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arXiv:2404.14776 · Physical review. B./Physical review. B

Dynamic transition of the density-matrix topology under parity-time symmetry

W. P. Wang, Wei Yi

Published Oct 17, 2024• 2 citations
arXivHTMLPDF

Abstract

Density-matrix topology, defined through the geometric property of the relevant modular Hamiltonian, can undergo transitions in the corresponding open-system dynamics. While symmetry considerations are crucial to ensure such a dynamic topological transition, we show that a hidden parity-time symmetry can further facilitate it. Considering the Lindbladian dynamics of a fermionic Gaussian state, we extract the time-evolved density-matrix topology from the single-particle correlation, whose dynamics is governed by a non-Hermitian damping matrix. We show that, for a parity-time symmetric damping matrix and a chiral symmetric correlation matrix, a dynamic transition in the density-matrix topology necessarily occurs in the parity-time unbroken regime where eigenvalues of the damping matrix are real. We illustrate our results using a concrete model, and map out the dynamic phase diagram. Remarkably, we find that the dynamic transition can also happen periodically in the parity-time symmetry broken regime.

DOI
10.1103/physrevb.110.155141

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Density-matrix topology, defined through the geometric property of the relevant modular Hamiltonian, can undergo transitions in the corresponding open-system dynamics. While symmetry considerations are crucial to ensure such a dynamic topological transition, we show that a hidden parity-time symmetry can further facilitate it. Considering the Lindbladian dynamics of a fermionic Gaussian state, we extract the time-evolved density-matrix topology from the single-particle correlation, whose dynamics is governed by a non-Hermitian damping matrix. We show that, for a parity-time symmetric damping matrix and a chiral symmetric correlation matrix, a dynamic transition in the density-matrix topology necessarily occurs in the parity-time unbroken regime where eigenvalues of the damping matrix are real. We illustrate our results using a concrete model, and map out the dynamic phase diagram. Remarkably, we find that the dynamic transition can also happen periodically in the parity-time symmetry broken regime.
Abstract The dynamical evolution of an open quantum system can be governed by the Lindblad equation of the density matrix. In this paper, we propose to characterize the density matrix topology by the topological invariant of its modular Hamiltonian. Since the topological classification of such Hamiltonians depends on their symmetry classes, a primary issue we address is determining the requirement for the Lindbladian operators, under which the modular Hamiltonian can preserve its symmetry class during the dynamical evolution. We solve this problem for the fermionic Gaussian state and for the modular Hamiltonian being a quadratic operator of a set of fermionic operators. When these conditions are satisfied, along with a nontrivial topological classification of the symmetry class of the modular Hamiltonian, a topological transition can occur as time evolves. We present two examples of dissipation-driven topological transitions where the modular Hamiltonian lies in the AIII class with U (1) symmetry and the DIII class without U (1) symmetry. By a finite size scaling, we show that this density matrix topology transition occurs at a finite time. We also present the physical signature of this transition.
We identify dynamic topological phenomena such as dynamic Chern numbers and dynamic quantum phase transitions in quantum quenches of the non-Hermitian Su-Schrieffer-Heeger Hamiltonian with parity-time (PT) symmetry. Their occurrences in the non-unitary dynamics are intimately connected with fixed points in the Brillouin zone, where the density matrices do not evolve in time. Based on our theoretical formalism characterizing topological properties of non-unitary dynamics, we prove the existence of fixed points for quenches between distinct static topological phases in the PT-symmetry-preserving regime, thus unveiling the interplay between dynamic topological phenomena and PT symmetry. Interestingly, non-Hermiticity of the driving Hamiltonian gives rise to rich dynamic topological phenomena which are different, either qualitatively or quantitatively, from their counterparts in unitary dynamics. Our work sheds light on dynamic topological phenomena in open systems and is readily accessible in experiments.
We investigate dissipation-driven topological phase transitions in one-dimensional quantum open systems governed by the Lindblad equation with linear dissipation operators, which ensure the density matrix retains its Gaussian form throughout the dynamics. By employing the modular Hamiltonian framework, we rigorously demonstrate that the Z2\rm{Z}_2Z2​ topological invariant characterizing steady states in one-dimensional class D systems is exclusively dependent on the dissipation operators, rather than the system Hamiltonian. Through a sudden quench protocol where the system evolves from the steady state of one Lindbladian to another, we reveal that topological transitions can occur at analytically predictable critical times, even when the initial and final steady states share identical topological indices. These transitions are shown, both analytically and numerically, to depend solely on dissipation parameters. Entanglement spectrum analysis demonstrates bulk-edge correspondence in non-equilibrium density matrices via coexisting single-particle gap closures (periodic boundaries) and topologically protected zero modes (open boundaries), directly underpinning the detection of dissipation-induced topology in quantum simulators.
In the presence of Lindblad decoherence, i.e., dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities between the eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for PT\mathcal{PT}PT-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators Lj{L}_{j}^{}Lj​ are pseudo-Hermitian, namely, $\ensuremath{\eta}{L}_{j}^{}{\ensuremath{\eta}}^{\ensuremath{-}1}={L}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}$ with $\ensuremath{\eta}$ being a linear and positive-definite metric, and respect the PT\mathcal{PT}PT symmetry as well. We demonstrate that the generalized density matrix ${\ensuremath{\rho}}_{\mathrm{G}}^{}(t)\ensuremath{\equiv}\ensuremath{\rho}(t)\ensuremath{\eta}$, instead of the normalized density matrix ${\ensuremath{\rho}}_{\mathrm{N}}^{}(t)\ensuremath{\equiv}\ensuremath{\rho}(t)/\mathrm{tr}\left[\ensuremath{\rho}(t)\right]$, should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of PT\mathcal{PT}PT-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence.
A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the GKSL equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called \textit{Lindbladian PT (L-PT) phase transition}, is closely related to two classes of exotic nonequilibrium many-body phenomena: \textit{continuous-time crystals} and \textit{non-reciprocal phase transitions}. In this review, we describe the recent advances in the study of L-PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L-PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L-PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L-PT phase transition point is typically \textit{a critical exceptional point}, where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L-PT symmetry. Finally, we discuss future research directions for L-PT phase transitions.
Topological edge states arise in non-Hermitian parity-time ([Formula: see text])-symmetric systems, and manifest themselves as bright or dark edge states, depending on the imaginary components of their eigenenergies. As the spatial probabilities of dark edge states are suppressed during the non-unitary dynamics, it is a challenge to observe them experimentally. Here we report the experimental detection of dark edge states in photonic quantum walks with spontaneously broken [Formula: see text] symmetry, thus providing a complete description of the topological phenomena therein. We experimentally confirm that the global Berry phase in [Formula: see text]-symmetric quantum-walk dynamics unambiguously defines topological invariants of the system in both the [Formula: see text]-symmetry-unbroken and -broken regimes. Our results establish a unified framework for characterizing topology in [Formula: see text]-symmetric quantum-walk dynamics, and provide a useful method to observe topological phenomena in [Formula: see text]-symmetric non-Hermitian systems in general.