RECORD
$PT$-symmetric non-Hermitian Hamiltonian and invariant operator in periodically driven $SU(1,1)$ system
Quantum Physics · Submitted 1 Jan 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is shown explicitly that the model can be realized by a periodically driven oscillator.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
We study in this paper the time evolution of $PT$-symmetric non-Hermitian Hamiltonian consisting of periodically driven $SU(1,1)$ generators. A non-Hermitian invariant operator is adopted to solve the Schr\"{o}dinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a scheme to construct the non-Hermitian invariant with a $PT$-symmetric but non-unitary transformation operator. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. A non-unitary time-evolution operator is found analytically. As an consequence of the non-unitarity the ket ($|\psi (t)\rangle $) and bra ($\langle \psi (t)|$) states are not normalized each other. While the inner product of two states can be evaluated with the help of a metric operator. It is shown explicitly that the model can be realized by a periodically driven oscillator.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study in this paper the time evolution of $PT$-symmetric non-Hermitian Hamiltonian consisting of periodically driven $SU(1,1)$ generators. | contribution 0.96 | 0.89 | 0.05 | asserted 0.93 |
| s1 | A non-Hermitian invariant operator is adopted to solve the Schr\"{o}dinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. | method 0.96 | 0.66 | 0.07 | asserted 1.00 |
| s2 | We propose a scheme to construct the non-Hermitian invariant with a $PT$-symmetric but non-unitary transformation operator. | method 0.65 | 0.88 | 0.12 | asserted 0.77 |
| s3 | The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. | method 0.96 | 0.86 | 0.12 | asserted 1.00 |
| s4 | We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. | result 1.00 | 0.94 | 0.17 | asserted 1.00 |
| s5 | A non-unitary time-evolution operator is found analytically. | result 0.95 | 0.96 | 0.17 | asserted 1.00 |
| s6 | As an consequence of the non-unitarity the ket ($|\psi (t)\rangle $) and bra ($\langle \psi (t)|$) states are not normalized each other. | result 0.55 | 0.76 | 0.09 | asserted 1.00 |
| s7 | While the inner product of two states can be evaluated with the help of a metric operator. | method 0.71 | 0.50 | 0.05 | asserted 0.74 |
| s8 | It is shown explicitly that the model can be realized by a periodically driven oscillator. | result 0.83 | 0.85 | 0.12 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
We study in this paper the time evolution of $PT$-symmetric non-Hermitian Hamiltonian consisting of periodically driven $SU(1,1)$ generators. A non-Hermitian invariant operator is adopted to solve the Schr\"{o}dinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a scheme to construct the non-Hermitian invariant with a $PT$-symmetric but non-unitary transformation operator. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. A non-unitary time-evolution operator is found analytically. As an consequence of the non-unitarity the ket ($|\psi (t)\rangle $) and bra ($\langle \psi (t)|$) states are not normalized each other. While the inner product of two states can be evaluated with the help of a metric operator. It is shown explicitly that the model can be realized by a periodically driven oscillator.
Human review
How the literature uses it
What this paper does with the arXiv papers it cites
What it cites · 1 sentence without a confident label
Uses the method or tool of 1 paper
-
Substituting the LR phase Eq.([ref]) into the general solution of the Schrödinger equation ([ref]) we have [equation] where [equation] Assume that the initial state at time $t=0$ is denoted by [equation] The state at time $t$ is generated by the time evolution operator such that [equation] where the time evolution operator is [equation] The time evolution operator is not unitary [equation] since the inverse operator is [equation] while the complex conjugate reads [equation] The bra state [equation] is not normalized with the corresponding ket state [equation] While it can be normalized with the help of metric operator [ref] [equation] In general the inner product of two states $|ψ( t) ⟩$ and $|φ( t) ⟩$ is evaluated as [equation]
Recovering Unitarity of Lee Model in Bi-Orthogonal Basis · The time-evolution operator · 0.87
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.