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arXiv:2405.07132 · Physical review. B./Physical review. B

Oscillating-mode gap: An indicator of phase transitions in open quantum many-body systems

Taiki Haga

Published Sep 9, 2024• 6 citations
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Abstract

It presents a significant challenge to elucidate the relationship between the phases of open quantum many-body systems and the spectral structure of their governing Liouvillian, which determines how the density matrix evolves. Previous studies have focused on the Liouvillian gap, defined as the decay rate of the most slowly decaying mode, as a key indicator of dissipative phase transition, noting its closure in symmetry-broken phases and opening in disordered phases. In this work, we propose an additional spectral gap, termed the oscillating-mode gap, defined as the decay rate of the most slowly decaying oscillating mode. Through the analysis of a prototype dissipative boson system, we demonstrate the necessity of both the Liouvillian gap and the oscillating-mode gap for the comprehensive characterization of the system's phases and the transitions between them.

DOI
10.1103/physrevb.110.104303

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It presents a significant challenge to elucidate the relationship between the phases of open quantum many-body systems and the spectral structure of their governing Liouvillian, which determines how the density matrix evolves. Previous studies have focused on the Liouvillian gap, defined as the decay rate of the most slowly decaying mode, as a key indicator of dissipative phase transition, noting its closure in symmetry-broken phases and opening in disordered phases. In this work, we propose an additional spectral gap, termed the oscillating-mode gap, defined as the decay rate of the most slowly decaying oscillating mode. Through the analysis of a prototype dissipative boson system, we demonstrate the necessity of both the Liouvillian gap and the oscillating-mode gap for the comprehensive characterization of the system's phases and the transitions between them.
A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.
The relaxation dynamics of an open quantum system is determined by the competition between the coherent Hamiltonian dynamics of a system and the dissipative dynamics due to interactions with environments. It is therefore of fundamental interest to understand the transition from the coherent to incoherent regimes. We find that hitherto unrecognized quasiparticles---incoherentons---describe this coherent-to-incoherent transition in eigenmodes of a Liouvillian superoperator that governs the dynamics of an open quantum many-body system. Here, an incoherenton is defined as an interchain bound state in an auxiliary ladder system that represents the density matrix of a system. The Liouvillian eigenmodes are classified into groups with different decay rates that reflect the number of incoherentons involved therein. We also introduce a spectral gap---quantum coherence gap---that separates the different groups of eigenmodes. We demonstrate the existence of incoherentons in a lattice boson model subject to dephasing, and show that the quantum coherence gap closes when incoherentons are deconfined, which signals a dynamical transition from incoherent relaxation with exponential decay to coherent oscillatory relaxation. Furthermore, we discuss how the decoherence dynamics of quantum many-body systems can be understood in terms of the generation, localization, and diffusion of incoherentons.
Dissipative phase transitions (DPT) are defined by sudden changes in the physical properties of nonequilibrium open quantum systems and they present characteristics that have no analog in closed and thermal systems. Several methods to detect and characterize DPT have been suggested in the literature, the most famous of which -- the Liouvillian gap\textit{Liouvillian gap}Liouvillian gap -- can be derived from a spectral analysis of the Liouvillian super-operator that governs the complex interplay between coherent and dissipative dynamics. Here, we consider the output current\textit{output current}output current, defined as the average total quantum jumps per unit time between the open quantum system and the environment. We propose that output current fluctuations, and in particular their dynamical correlations, their power spectrum, and their characteristic timescale can provide valuable information about DPT, confirming a dramatic change of behavior at the critical point. We validate our proposal using the dissipative XYZ model and the nonlinear driven-dissipative Kerr model, showing good agreement with previous estimates of the location of the critical point. Compared to previous approaches, our proposal could be already experimentally tested in optical systems, providing a practical method to detect criticality in quantum open systems.
We introduce the , a nonperiodic analog of the discrete time crystal that arises in periodically driven dissipative quantum many-body systems. This phase is defined by two key features: (1) spatial long-range order arising from the spontaneous breaking of an internal symmetry and (2) temporally chaotic oscillations of the order parameter, whose lifetime diverges with system size. In other words, a time glass is a state of matter in which all components evolve in a synchronized yet chaotic manner. To characterize the time glass phase, we focus on the spectral gap of the one-cycle (Floquet) Liouvillian, which determines the decay rate of the slowest relaxation mode. Theoretical arguments and numerical studies of periodically driven dissipative Ising models show that, in the time glass phase, the Liouvillian gap remains finite in the thermodynamic limit, in contrast to time crystals where the gap closes exponentially with system size. We further demonstrate that the Liouvillian gap converges to the decay rate of the order-parameter autocorrelation derived from the classical (mean-field) dynamics in the thermodynamic limit. This result establishes a direct correspondence between microscopic spectral features and emergent macroscopic dynamics in driven dissipative quantum systems. At first glance, the existence of a nonzero Liouvillian gap appears incompatible with the presence of indefinitely persistent chaotic oscillations. We resolve this apparent paradox by showing that the quantum Rényi divergence between a localized coherent initial state and the highly delocalized steady state grows unboundedly with system size. This divergence allows long-lived transients to persist even in the presence of a finite Liouvillian gap.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
In specific open systems with collective dissipation the Liouvillian can be mapped to a non-Hermitian Hamiltonian. We here consider such a system where the Liouvillian is mapped to an XXZ Richardson-Gaudin integrable model and detail its exact Bethe ansatz solution. While no longer Hermitian, the Hamiltonian is pseudo-Hermitian/PT\text{pseudo-Hermitian}/\mathcal{P}\mathcal{T}pseudo-Hermitian/PT symmetric, and as the strength of the coupling to the environment is increased, the spectrum in a fixed symmetry sector changes from a broken pseudo-Hermitian phase with complex conjugate eigenvalues to a pseudo-Hermitian phase with real eigenvalues, passing through a series of exceptional points and associated dissipative quantum phase transitions. The homogeneous limit supports a nontrivial steady state, and away from this limit this state gives rise to a slow logarithmic growth of the decay rate (spectral gap) with system size. Using the exact solution, it is furthermore shown how at large coupling strengths the ratio of the imaginary to the real part of the eigenvalues becomes approximately quantized in the remaining symmetry sectors.