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arXiv:1811.04638 · Physical Review A

Quantum geometric tensor in PT -symmetric quantum mechanics

Da-Jian Zhang, Qinghai Wang, Jiangbin Gong

Published Apr 4, 2019• 70 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.99.042104

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A series of geometric concepts are formulated for PT\mathcal{PT}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 PT\mathcal{PT}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 PT\mathcal{PT}PT symmetry breaking in PT\mathcal{PT}PT-symmetric systems.
A new type of quantum theory known as time-dependent P T -symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a P T -symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent P T -symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a P T -symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of P T -symmetric systems with time-varying system’s parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [ Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.
Differential geometry of time-dependent PT\mathcal{PT}PT-symmetric quantum mechanics
Time-dependent PT\mathcal{PT}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\mathcal{PT}PT-symmetric quantum mechanics. We not only find a geometric phase factor emerging naturally from cyclic evolutions of PT\mathcal{PT}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\mathcal{PT}PT-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.
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Abstract The quantum geometric tensor (QGT) is a fundamental quantity for characterizing the geometric properties of quantum states and plays an essential role in elucidating various physical phenomena. The traditional QGT, defined only for pure states, has limited applicability in realistic scenarios where mixed states are common. To address this limitation, we generalize the definition of the QGT to mixed states using the purification bundle and the covariant derivative. Notably, our proposed definition reduces to the traditional QGT when mixed states approach pure states. In our framework, the real and imaginary parts of this generalized QGT correspond to the Bures metric and the mean gauge curvature, respectively, endowing it with a broad range of potential applications. Additionally, using our proposed mixed-state QGT, we derive the geodesic equation applicable to mixed states. This work establishes a unified framework for the geometric analysis of both pure and mixed states, thereby deepening our understanding of the geometric properties of quantum states.
In this paper, we extend the quantum geometric tensor for parameter-dependent curved spaces to higher dimensions, and introduce an equivalent definition that generalizes the Zanardi, et al, formulation of the tensor. The parameter-dependent metric modifies the behavior of both the quantum metric tensor and Berry curvature in a purely geometric way. Our focus is on understanding the distinctions in higher dimensions that emerge when using the generalized tensor compared to the conventional one. Through a comparative analysis, illustrated with examples in two dimensions, we highlight unique quantum geometric properties for both the quantum metric tensor and the Berry curvature. Additionally, we explore differences between analytical and perturbative approaches in solving the problems.
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