We study the quench dynamics of noninteracting fermionic quantum many-body systems that are subjected to Markovian drive and dissipation and are described by a quadratic Liouvillian which has parity-time (PT) symmetry. In recent work, we have shown that such systems relax locally to a maximum entropy ensemble that we have dubbed the PT-symmetric generalized Gibbs ensemble (PTGGE), in analogy to the generalized Gibbs ensemble that describes the steady state of isolated integrable quantum many-body systems after a quench. Here, using driven-dissipative versions of the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain as paradigmatic model systems, we corroborate and substantially expand upon our previous results. In particular, we confirm the validity of a dissipative quasiparticle picture at finite dissipation by demonstrating light cone spreading of correlations and the linear growth and saturation to the PTGGE prediction of the quasiparticle-pair contribution to the subsystem entropy in the PT-symmetric phase. Further, we introduce the concept of directional pumping phases, which is related to the non-Hermitian topology of the Liouvillian and based upon qualitatively different dynamics of the dual string order parameter and the subsystem fermion parity in the SSH model and the Kitaev chain, respectively: depending on the postquench parameters, there can be pumping of string order and fermion parity through both ends of a subsystem corresponding to a finite segment of the one-dimensional lattice, through only one end, or there can be no pumping at all. We show that transitions between dynamical pumping phases give rise to a new and independent type of dynamical critical behavior of the rates of directional pumping, which are determined by the soft modes of the PTGGE. Published by the American Physical Society 2024
We study the quench dynamics of noninteracting fermionic quantum many-body systems that are subjected to Markovian drive and dissipation and are described by a quadratic Liouvillian which has parity-time (PT) symmetry. In recent work, we have shown that such systems relax locally to a maximum entropy ensemble that we have dubbed the PT-symmetric generalized Gibbs ensemble (PTGGE), in analogy to the generalized Gibbs ensemble that describes the steady state of isolated integrable quantum many-body systems after a quench. Here, using driven-dissipative versions of the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain as paradigmatic model systems, we corroborate and substantially expand upon our previous results. In particular, we confirm the validity of a dissipative quasiparticle picture at finite dissipation by demonstrating light cone spreading of correlations and the linear growth and saturation to the PTGGE prediction of the quasiparticle-pair contribution to the subsystem entropy in the PT-symmetric phase. Further, we introduce the concept of directional pumping phases, which is related to the non-Hermitian topology of the Liouvillian and based upon qualitatively different dynamics of the dual string order parameter and the subsystem fermion parity in the SSH model and the Kitaev chain, respectively: depending on the postquench parameters, there can be pumping of string order and fermion parity through both ends of a subsystem corresponding to a finite segment of the one-dimensional lattice, through only one end, or there can be no pumping at all. We show that transitions between dynamical pumping phases give rise to a new and independent type of dynamical critical behavior of the rates of directional pumping, which are determined by the soft modes of the PTGGE. Published by the American Physical Society 2024
The construction of the generalized Gibbs ensemble, to which isolated integrable quantum many-body systems relax after a quantum quench, is based upon the principle of maximum entropy. In contrast, there are no universal and model-independent laws that govern the relaxation dynamics and stationary states of open quantum systems, which are subjected to Markovian drive and dissipation. Yet, as we show, relaxation of driven-dissipative systems after a quantum quench can, in fact, be determined by a maximum entropy ensemble, if the Liouvillian that generates the dynamics of the system has parity-time symmetry. Focusing on the specific example of a driven-dissipative Kitaev chain, we show that, similar to isolated integrable systems, the approach to a parity-time symmetric generalized Gibbs ensemble becomes manifest in the relaxation of local observables and the dynamics of subsystem entropies. In contrast, the directional pumping of fermion parity, which is induced by nontrivial non-Hermitian topology of the Kitaev chain, represents a phenomenon that is unique to relaxation dynamics in driven-dissipative systems. Upon increasing the strength of dissipation, parity-time symmetry is broken at a finite critical value, which thus constitutes a sharp dynamical transition that delimits the applicability of the principle of maximum entropy. We show that these results, which we obtain for the specific example of the Kitaev chain, apply to broad classes of noninteracting fermionic models, and we discuss their generalization to a noninteracting bosonic model and an interacting spin chain.
