Differential geometry of time-dependent PT-symmetric quantum mechanics · arXivDesk
1811.04640Nov 12, 20189 pages (no figure), as an accompanying paper of a preprint titled "Quantum Geometric Tensor in PT-Symmetric Quantum Mechanics" by the same authors
Differential geometry of time-dependent PT-symmetric quantum mechanics
Time-dependent PT-symmetric quantum mechanics is featured by a varying inner-product metric and has stimulated a number of interesting studies beyond conventional quantum mechanics. In this paper, we explore geometric aspects of time-dependent PT-symmetric quantum mechanics. We not only find a geometric phase factor emerging naturally from cyclic evolutions of PT-symmetric systems, but also formulate a series of differential geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. Our findings constitute a useful, perhaps indispensible, tool to tackle physical problems involving PT
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-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometrical phase [Phys. Rev. Lett. 91, 187902 (2003)], consisting of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase identified in this work.