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arXiv:1805.04968

Symmetries and invariants for non-Hermitian Hamiltonians

Martínez, M. A. Simón, Álvaro Buendía, Juan Gonzalo Muga

Published May 13, 2018• 2 citations
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Abstract

We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation $AHA^\dagger$ that leaves the Hamiltonian $H$ unchanged represents a symmetry of the Hamiltonian, which implies the commutativity $[H,A]=0$, and a conservation law, namely the invariance of expectation values of $A$. For non-Hermitian Hamiltonians, $H^\dagger$ comes into play as a distinct operator that complements $H$ in generalized unitarity relations. The above description of symmetries has to be extended to include also $A$-pseudohermiticity relations of the form $AH=H^\dagger A$. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of $A$ operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.

DOI
10.48550/arxiv.1805.04968

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We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA†AHA^\daggerAHA† that leaves the Hamiltonian HHH unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0[H,A]=0[H,A]=0, and a conservation law, namely the invariance of expectation values of AAA. For non-Hermitian Hamiltonians, H†H^\daggerH† comes into play as a distinct operator that complements HHH in generalized unitarity relations. The above description of symmetries has to be extended to include also AAA-pseudohermiticity relations of the form AH=H†AAH=H^\dagger AAH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of AAA operators forming Klein's 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.
Abstract A non-Abelian group of 16 symmetry operations on (generally) non-Hermitian discrete Hamiltonians represented by matrices is studied. The symmetry operations are described by unitary/antiunitary superoperators that arise when combining three basic generating operations with simple ‘geometric’ interpretations. The corresponding Hamiltonian symmetries occur when the Hamiltonian remains invariant under the superoperator action. These symmetries include PT-symmetry and Hermiticity as particular cases. The interplay between the group of symmetry operations and Hamiltonian symmetries is analyzed systematically by introducing the concepts of equivalent operations and associated symmetries. Spectral properties implied by some of the symmetries are described.
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.
Abstract We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.
We extend the definition of generalized parity PPP, charge-conjugation CCC and time-reversal TTT operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that TPTPTP and CTPCTPCTP are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian HHH to admit a PPP-reflecting symmetry which generates the PPP-pseudounitary and the PPP-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the PPP-unitary evolution of a physical system are also given.
It is shown that if a Hamiltonian H is Hermitian, then there always exists an operator having the following properties: (i) is linear and Hermitian; (ii) commutes with H ; (iii) 2 = 1; (iv) the n th eigenstate of H is also an eigenstate of with eigenvalue (−1) n . Given these properties, it is appropriate to refer to as the parity operator and to say that H has parity symmetry, even though may not refer to spatial reflection. Thus, if the Hamiltonian has the form H = p 2 + V ( x ), where V ( x ) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has symmetry. This shows that symmetry is a generalization of Hermiticity: all Hermitian Hamiltonians of the form H = p 2 + V ( x ) have symmetry, but not all -symmetric Hamiltonians of this form are Hermitian.