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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, $$\theta^1$$ and $$\theta^2$$, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM, $$\theta^1$$ is complex and $$\theta^2$$ is its real part. The imaginary part of $$\theta^1$$ 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. \textcolor{red}{At certain critical points, the generalized IGP may exhibit discrete jumps at finite temperatures, signaling a geometric phase transition. We illustrate the IGP of PTQM through two examples and compare their differences}.
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Geometry of variational methods: dynamics of closed quantum systems
We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-Kähler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.
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- Award ID(s):
- 1734011
- PAR ID:
- 10248686
- Date Published:
- Journal Name:
- SciPost Physics
- Volume:
- 9
- Issue:
- 4
- ISSN:
- 2542-4653
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.21468/SciPostPhys.9.4.048
