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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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A unified approach of detecting phase transition in time-varying complex networks
Abstract Deciphering the non-trivial interactions and mechanisms driving the evolution of time-varying complex networks (TVCNs) plays a crucial role in designing optimal control strategies for such networks or enhancing their causal predictive capabilities. In this paper, we advance the science of TVCNs by providing a mathematical framework through which we can gauge how local changes within a complex weighted network affect its global properties. More precisely, we focus on unraveling unknown geometric properties of a network and determine its implications on detecting phase transitions within the dynamics of a TVCN. In this vein, we aim at elaborating a novel and unified approach that can be used to depict the relationship between local interactions in a complex network and its global kinetics. We propose a geometric-inspired framework to characterize the network’s state and detect a phase transition between different states, to infer the TVCN’s dynamics. A phase of a TVCN is determined by its Forman–Ricci curvature property. Numerical experiments show the usefulness of the proposed curvature formalism to detect the transition between phases within artificially generated networks. Furthermore, we demonstrate the effectiveness of the proposed framework in identifying the phase transition phenomena governing the training and learning processes of artificial neural networks. Moreover, we exploit this approach to investigate the phase transition phenomena in cellular re-programming by interpreting the dynamics of Hi-C matrices as TVCNs and observing singularity trends in the curvature network entropy. Finally, we demonstrate that this curvature formalism can detect a political change. Specifically, our framework can be applied to the US Senate data to detect a political change in the United States of America after the 1994 election, as discussed by political scientists.
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- Award ID(s):
- 1932620
- PAR ID:
- 10481602
- Publisher / Repository:
- https://www.nature.com/articles/s41598-023-44791-3
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 13
- Issue:
- 1
- ISSN:
- 2045-2322
- 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/10.1038/s41598-023-44791-3
