Lindbladian PT phase transitions

TL;DR

This review defines Lindbladian PT (L-PT) phase transitions as cooperative, nonequilibrium transitions in GKSL dynamics, where PT symmetry at the Lindbladian level induces a nonlinear PT symmetry in mean-field equations, leading to persistent oscillations and a CEP at the transition. It develops MF theories for collective-spin and long-range/density-conserving bosonic systems, links L-PT transitions to continuous-time crystals and nonreciprocal phenomena, and demonstrates that the transition is typically governed by a Lindbladian exceptional point in the thermodynamic limit. Beyond MF, it analyzes quantum properties such as purity and entanglement indicators, showing pronounced changes at critical points and near CEPs, with implications for quantum metrology. The work also outlines extensions to broader dissipative settings and discusses experimental prospects in driven-dissipative spin and cavity-QED platforms. Overall, it establishes a structured framework connecting symmetry, spectral properties, and nonequilibrium phases in open quantum systems with potential for quantum technological applications.

Abstract

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.

Figures (13)

• Figure 1: Connections between L-$\mathcal{PT}$ phase transitions and other physical phenomena, symmetry and applications.
• Figure 2: A basic PT-symmetric example with balanced gain and loss: the parity operator exchanges two particles, while the time-reversal operator swaps gain and loss. [Inspired by Ref.ozdemir2019parity]
• Figure 3: (Top) Lindbladian eigenvalues and (Bottom) their corresponding dynamics of an observable $\braket{O}$ for (a) typical case and (b) persistent oscillations including CTCs.
• Figure 4: Dissipative continuous phase transition with spontaneous breaking of a weak unitary symmetry: Beyond the critical point, multiple symmetry-broken steady states emerge. This indicates that the Lindbladian gap $\Delta_{\rm L}$ vanishes and an order parameter $|\braket{O}|$ takes a finite value. [Inspired by Refs.Mingantifazio2024many]
• Figure 5: Numerical calculation for the DDM. (a) Illustration of the DDM. (b) The magnetization dynamics for $S=20,40,80$ and mean-field analysis ($S=\infty$). (c) Top: The Lindbladian gap in the TISS. Middle: The normalized magnetization $\braket{m_{z}}$ in the TISS. Viewing $\braket{m_{z}}$ as the order parameter, the Lindbladian gap closes in the disordered phase ($\braket{m_{z}}=0$), while it remains open in the ordered phase ($\braket{m_{z}}\neq0$). This behavior is different from the conventional spectral theory for DPTs with unitary symmetry breaking. Bottom: Mean-field dynamics (green arrow), stable (orange), and unstable (black) fixed points, and the representative collective excitation modes (blue and red or pink arrows). The transition point is a CEP. (d) Purity and (e) Kitagawa-Ueda spin squeezing parameter $\xi_{\rm KU}$ in the TISS. (f) The non-zero imaginary part with the maximum real part of the Lindbladian eigenvalues and (g) the Lindbladian gap for $S=200$ (blue dot) and the mean-field solution (light blue line) with $g=1$. (h) Top: the $S$-dependence of the maximum real parts. Bottom: imaginary parts of eigenvalues with maximum real parts. (i) The $S$-dependence of (top) $|\kappa-\kappa_c|$ and (bottom) real part for the LEP with maximum real part. The dashed line shows the possible $S^{-0.62}$, $S^{-0.35}$ scaling, respectively. (j) $|\rho_{ss}-PT\rho_{ss} (PT)^{-1}|$ for (left) $\kappa/g=0.5$ (right) $\kappa/g=1.5$ and $S=10$ where $|\rho|$ implies the matrix taking the absolute value for each element of the density matrix $\rho$. Here, elements are computed on the $S_{z}$ basis.

Theorems & Definitions (2)

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