Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1812.09462 · Europhysics Letters (EPL)

Anyonic $\boldsymbol{\mathcal{PT}}$ symmetry, drifting potentials and non-Hermitian delocalization

Stefano Longhi, E. Pinotti

Published Jan 30, 2019• 8 citations
arXivHTMLPDF

Abstract

We consider wave dynamics for a Schrödinger equation with a non-Hermitian Hamiltonian H satisfying the generalized (anyonic) parity-time symmetry PT H = exp ⁡ ( 2 i φ ) HPT , where P and T are the parity and time-reversal operators. For a stationary potential, the anyonic phase φ just rotates the energy spectrum of H in a complex plane, however, for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilean invariance when φ ≠ 0 . In particular, in the unbroken PT phase the drift makes a scattering potential barrier reflectionless, whereas for a potential well the number of bound states decreases as the drift velocity increases because of a non-Hermitian delocalization transition.

DOI
10.1209/0295-5075/125/10006

Related papers

The anyonic recovery of parity-time symmetry in coupled circuits system

Xin-Chang Liu, Dan-Dan Sun, Xiaodong Shi +3 more

Abstract Hermiticity is the fundamental property of the physical systems, for which it should obey the laws of energy conservation and time-reversal symmetry. Parity-time symmetry is introduced to study the non-Hermitian system with real energy spectra. Considering the coupling dissipation of the practical system, there would be phase-related terms on the non-diagonal elements of the Hermitian matrix. Here in this work, we present a coupled circuit model of the system and investigate the dynamics of parity-time-anyonic Hamiltonian related to an arbitrary phase of the system. We find that parity-time symmetry can be achieved under the tunable phases in the quasi-parity-time symmetric circuit system.

Europhysics Letters (EPL)•Nov 28, 2022•0 citations

Topological metal-insulator transitions in one-dimensional non-Hermitian quasicrystals: beyond PT-symmetry

Guangjie Zhang, Bing Shao, Longwen Zhou +2 more

One-dimensional non-Hermitian quasicrystals with parity and time-reversal (PT) symmetry can simultaneously exhibit localization-delocalization transition, topological phase transition, and PT-symmetry-breaking transition. This motivates this work to investigate how the absence of PT symmetry impacts topological metal-insulator transitions in non-Hermitian quasicrystals. We propose a non-Hermitian quasiperiodic model that generally does not preserve PT symmetry and demonstrate that, in most parameter regions, such a system supports triple phase transitions that encompass localization, topology, and degeneracy-breaking. The system may also exhibit a particular type of localization-delocalization transition analogous to the Hermitian case, namely, without activating topological phase transitions or degeneracy-breaking transitions. Our work extends the topological metal-insulator transitions previously studied in PT-symmetric systems to a more general class of non-Hermitian setting, and further reveals that non-Hermitian systems can host distinct types of localization behavior.

cond-mat.dis-nnCross-listedarXiv:2605.25072•May 26, 2026

Phase transitions and spectral singularities in a class of one-dimensional parity-time-symmetric complex potentials

Jinlin Fan, Feilong Wang, Ruolin Chai Zhibin Zhao +1 more

We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time ($\mathcal{PT}$) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.

quant-pharXiv:2511.09010•Nov 13, 2025

P, T, PT, and CPT invariance of Hermitian Hamiltonians

Zafar Ahmed

Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the PT symmetry is unbroken. PT-symmetry is called unbroken or exact if the energy-eigenstates are also simultaneous eigenstates of PT. On the other hand when PT-symmetry is spontaneously broken the energy-eigenvalues are complex conjugate pairs. Multipronged investigations supporting

•Jan 1, 2003•19 citations

A family of complex potentials with real spectrum

Francisco Marcelo Fernández, R. Guardiola, J Ros +1 more

We consider a two-parameter non-Hermitian quantum mechanical Hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations. Numerical investigation shows that for some values of the potential parameters the Hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.

arXiv:quant-ph/9812026•Journal of Physics A Mathematical and General•Jan 1, 1999•144 citations

