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

Progress of Theoretical and Experimental Physics

On the time-reversal symmetry in pseudo-Hermitian systems

B. Choutri, Omar Cherbal, F.-Z. Ighezou, D. A. Trifonov

Published Nov 24, 2014• 2 citations
Publisher

Abstract

In a recent paper [M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012)] Sato and his collaborators established a generalization of the Kramers degeneracy structure to pseudo-Hermitian Hamiltonian systems, admitting even time-reversal symmetry, |$T^{2}=1$|⁠. This extension is achieved using the mathematical structure of split-quaternions instead of quaternions, usually adopted in the case of Hermitian Hamiltonians with odd time-reversal symmetry, |$T^{2}=-1$|⁠. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian |$H$| that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such |$H$| with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart |$h$|⁠, so that the generalized Kramers degeneracy of |$H$| is in fact crypto-Hermitian Kramers degeneracy. We study in greater detail a new example of the pseudo-Hermitian split-quaternionic four-level Hamiltonian system, which admits an indefinite metric operator and time-reversal symmetry and, as a consequence, a generalized Kramers degeneracy structure. We provide a complete solution of the eigenvalue problem, construct pseudo-Hermitian ladder operators closing the normal and abnormal pseudo-fermionic algebras, and show that this system fulfills a crypto-Hermitian degeneracy.

DOI
10.1093/ptep/ptu157

Related papers

Time-Reversal Symmetry in Non-Hermitian Systems

M. Sato, K. Hasebe, K. Esaki +1 more

For ordinary Hermitian Hamiltonians, the states support the mathematical structure of quaternion and show the Kramers degeneracy when the system has a half-odd-integer spin and the time-reversal operator obeys Θ 2 = -1, while no such structure exists when Θ 2 = +1.Here, we point out that for non-Hermitian systems, there exists a similar mathematical structure called split-quaternion even when Θ 2 = +1.It is also found that a degeneracy similar to the Kramers degeneracy follows from the split-quaternion structure if the Hamiltonian is pseudo-Hermitian with the metric operator η satisfying {η, Θ} = 0. Furthermore, we show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split-quaternion in a class of non-Hermitian Hamiltonians.As concrete examples, we examine in detail N × N Hamiltonians with N = 2 and 4, which are non-Hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.

arXiv:1106.1806•Progress of Theoretical Physics•Jun 1, 2012•81 citations

Time-Reversal Symmetry in Non-Hermitian Systems

Masatoshi Sato, Kazuki Hasebe, Kenta Esaki +1 more

For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys \Theta^2=-1, but no such a degeneracy exists when \Theta^2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when \Theta^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.

•Jan 1, 2013•65 citations

On the pseudo-Hermitian nondiagonalizable Hamiltonians

G. Scolarici, Luigi Solombrino

We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, and we establish for such a class the equivalence between the pseudo-Hermiticity property and the existence of an antilinear involutory symmetry. Moreover, we prove that this class actually coincides with the one of (possibly nondiagonalizable) weak pseudo-Hermitian operators, and that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a definite inner product. Finally, we show that a typical degeneracy of the real eigenvalues (which reduces to the well-known Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an antilinear symmetry like the time-reversal operator T. Some consequences and applications are briefly discussed.

arXiv:quant-ph/0211161•Journal of Mathematical Physics•Sep 25, 2003•44 citations

Metric operators for non-Hermitian quadraticsu(2) Hamiltonians

Paulo E G Assis

A class of non-Hermitian quadratic su (2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realizations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified, the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here, Lie-algebraic methods which were used to investigate the generalized Swanson Hamiltonian ( Assis and Fring 2009 J. Phys. A: Math. Theor . 42 015203 ) are employed to identify the class of quadratic Hamiltonians that allow for such a mapping to the Hermitian counterpart. Whereas for the linear su (2) system, every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su (2) generators, the situation is more complicated for quadratic Hamiltonians. Therefore, the possibility of more elaborate similarity transformations, including quadratic exponents, is also explored in detail. The existence of finite-dimensional representations for the su (2) Hamiltonian, as opposed to the su (1, 1) studied before, allows for comparison with explicit diagonalization results for finite matrices. Finally, the similarity transformations constructed are compared with the analogue of Swanson's method for exact diagonalization of the problem, establishing a simple relation between both approaches.

