Quantum geometric tensor in PT -symmetric quantum mechanics

Abstract

A series of geometric concepts are formulated for 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 system's parameter manifold. This results in a unified conceptual framework to understand and explore physical properties of PT -symmetric systems from a geometric perspective. To illustrate the usefulness of the extended QGT, we show how its real part, i.e., the metric tensor, can be exploited as a tool to detect quantum phase transitions as well as spontaneous PT -symmetry breaking in PT -symmetric systems.Given a family of Hamiltonians depending smoothly on a manifold of parameters, e.g., external field strengths, a problem of great importance is how to characterize geometric aspects of the eigenstates of the Hamiltonians. In standard quantum mechanics (QM) where Hamiltonians are Hermitian operators, the solution to the problem is to use the quantum geometric tensor (QGT), of which the imaginary part determines the Berry curvature [1] and the real part induces a Riemannian metric tensor [2] on the manifold. The QGT has played an indispensable role in various frontier topics of quantum computation, quantum information, and condensed-matter physics [3,4], where both its imaginary and real parts serve as versatile tools.Since the pioneering work of Bender and Boettcher [5], however, it has been realized that Hamiltonians can be non-Hermitian but still possess real spectra due to paritytime reversal (PT ) symmetry. This has led to a complex extension of standard QM called PT -symmetric QM (PT QM) [6,7], with one main conceptual advance being the introduction of a nontrivial inner-product metric to define its Hilbert space. Over the past decade, PTsymmetric systems have been experimentally realized by spatially engineering gain-loss structures [8], thus further boosting PT QM as an important research area. A current stream of development is towards extensive studies of physical properties, especially topological properties, of PT -symmetric systems [9][10][11][12][13][14][15][16][17].Unfortunately, a systematic geometric concept like the QGT is still elusive in PT QM. Without such a concept, it is difficult to extend geometric understandings, such as those of quantum phase transitions (QPTs) [18][19][20][21][22][23][24][25], to PT -symmetric systems. On the other hand, the interplay between the nontrivial inner-product metric and geometric aspects of PT QM is rarely understood to date. In particular, in the course of varying parameters of a PT -symmetric system, the inner-product metric varies as well [26]. How this feature impacts on previous geometric perspectives in standard QM (such as curvature and metric tensor) remains unknown.In this Letter, we report the finding of an extended QGT in PT QM, of which the imaginary part gives a Berry curvature whereas the real part induces a metric tensor on system's parameter manifold. Thi...

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“…It should be mentioned that there are other proposals of making use of geometric entities to study QPTs, such as the ones based on the fidelity [63][64][65][66][67][68], the ones based on the Berry phase [69,70], the ones based on the metric tensor [71,72], and the one based on the quantum geometric tensor [36] (see the review paper [73] for more information). Generally speaking, all of these proposals are unable to provide a geometrically visible characterization of QPTs as did in our proposal, for the obvious reason that the geometric entities adopted there are defined in a mathematically abstract way.…”