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arXiv:2109.00425 · Physical Review A

Physics of PT -symmetric quantum systems at finite temperatures

Qian Du, Kui Cao, Su-Peng Kou

Published Sep 7, 2022• 12 citations
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Abstract

We study parity-time-symmetric non-Hermitian quantum systems at finite temperature, where the Boltzmann distribution law fails to hold. To characterize their abnormal physical properties, a quantum statistical theory (the so-called quantum Liouvillian statistical theory) is developed, in which the Boltzmann distribution law is replaced by the Liouvillian-Boltzmann distribution law. Using the theory, we derive analytical results of thermodynamic properties for parity-time-symmetric non-Hermitian quantum systems and find that a ``continuous'' thermodynamic phase transition occurs at the exceptional point, where a zero-temperature anomaly exists.

DOI
10.1103/physreva.106.032206

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•May 31, 2011•0 citations
We study parity-time-symmetric non-Hermitian quantum systems at finite temperature, where the Boltzmann distribution law fails to hold. To characterize their abnormal physical properties, a quantum statistical theory (the so-called quantum Liouvillian statistical theory) is developed, in which the Boltzmann distribution law is replaced by the Liouvillian-Boltzmann distribution law. Using the theory, we derive analytical results of thermodynamic properties for parity-time-symmetric non-Hermitian quantum systems and find that a ``continuous'' thermodynamic phase transition occurs at the exceptional point, where a zero-temperature anomaly exists.
Abstract As a very simple model, the Ising model plays an important role in statistical physics. In the paper, with the help of quantum Liouvillian statistical theory, we study the one-dimensional non-Hermitian Ising model at finite temperature and give its analytical solutions. We find that the non-Hermitian Ising model shows quite different properties from those of its Hermitian counterpart. For example, the ‘pseudo-phase transition’ is explored between the ‘topological’ phase and the ‘non-topological’ phase, at which the Liouvillian energy gap is closed rather than the usual energy gap. In particular, we point out that the one-dimensional non-Hermitian Ising model at finite temperature can be equivalent to an effective anisotropic XY model in the transverse field. This work will help people understand quantum statistical properties of non-Hermitian systems at finite temperatures.
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.
Abstract We propose a new type of quantum thermodynamic cycle whose efficiency is greater than the one of the classical Carnot cycle for the same conditions for a system when viewed as homogeneous. In our model, this type of cycle only exists in the low-temperature regime in the spontaneously broken parity-time-reversal (PT) symmetry regime of a non-Hermitian quantum theory and does not manifest in the PT-symmetric regime. We discuss this effect for an ensemble based on a model of a single boson coupled in a non-Hermitian way to a bath of different types of bosons with and without a time-dependent boundary. The cycle cannot be set up when considering our system as heterogeneous, i.e. undergoing a first-order phase transition. Within that interpretation, we find that the entropy is vanishing throughout the spontaneously broken PT-regime.
Parity-time (PT\mathcal{P}\mathcal{T}PT) symmetry is extensively studied within non-Hermitian physical systems as it has the capability to ensure the presence of real eigenenergies of the systems characterized by non-Hermitian Hamiltonians. Previous studies have been mostly based on the Bloch band theory. Recent developments have unveiled intriguing phenomena which can only be explained by non-Bloch band theory. In this paper, we observe the PT\mathcal{P}\mathcal{T}PT phase transition in photonic quantum walks, which can only be explained by the non-Bloch band theory, which relies on the open boundary condition. Furthermore, we explore the impact of disorder on PT\mathcal{P}\mathcal{T}PT symmetry and demonstrate disorder-induced PT\mathcal{P}\mathcal{T}PT phase transitions. Our work provides a valuable perspective for further explorations on non-Hermitian physics.
Recently, impressive progress has been made in the study of non-Hermitian systems with parity-time symmetry, such as observations of topological properties of physical systems and criticality at exceptional points. A crucial aspect of parity-time symmetric nonunitary dynamics is the information flow between the system and the environment. In this paper, we use the physical quantity, distinguishability between quantum states, to uniformly quantify the information flow between low-dimensional and high-dimensional parity-time symmetric non-Hermitian systems and environments. The numerical results show that the oscillation of quantum state distinguishability and complete information retrieval and can be obtained in the parity-time-unbroken phase. However, the information decays exponentially in the paritytime-broken phase. The exceptional point marks the criticality between reversibility and irreversibility of information flow, and the distinguishability between quantum states exhibits the behavior of power-law decay. Understanding these unique phenomena in nonunitary quantum dynamics provides an important perspective for the study of open quantum systems and contributes to their application in quantum information.