level statistics of the non-Hermitian quantum models.
This thesis is a collection of three exact results on correlation and response functions in integrable systems. In the first part we study transport (two-point functions) in the 1D Hubbard model. First, we analyze the limit of large on-site repulsion and characterize spin transport as a function of temperature. Then, we consider the case where the model displays non-abelian symmetries, e.g., spin SU(2), and argue that in this case transport is anomalous with dynamical scaling exponent z=3/2 and follows \acrshort{kpz} scaling. In the next two parts we focus instead on (perturbative) nonlinear response functions, which can be expressed as n-point correlators with n>2. Given the lack of a systematic understanding of the information which can be extracted from these, we compute them in integrable 1D systems. We therefore focus on two distinct scenarios. In the first scenario, we consider a finite temperature system perturbed by some inhomogeneous field coupling to local charges --- thus the perturbations can only accelerate quasiparticles, but not create/annihilate them. In this context, we develop a diagrammatic framework based on generalized hydrodynamics that allows a systematic calculation of nonlinear response functions. We show that, in the hydrodynamic limit, nonlinear response functions qualitatively distinguish between non-interacting and interacting systems. In the second scenario, we instead consider a zero temperature system whose ground state coincides with the quasiparticle vacuum. In this simpler setting we consider more general perturbations that can also create and annihilate quasiparticles. Focusing on the Ising chain as a paradigmatic model we show that the long-time limit of four-point functions (third-order response) grows linearly in time. We interpret these divergences in terms of semiclassical processes where quasiparticles propagate ballistically and scatter when their trajectories intersect.
Unraveling the mechanisms of ergodicity breaking in complex quantum systems is a central pursuit in nonequilibrium physics. In this work, we investigate a one-dimensional spin model featuring a tunable long-range hopping term, Hn, which introduces nonlocal interactions and bridges the gap between Hermitian and non-Hermitian regimes. Through a systematic analysis of level-spacing statistics, Krylov complexity, and entanglement entropy, we demonstrate that Hn acts as a universal control parameter driving the transition from integrability to quantum chaos. Specifically, increasing the strength of Hn induces a crossover from Poissonian to Gaussian Orthogonal Ensemble statistics in the Hermitian limit, and similarly triggers chaotic dynamics in the non-Hermitian case. Most remarkably, despite the onset of global chaos, we identify a tower of exact nonthermal eigenstates that evade thermalization. These states survive as robust quantum many-body scars, retaining low entanglement and coherent dynamics even under strong non-Hermitian perturbations. Our findings reveal a universal mechanism by which long-range and non-Hermitian effects reshape quantum ergodicity, offering new pathways for preserving quantum coherence in complex many-body systems.
Many experimentally relevant quantum spin chains are approximately integrable, and support long-lived quasiparticle excitations. A canonical example of integrable model of quantum magnetism is the XXZ spin chain, for which energy spreads ballistically, but, surprisingly, spin transport can be diffusive or superdiffusive. We review the transport properties of this model using an intuitive quasiparticle picture that relies on the recently introduced framework of generalized hydrodynamics. We discuss how anomalous linear response properties emerge from hierarchies of quasiparticles both in integrable and near-integrable limits, with an emphasis on the role of hydrodynamic fluctuations. We also comment on recent developments including non-linear response, full-counting statistics and far-from-equilibrium transport. We provide an overview of recent numerical and experimental results on transport in XXZ spin chains.
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].
Recently, it has become clear that non-Hermitian phenomena can be observed not only in open quantum systems experiencing gain and loss but also in equilibrium single-particle properties of strongly correlated systems. However, the circumstances and requirements for the emergence of non-Hermitian phenomena in each field are entirely different. While the implementation of postselection is a significant obstacle to observe the dynamics governed by a non-Hermitian Hamiltonian in open quantum systems, it is unnecessary in strongly correlated systems. Until now, a relation between both descriptions of non-Hermitian phenomena has not been revealed. In this Letter, we close this gap and demonstrate that the non-Hermitian Hamiltonians emerging in both fields are identical, and we clarify the conditions for the emergence of a non-Hermitian Hamiltonian in strongly correlated materials. Using this knowledge, we propose a method to analyze non-Hermitian properties without the necessity of postselection by studying specific response functions of open quantum systems and strongly correlated systems.
In the thesis we present an analytic approach towards exact description for steady state density operators of nonequilibrium quantum dynamics in the framework of open systems. We employ the so-called quantum Markovian semi-group evolution, i.e. a general form of time-autonomous positivity and trace-preserving dynamical equation for reduced density operators, by only allowing unitarity-breaking dissipative terms acting at the boundaries of a system. Such setup enables to simulate macroscopic reservoirs for different values of effective thermodynamic potentials, causing incoherent transitions between quantum states which are modeled with aid of the Lindblad operators. This serves as a simple minimalistic model for studying quantum transport properties, either in the linear response domain or in more general regimes far from canonical equilibrium. We are mainly exploring possibilities of identifying nonequilibrium situations which are amenable to exact description within matrix product state representation, by exclusively focusing on steady states, i.e. fixed points of the Lindblad equation, of certain prototypic interacting integrable spin chains driven by incoherent polarizing processes. Finally, we define a concept of pseudo-local extensive almost-conserved quantities by allowing a violation of time-invariance up to boundary-localized terms. We elucidate the role of such quantities on non-ergodic behaviour of temporal correlation functions rendering anomalous transport properties. It turns out that such conservation laws can be generated by means of boundary universal quantum transfer operators of the fundamental integrable models.