Quantum Geometric Tensor in $\mathcal{PT}$-Symmetric Quantum Mechanics

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 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, i.e., 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.

• The paper introduces an extended Quantum Geometric Tensor that applies to non-Hermitian, PT-symmetric systems.
• It employs Berry curvature and fidelity susceptibility to derive a metric tensor, pivotal for detecting quantum phase transitions.
• Numerical simulations on models like the dimerized XY spin chain validate the framework and highlight critical behaviors.

Quantum Geometric Tensor in $\mathcal{PT}$-Symmetric Quantum Mechanics

Introduction

The paper "Quantum Geometric Tensor in $\mathcal{PT}$-Symmetric Quantum Mechanics" explores an innovative approach to understanding the geometric aspects of $\mathcal{PT}$-symmetric quantum systems. By extending the concept of the Quantum Geometric Tensor (QGT) from Hermitian to non-Hermitian systems, the authors provide a framework that unifies various geometric entities into a single construct applicable to $\mathcal{PT}$-symmetric quantum mechanics (QM). This work builds upon the idea that non-Hermitian Hamiltonians can adhere to parity-time ($\mathcal{PT}$) symmetry conditions, offering real spectra despite their non-Hermitian nature.

Quantum Geometric Tensor in $\mathcal{PT}$-symmetric QM

In classical quantum mechanics, geometric concepts such as the Berry curvature and the quantum metric arise from the QGT, which is inherently linked to the Hermitian nature of the system's Hamiltonian. The novel contribution of this work is the formulation of an extended QGT applicable to $\mathcal{PT}$-symmetric systems, where Hamiltonians are allowed to be non-Hermitian. The extended QGT maintains the separation into real and imaginary components: the real part provides a metric tensor on the parameter manifold, while the imaginary part describes a Berry curvature.

Implementation of Extended QGT

Here is a procedural outline for implementing the extended QGT in $\mathcal{PT}$-symmetric QM:

1. Hamiltonian Preparation: Define a family of $\mathcal{PT}$-symmetric Hamiltonians $H(\lambda)$ parametrized by $\lambda$. These Hamiltonians must satisfy specific symmetry conditions, i.e., $$W(\lambda)H(\lambda) = H^\dagger(\lambda)W(\lambda)$$ for a suitable metric operator $W(\lambda)$.
2. Quantum Evolution Consideration: Consider the parameter manifold $M$ over which $H(\lambda)$ is defined. Classify the regions of this manifold based on the symmetry—unbroken ($\mathcal{PT}$ symmetric) and broken ($\mathcal{PT}$ broken) regions.
3. Geometric Phase and Berry Curvature Calculation: Use the adiabatic evolution of eigenstates in the unbroken regime to define the Berry connection and compute the Berry curvature, integral to identifying geometric phases in parameter space.
4. Metric Tensor Derivation: Utilize the fidelity susceptibility—as derived from the QGT—to determine the metric tensor, facilitating the analysis of quantum criticality and phase transitions within $\mathcal{PT}$ symmetric regions.
5. Numerical Analysis: Implement numerical simulations to explore critical points using the metric tensor, specifically focusing on its singularities which indicate phase transitions or spontaneous symmetry breakings.

Applications and Implications

The utility of the extended QGT lies in its ability to detect quantum critical points where phase transitions occur, even in complex non-Hermitian systems. Its applications span quantum information, where understanding global geometric phases can lead to more robust quantum computation protocols, and condensed matter physics, where nontrivial topological states are explored.

Example: Dimerized $XY$ Model

To demonstrate the practical use of the extended QGT, consider the example of a dimerized $XY$ spin chain in an alternating complex magnetic field. Here, the metric tensor derived from the QGT helps identify critical points in the system's parameter space. By examining the behavior of the metric tensor across these points, one can pinpoint quantum phase transitions, providing critical insights into the system's physical properties.

Conclusion

Expanding the QGT to $\mathcal{PT}$-symmetric systems represents a significant step in the geometric analysis of non-Hermitian quantum mechanics. By providing both theoretical groundwork and practical tools for implementation, this work bridges the gap between abstract geometric concepts and their tangible applications in understanding and manipulating quantum systems. The prospect of generalizing these methods to broader classes of non-Hermitian systems holds promise for future research and technological advancements in quantum sciences.