Generalized gauge transformation with $PT$-symmetric non-unitary operator and classical correspondence of non-Hermitian Hamiltonian for a periodically driven system

• URL: http://arxiv.org/abs/2209.01393v1
• Date: Sat, 3 Sep 2022 10:29:29 GMT
• Title: Generalized gauge transformation with $PT$-symmetric non-unitary operator and classical correspondence of non-Hermitian Hamiltonian for a periodically driven system
• Authors: Yan Gu, Xiao-Lei Hao, J.-Q. Liang
• Abstract summary: Biorthogonal sets of eigenstates appear necessarily as a consequence of non-Hermitian Hamiltonian. The classical version of the non-Hermitian Hamiltonian becomes a complex function of canonical variables and time. With the change of position-momentum to angle-action variables it is revealed that the non-adiabatic Hannay's angle $Delta theta_H$ and Berry phase satisfy precisely the quantum-classical correspondence.
• Score: 1.4287758028119788
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We in this paper demonstrate that the $PT$-symmetric non-Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time-dependent transformation operator has to be $PT$ symmetric and non-unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of non-Hermitian Hamiltonian. We obtain analytically the wave functions and associated non-adiabatic Berry phase $\gamma_{n}$ for the $n$th eigenstate. The classical version of the non-Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with $PT$ symmetric canonical-variable transfer in the classical gauge transformation. Moreover, with the change of position-momentum to angle-action variables it is revealed that the non-adiabatic Hannay's angle $\Delta \theta_{H}$ and Berry phase satisfy precisely the quantum-classical correspondence,$\gamma_{n}=$ $(n+1/2)\Delta \theta_{H}$.

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