Abstract We formulate and discuss the explicit computation of dynamic correlation functions in open quadratic fermionic systems which are driven and dissipated by Lindblad jump processes that are linear in canonical fermionic operators. Dynamic correlators are interpreted in terms of a local quantum quench where the pre-quench state is a non-equilibrium steady state, i.e. a fixed point of the Liouvillian. As an example, we study the XY spin 1/2 chain and quadratic Majorana chains with boundary Lindblad driving, whose dynamics exhibit asymmetric (skewed) light cone behaviour. We also numerically treat the two-dimensional XY model and the XY spin chain with additional Dzyaloshinskii–Moriya interactions. The latter exhibits a new non-thermal phase transition which can be understood in terms of bifurcations of the quasi-particle dispersion relation. Finally, considering in some detail the periodic quadratic fermionic ring with dissipation at a single (arbitrary) site, we present analytical expressions for first order corrections (in the strength of dissipation) to the spectrum and non-equilibrium steady state (NESS) correlation functions.
Open driven quantum systems have defined a powerful paradigm of nonequilibrium phases and phase transitions; however, quantum phase transitions are generically not expected in this setting due to the decohering effect of dissipation. In this Letter, we consider a quantum Ising model subject to bulk dissipation (at rate Γ) and show that, although the correlation length remains finite (hence no phase transition), it develops a pronounced peak close to the ground-state quantum critical point. While standard techniques fail in this regime, we develop a versatile analytical approach that becomes exact with vanishing dissipation (Γ→0 but finite Γt). On a technical level, our approach builds on previous work where the state of the system is described by a slowly evolving generalized Gibbs ensemble that accounts for the integrability of the Hamiltonian while treating dissipation perturbatively. Finally, we demonstrate a kind of universality in that integrability-breaking perturbations of the Hamiltonian lead to the same behavior. To this end, we first show that the steady state of a chaotic Ising Hamiltonian under local Markovian dissipation that preserves the Ising symmetry, and in the limit Γ→0, is identical to that of quench dynamics in the absence of dissipation. This intriguing connection then allows us to draw on recent findings about quantum phase transition signatures in quench dynamics.
PT-Symmetric Effective Model for Nonequilibrium Phase Transitions in a Dissipative Fermionic Mott Insulator Chain
Abstract Nonequilibrium phase transitions in open dissipative systems can be described as instabilities in the spectra and wavefunctions of effective non-Hermitian Hamiltonians invariant under simultaneous parity ( P P ) and time-reversal ( T T ) transformations. The degree of non-Hermiticity reflects the strength of the external drive and dissipation, and the transition is described as a loss of the PT PT symmetry of the solutions corresponding to stationary low-drive dynamics. This approach has been successfully applied to spin, superconducting, and Mott insulator systems. However, the microscopic foundations for the employed phenomenological models are currently lacking. Here we propose a microscopic mechanism leading to the PT PT -symmetric effective model in the context of the nonequilibrium Mott transition in a dissipative Hubbard chain. Our model comprises a half-filled fermionic Hubbard chain subject to a constant electric field. The dissipation is introduced via the electron-phonon coupling. We obtain the explicit expressions for the non-Hermitian parameter in terms of the electron-phonon coupling strength and driving field. Analyzing the implications of microscopic model, we find a re-entrant Mott insulator with the increasing electric field for phonon density of states that increases slower than the square of the energy (such as in one or two dimensions), or varies non-monotonously with energy.
