arXiv:2401.07442 · Physical review. B./Physical review. B

Interferometric geometric phases of PT -symmetric quantum mechanics

Xin Wang, Zheng Zhou, Jia-Chen Tang, Xu-Yang Hou, Hao Guo, Chih-Chun Chien

Published Jun 10, 2024• 0 citations

Abstract

We present a generalization of the geometric phase to pure and thermal states in $\mathcal{PT}$-symmetric quantum mechanics (PTQM) based on the approach of the interferometric geometric phase (IGP). The formalism first introduces the parallel-transport conditions of quantum states and reveals two geometric phases, ${\ensuremath{\theta}}^{1}$ and ${\ensuremath{\theta}}^{2}$, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM, ${\ensuremath{\theta}}^{1}$ is complex and ${\ensuremath{\theta}}^{2}$ is its real part. The imaginary part of ${\ensuremath{\theta}}^{1}$ plays an important role when we generalize the IGP to thermal states in PTQM. The generalized IGP modifies the thermal distribution of a thermal state, thereby introducing effective temperatures. At certain critical points, the generalized IGP may exhibit discrete jumps at finite temperatures, signaling a geometric phase transition. We illustrate this phenomenon in PTQM through two examples and compare their differences.

DOI: 10.1103/physrevb.109.245411

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arXiv:2401.07442 · Physical review. B./Physical review. B

Interferometric geometric phases of PT -symmetric quantum mechanics

Xin Wang, Zheng Zhou, Jia-Chen Tang, Xu-Yang Hou, Hao Guo, Chih-Chun Chien

Published Jun 10, 2024• 0 citations

arXiv HTML PDF

Abstract

We present a generalization of the geometric phase to pure and thermal states in

\mathcal{PT}

-symmetric quantum mechanics (PTQM) based on the approach of the interferometric geometric phase (IGP). The formalism first introduces the parallel-transport conditions of quantum states and reveals two geometric phases,

{\ensuremath{\theta}}^{1}

and

{\ensuremath{\theta}}^{2}

, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM,

{\ensuremath{\theta}}^{1}

is complex and

{\ensuremath{\theta}}^{2}

is its real part. The imaginary part of

{\ensuremath{\theta}}^{1}

plays an important role when we generalize the IGP to thermal states in PTQM. The generalized IGP modifies the thermal distribution of a thermal state, thereby introducing effective temperatures. At certain critical points, the generalized IGP may exhibit discrete jumps at finite temperatures, signaling a geometric phase transition. We illustrate this phenomenon in PTQM through two examples and compare their differences.

DOI: 10.1103/physrevb.109.245411

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We present a generalization of the geometric phase to pure and thermal states in

\mathcal{PT}

{\ensuremath{\theta}}^{1}

and

{\ensuremath{\theta}}^{2}

, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM,

{\ensuremath{\theta}}^{1}

is complex and

{\ensuremath{\theta}}^{2}

is its real part. The imaginary part of

{\ensuremath{\theta}}^{1}

A series of geometric concepts are formulated for

\mathcal{PT}

-symmetric quantum mechanics and they are further unified into one entity, i.e., an extended quantum geometric tensor (QGT). The imaginary part of the extended QGT gives a Berry curvature whereas the real part induces a metric tensor on the system's parameter manifold. This results in a unified conceptual framework to understand and explore physical properties of

\mathcal{PT}

-symmetric systems from a geometric perspective. To illustrate the usefulness of the extended QGT, we show how its real part, the metric tensor, can be exploited as a tool to detect quantum phase transitions as well as spontaneous

\mathcal{PT}

symmetry breaking in

\mathcal{PT}

-symmetric systems.

Unitary evolution in

\mathcal{P}

\mathcal{T}

-symmetric quantum mechanics (QM) with a time-dependent metric is found to yield an interesting class of adiabatic processes. As an explicit example, a Berry-like phase associated with a

\mathcal{P}

\mathcal{T}

-symmetric two-level system is derived and is interpreted as the flux of a fictitious monopole with a tunable charge plus a singular string component with nontrivial phase contributions. To gain more insight, the Hermitian analog of our non-Hermitian problem is also analyzed, which results in an intriguing class of geometric-phase problems in conventional QM as well, where the Hamiltonian includes a perturbative term that is proportional to the rate of change in adiabatic parameters.