Abstract Parity Time Reversal (PT) phase transition is a typical characteristic of most of the PT symmetric non-Hermitian (NH) systems. Depending on the theory, a particular system and spacetime dimensionality PT phase transition has various interesting features. In this article we review some of our works on PT phase transitions in quantum mechanics (QM) as well as in Quantum Field theory (QFT). We demonstrate typical characteristics of PT phase transition with the help of several analytically solved examples. In one dimensional QM, we consider examples with exactly as well as quasi exactly solvable (QES) models to capture essential features of PT phase transition. The discrete symmetries have rich structures in higher dimensions which are used to explore the PT phase transition in higher dimensional systems. We consider anisotropic SHOs in two and three dimensions to realize some connection between the symmetry of original hermitian Hamiltonian and the unbroken phase of the NH system. We consider the 2+1 dimensional massless Dirac particle in the external magnetic field with PT symmetric non-Hermitian spin-orbit interaction in the background of the Dirac oscillator potential to show the PT phase transition in a relativistic system. A small mass gap, consistent with the other approaches and experimental observations is generated only in the unbroken phase of the system. Finally we develop the NH formulation in an SU(N) gauge field theoretic model by using the natural but unconventional Hermiticity properties of the ghost fields. Deconfinement to confinement phase transition has been realized as PT phase transition in such a non-hermitian model.
Abstract Parity Time Reversal (PT) phase transition is a typical characteristic of most of the PT symmetric non-Hermitian (NH) systems. Depending on the theory, a particular system and spacetime dimensionality PT phase transition has various interesting features. In this article we review some of our works on PT phase transitions in quantum mechanics (QM) as well as in Quantum Field theory (QFT). We demonstrate typical characteristics of PT phase transition with the help of several analytically solved examples. In one dimensional QM, we consider examples with exactly as well as quasi exactly solvable (QES) models to capture essential features of PT phase transition. The discrete symmetries have rich structures in higher dimensions which are used to explore the PT phase transition in higher dimensional systems. We consider anisotropic SHOs in two and three dimensions to realize some connection between the symmetry of original hermitian Hamiltonian and the unbroken phase of the NH system. We consider the 2+1 dimensional massless Dirac particle in the external magnetic field with PT symmetric non-Hermitian spin-orbit interaction in the background of the Dirac oscillator potential to show the PT phase transition in a relativistic system. A small mass gap, consistent with the other approaches and experimental observations is generated only in the unbroken phase of the system. Finally we develop the NH formulation in an SU(N) gauge field theoretic model by using the natural but unconventional Hermiticity properties of the ghost fields. Deconfinement to confinement phase transition has been realized as PT phase transition in such a non-hermitian model.
Abstract Parity-time-reversal (PT) symmetric quantum mechanics promotes the increasing research interest of non-Hermitian (NH) systems for the theoretical value, novel properties, and links to open and dissipative systems in various areas. Recently, anti-PT–symmetric systems and their featured properties have started to be investigated. In this work, we develop the PT and anti-PT symmetry to PT-arbitrary-phase symmetry (or PT- φ symmetry) for the first time, being analogous to bosons, fermions and anyons. It can also be seen as a complex extension of the PT symmetry, unifying the PT and anti-PT symmetries and having intermediate properties between them. Many of the established concepts and mathematics in the PT-symmetric system are still compatible. We mainly investigate quantum simulation of this novel NH system in two dimensions in detail and discuss for higher-dimensional cases in general using the linear combinations of unitaries in the scheme of duality quantum computing, enabling implementations and experimental investigations of novel properties on both small quantum devices and near-term quantum computers.
Most quantum physicists believe that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under matrix transposition and complex conjugation) to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the non-Dirac-hermitian Hamiltonian H = p2+ix3 has a real positive discrete spectrum and generates unitary time evolution and defines a fully consistent and physical quantum theory. Evidently, Dirac Hermiticity is too restrictive. While H = p2+ix3 is not Dirac Hermitian, it is PT symmetric (invariant under combined space reflection P and time reversal T). Another PT-symmetric Hamiltonian whose energy levels are real, positive and discrete is H = p2−x4, which contains an upside-down potential. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics and quantum field theory are extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past two years some of these properties have been verified in laboratory experiments. Here, we first discuss PT-symmetric Hamiltonians at a simple intuitive level and explain why the energy levels of such Hamiltonians may be real, positive, and discrete. Second, we describe a recent experiment in which the PT phase transition was observed. Third, we briefly mention that PT-symmetric theories can be useful at a fundamental level. While the double-scaling limit of an O(N)-symmetric gφ4 quantum field theory appears to be inconsistent because the critical value of g is negative, this limit is in fact not inconsistent because the critical theory is PT symmetric.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].
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 Quantum systems governed by non-Hermitian Hamiltonians with PT symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that PT symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to PT -symmetric effective Lagrangians. We show how PT symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples PT -symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the PT norm that occurs in Hamiltonian quantum theory. We discuss examples where PT -symmetric field theories emerge from Hermitian field theories due to effects of renormalisation. We also consider the effects of renormalisation on field theories that are non-Hermitian but PT -symmetric from the start.
A quantum system governed by a non-Hermitian Hamiltonian may exhibit zero-temperature phase transitions that are driven by interactions, just as its Hermitian counterpart, raising the fundamental question of how non-Hermiticity affects quantum criticality. In this context, we consider a non-Hermitian system consisting of an X Y model with a complex-valued four-state clock interaction that may or may not have parity-time-reversal ( PT ) symmetry. When the PT symmetry is broken, and time-evolution becomes nonunitary, a scaling behavior similar to the Berezinskii-Kosterlitz-Thouless phase transition ensues, but in a highly unconventional way, as the line of fixed points is absent. From the analysis of the d -dimensional renormalization group equations, we obtain that the unconventional behavior in the PT broken regime follows from the collision of two fixed points in the d → 2 limit, leading to walking behavior or pseudocriticality. For d = 2 + 1 the near critical behavior is characterized by a correlation length exponent ν = 3 / 8 , a value smaller than the mean-field one. These results are in sharp contrast with the PT -symmetric case where only one fixed point arises for 2 < d < 4 and in d = 1 + 1 three lines of fixed points occur with a continuously varying critical exponent ν . Published by the American Physical Society 2024