In recent years the out-of-equilibrium dynamics of many-body quantum systems has attracted growing interest, both for its fundamental implications in the understanding of quantum thermalization and for potential applications in quantum information and quantum computation. In particular, quantum quenches provide a natural setting to investigate the mechanisms of correlation propagation and entanglement growth far from equilibrium. Interest in these phenomena stems not only from the possibility of testing theoretical predictions coming from emergent hydrodynamic descriptions, but also from understanding whether, and to what extent, such descriptions remain valid in the presence of spatial gradients or interfaces. Two paradigmatic frameworks in this context are the Generalized Hydrodynamics (GHD) and quasi-particle picture. The former is an effective theory devised to describe the long-time evolution after a quench in the thermodynamic limit. Within this perspective, inhomogeneous quenches are a crucial testbed for extending hydrodynamic theories to more realistic situations where, for example, scattering at an interface can occur. This is the main goal of the present thesis. The reference physical model is the one-dimensional Kitaev chain, an integrable spinless p-wave superconductor that exhibits both a topological and a trivial phase. The bilinear structure of the Hamiltonian in the fermionic creation and annihilation operators admits an exact rewriting in terms of Majorana fermions, in which the Hamiltonian remains quadratic. This representation allows a compact description of the ground-state properties via the Majorana correlation matrix (Gamma) at time t=0: the entries of Gamma completely encode the quantum information of the Gaussian ground state. The time evolution of Gamma is studied in the Heisenberg picture, where unitary dynamics correspond to an orthogonal rotation of Gamma implemented by an orthogonal matrix determined by the Kitaev Hamiltonian in the Majorana basis. Consequently, the entanglement entropy dynamics following a quench, both homogeneous and inhomogeneous, can be simulated through simple matrix products, making the approach numerically efficient even for sizable systems. The evolution code for Gamma was validated on a homogeneous quench with periodic boundary conditions: subsystem-size (l) curves exhibit the expected finite-size scaling, (Δ \propto 1/l), and numerical growth rates in the linear regime (t < v/2l), extracted by linear fits at fixed l, match the quasi-particle predictions for various values of the new chemical potential. The rate discrepancy behaves asymptotically as (Δ_{rate}} \propto 1/t_{\min}\) for large (t_{\min}\), up to oscillations from sub-extensive terms. The physical problem of interest is then an inhomogeneous quench of the chemical potential on the infinite Kitaev chain ground state with open boundaries, (μ_0 to μ_L, μ_R), producing a central interface. Initial and final Hamiltonians are free in the quasiparticle sense (ballistic propagation), but the presence of two distinct energy bands implies interface scattering. Simulations varying only μ_R (within the same trivial phase) show a nonmonotonic dependence of the entanglement growth rate, with a maximum at a particular μ_R. To explain this, an analytic ansatz for the stationary Majorana wavefunction was constructed: plane waves multiplied by a Majorana spinor, with bulk spinors obtained from the homogeneous PBC problem via the time-independent Schrödinger equation. Imposing matching at the interface yields four nontrivial equations for two unknowns (overdetermined in general). In the special cases (Δ ̸= t) two equations become trivial, the system is determined, and one obtains an explicit transmission coefficient. Combining this result with known quasi-particle formulas for two semi-infinite chains yields a hydrodynamic prediction for entanglement growth with interface scattering. Energy conservation across the interface selects the momenta that can be transmitted and thus contribute to inter-region correlations. Numerical tests for several total lengths (L) confirm the finite-size scaling (Δ_{\mathrm{e.e.}} \propto 1/L\), and show excellent agreement between numerical and theoretical growth rates for both quenches restricted to the right half and quenches applied to both halves. For the right-only case the rate discrepancy again follows (Δ_{rate} \propto 1/t_{min}\) as tmin→∞. Possible future directions include: the study of the general inhomogeneous quench with (Δ ̸= t) via a direct extension of the analytic ansatz; the analysis of quenches starting from the topological ground state; the investigation of universal effects near the quantum critical point (μ=2t); the introduction of multiple interfaces or regions with spatially varying parameters to explore interference phenomena between quasiparticle fronts (e.g. periodic alternation (μ_1,μ_2,μ_1,μ_2,....) and, finally, the extension to interacting models where the Gaussian formalism is no longer applicable.
We investigate generalized thermalization in an isolated free fermionic chain evolving from an out of equilibrium initial state through a sudden quench. We consider the quench where a fermionic chain is broken into two disjoint chains. We focus on the evolution of the local observables namely, occupation number, nearest neighbor hopping, information sharing and out-of-time-order correlations after the quench and study the relaxation of the observable, leading to generalized Gibbs ensemble for the system in the thermodynamic limit though it has been argued that noninteracting or free fermionic models in general do not relax to the generalized Gibbs ensemble (GGE). We obtain the light cone formed by the evolution of the observables along the fermionic lattice chain due to the sudden quench which abides by the Lieb-Robinson bound in quantum systems. We also analytically study a simpler model which captures the essential features of the system. Our analysis strongly suggest that the internal interactions within the system do not remain of much importance once the quench is sufficiently strong.