arXiv:1012.0194•Journal of Physics A Mathematical and Theoretical•Jun 1, 2011•10 citations

Extended PT - and CPT -symmetric representations of fermionic algebras

Omar Cherbal, D. A. Trifonov

In a recent paper, Bender and Klevansky [Phys. Rev. A 84, 024102 (2011)] considered $\mathcal{PT}$-symmetric matrix representations for fermionic operator algebras of the form ${\ensuremath{\xi}}^{2}={\overline{\ensuremath{\xi}}}^{2}=0$, $\ensuremath{\xi}\overline{\ensuremath{\xi}}+\overline{\ensuremath{\xi}}\ensuremath{\xi}=\ensuremath{\varepsilon}1$, where $\overline{\ensuremath{\xi}}$ is the $\mathcal{PT}$ transform of $\ensuremath{\xi}$. They constructed such algebras for $\ensuremath{\varepsilon}=\ensuremath{-}1$ and established that it is not possible to construct a matrix representation for the standard fermionic algebra ($\ensuremath{\varepsilon}=1$). Bender and Klevansky used the formalism developed by Jones-Smith and Mathur [Phys. Rev. A 82, 042101 (2010)] which extends $\mathcal{PT}$-symmetric quantum mechanics to the case of odd time-reversal symmetry (fermionic case). By using the same formalism, we show that $\mathcal{PT}$-symmetric matrix representations exist for both standard ($\ensuremath{\varepsilon}=1)$ and abnormal ($\ensuremath{\varepsilon}=\ensuremath{-}1)$ fermionic algebras if one takes $\overline{\ensuremath{\xi}}$ as adjoint $\ensuremath{\xi}$ with respect to the $\mathcal{CPT}$ and $\mathcal{PT}$ inner products, respectively. This general result is illustrated for the example of a typical quaternionic four-level model by an explicit construction of the fermionic creation and annihilation operators which satisfy all the criteria of $\mathcal{PT}$ quantum mechanics for the odd time-reversal symmetry.