Non-Hermitian quantum Hamiltonians with PT symmetry

Katherine Jones-Smith, Harsh Mathur

We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under $\mathcal{P}\mathcal{T}$, where $\mathcal{P}$ is the parity and $\mathcal{T}$ denotes time reversal, for the case that time-reversal symmetry is odd (${\mathcal{T}}^{2}=\ensuremath{-}1$), generalizing prior work for the even case (${\mathcal{T}}^{2}=1$). We discover an analog of Kramer's theorem for $\mathcal{P}\mathcal{T}$ quantum mechanics, present a prototypical example of a $\mathcal{P}\mathcal{T}$ quantum system with odd time reversal, and discuss potential applications of the formalism.

arXiv:0908.4255•Physical Review A•Oct 6, 2010•77 citations
Anyonic PT\boldsymbol{\mathcal{PT}}PT symmetry, drifting potentials and non-Hermitian delocalization
We consider wave dynamics for a Schrödinger equation with a non-Hermitian Hamiltonian H satisfying the generalized (anyonic) parity-time symmetry PT H = exp ⁡ ( 2 i φ ) HPT , where P and T are the parity and time-reversal operators. For a stationary potential, the anyonic phase φ just rotates the energy spectrum of H in a complex plane, however, for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilean invariance when φ ≠ 0 . In particular, in the unbroken PT phase the drift makes a scattering potential barrier reflectionless, whereas for a potential well the number of bound states decreases as the drift velocity increases because of a non-Hermitian delocalization transition.
Abstract Hermiticity is the fundamental property of the physical systems, for which it should obey the laws of energy conservation and time-reversal symmetry. Parity-time symmetry is introduced to study the non-Hermitian system with real energy spectra. Considering the coupling dissipation of the practical system, there would be phase-related terms on the non-diagonal elements of the Hermitian matrix. Here in this work, we present a coupled circuit model of the system and investigate the dynamics of parity-time-anyonic Hamiltonian related to an arbitrary phase of the system. We find that parity-time symmetry can be achieved under the tunable phases in the quasi-parity-time symmetric circuit system.
One-dimensional non-Hermitian quasicrystals with parity and time-reversal (PT) symmetry can simultaneously exhibit localization-delocalization transition, topological phase transition, and PT-symmetry-breaking transition. This motivates this work to investigate how the absence of PT symmetry impacts topological metal-insulator transitions in non-Hermitian quasicrystals. We propose a non-Hermitian quasiperiodic model that generally does not preserve PT symmetry and demonstrate that, in most parameter regions, such a system supports triple phase transitions that encompass localization, topology, and degeneracy-breaking. The system may also exhibit a particular type of localization-delocalization transition analogous to the Hermitian case, namely, without activating topological phase transitions or degeneracy-breaking transitions. Our work extends the topological metal-insulator transitions previously studied in PT-symmetric systems to a more general class of non-Hermitian setting, and further reveals that non-Hermitian systems can host distinct types of localization behavior.
We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time (PT\mathcal{PT}PT) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.
Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the PT symmetry is unbroken. PT-symmetry is called unbroken or exact if the energy-eigenstates are also simultaneous eigenstates of PT. On the other hand when PT-symmetry is spontaneously broken the energy-eigenvalues are complex conjugate pairs. Multipronged investigations supporting
We consider a two-parameter non-Hermitian quantum mechanical Hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations. Numerical investigation shows that for some values of the potential parameters the Hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.
We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under PT\mathcal{P}\mathcal{T}PT, where P\mathcal{P}P is the parity and T\mathcal{T}T denotes time reversal, for the case that time-reversal symmetry is odd (${\mathcal{T}}^{2}=\ensuremath{-}1$), generalizing prior work for the even case (T2=1{\mathcal{T}}^{2}=1T2=1). We discover an analog of Kramer's theorem for PT\mathcal{P}\mathcal{T}PT quantum mechanics, present a prototypical example of a PT\mathcal{P}\mathcal{T}PT quantum system with odd time reversal, and discuss potential applications of the formalism.