Physical Review A•May 23, 2012•12 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
In a recent paper [M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012)] Sato and his collaborators established a generalization of the Kramers degeneracy structure to pseudo-Hermitian Hamiltonian systems, admitting even time-reversal symmetry, |T2=1T^{2}=1T2=1|⁠. This extension is achieved using the mathematical structure of split-quaternions instead of quaternions, usually adopted in the case of Hermitian Hamiltonians with odd time-reversal symmetry, |T2=−1T^{2}=-1T2=−1|⁠. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian |HHH| that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such |HHH| with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart |hhh|⁠, so that the generalized Kramers degeneracy of |HHH| is in fact crypto-Hermitian Kramers degeneracy. We study in greater detail a new example of the pseudo-Hermitian split-quaternionic four-level Hamiltonian system, which admits an indefinite metric operator and time-reversal symmetry and, as a consequence, a generalized Kramers degeneracy structure. We provide a complete solution of the eigenvalue problem, construct pseudo-Hermitian ladder operators closing the normal and abnormal pseudo-fermionic algebras, and show that this system fulfills a crypto-Hermitian degeneracy.
For ordinary Hermitian Hamiltonians, the states support the mathematical structure of quaternion and show the Kramers degeneracy when the system has a half-odd-integer spin and the time-reversal operator obeys Θ 2 = -1, while no such structure exists when Θ 2 = +1.Here, we point out that for non-Hermitian systems, there exists a similar mathematical structure called split-quaternion even when Θ 2 = +1.It is also found that a degeneracy similar to the Kramers degeneracy follows from the split-quaternion structure if the Hamiltonian is pseudo-Hermitian with the metric operator η satisfying {η, Θ} = 0. Furthermore, we show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split-quaternion in a class of non-Hermitian Hamiltonians.As concrete examples, we examine in detail N × N Hamiltonians with N = 2 and 4, which are non-Hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.
For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys \Theta^2=-1, but no such a degeneracy exists when \Theta^2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when \Theta^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.
We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, and we establish for such a class the equivalence between the pseudo-Hermiticity property and the existence of an antilinear involutory symmetry. Moreover, we prove that this class actually coincides with the one of (possibly nondiagonalizable) weak pseudo-Hermitian operators, and that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a definite inner product. Finally, we show that a typical degeneracy of the real eigenvalues (which reduces to the well-known Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an antilinear symmetry like the time-reversal operator T. Some consequences and applications are briefly discussed.
A class of non-Hermitian quadratic su (2) Hamiltonians having an anti-linear symmetry is constructed. This is achieved by analysing the possible symmetries of such systems in terms of automorphisms of the algebra. In fact, different realizations for this type of symmetry are obtained, including the natural occurrence of charge conjugation together with parity and time reversal. Once specified, the underlying anti-linear symmetry of the Hamiltonian, the former, if unbroken, leads to a purely real spectrum and the latter can be mapped to a Hermitian counterpart by, amongst other possibilities, a similarity transformation. Here, Lie-algebraic methods which were used to investigate the generalized Swanson Hamiltonian ( Assis and Fring 2009 J. Phys. A: Math. Theor . 42 015203 ) are employed to identify the class of quadratic Hamiltonians that allow for such a mapping to the Hermitian counterpart. Whereas for the linear su (2) system, every Hamiltonian of this type can be mapped to a Hermitian counterpart by a transformation which is itself an exponential of a linear combination of su (2) generators, the situation is more complicated for quadratic Hamiltonians. Therefore, the possibility of more elaborate similarity transformations, including quadratic exponents, is also explored in detail. The existence of finite-dimensional representations for the su (2) Hamiltonian, as opposed to the su (1, 1) studied before, allows for comparison with explicit diagonalization results for finite matrices. Finally, the similarity transformations constructed are compared with the analogue of Swanson's method for exact diagonalization of the problem, establishing a simple relation between both approaches.
In a recent paper, Bender and Klevansky [Phys. Rev. A 84, 024102 (2011)] considered PT\mathcal{PT}PT-symmetric matrix representations for fermionic operator algebras of the form ${\ensuremath{\xi}}^{2}={\overline{\ensuremath{\xi}}}^{2}=0$, $\ensuremath{\xi}\overline{\ensuremath{\xi}}+\overline{\ensuremath{\xi}}\ensuremath{\xi}=\ensuremath{\varepsilon}1$, where $\overline{\ensuremath{\xi}}$ is the PT\mathcal{PT}PT transform of $\ensuremath{\xi}$. They constructed such algebras for $\ensuremath{\varepsilon}=\ensuremath{-}1$ and established that it is not possible to construct a matrix representation for the standard fermionic algebra ($\ensuremath{\varepsilon}=1$). Bender and Klevansky used the formalism developed by Jones-Smith and Mathur [Phys. Rev. A 82, 042101 (2010)] which extends PT\mathcal{PT}PT-symmetric quantum mechanics to the case of odd time-reversal symmetry (fermionic case). By using the same formalism, we show that PT\mathcal{PT}PT-symmetric matrix representations exist for both standard ($\ensuremath{\varepsilon}=1)$ and abnormal ($\ensuremath{\varepsilon}=\ensuremath{-}1)$ fermionic algebras if one takes $\overline{\ensuremath{\xi}}$ as adjoint $\ensuremath{\xi}$ with respect to the CPT\mathcal{CPT}CPT and PT\mathcal{PT}PT inner products, respectively. This general result is illustrated for the example of a typical quaternionic four-level model by an explicit construction of the fermionic creation and annihilation operators which satisfy all the criteria of PT\mathcal{PT}PT quantum mechanics for the odd time-reversal symmetry.
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.