Classifying and measuring the geometry of the quantum ground state manifold

Abstract: From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics many physical processes depend on the Berry curvature. However, recent advances in quantum information theory have highlighted the role of its symmetric counterpart, the quantum metric tensor. In this paper, we perform a detailed analysis of the ground state Riemannian geometry induced by the metric tensor, using the quantum XY chain in a transverse field as our primary example. We focus on a particular geometric invariant -- the Gaussian curvature -- and show how both integrals of the curvature within a given phase and singularities of the curvature near phase transitions are protected by critical scaling theory. For cases where the curvature is integrable, we show that the integrated curvature provides a new geometric invariant, which like the Chern number characterizes individual phases of matter. For cases where the curvature is singular, we classify three types -- integrable, conical, and curvature singularities -- and detail situations where each type of singularity should arise. Finally, to connect this abstract geometry to experiment, we discuss three different methods for measuring the metric tensor, namely via integrating a properly weighted noise spectral function and by using leading order responses of the work distribution to ramps and quenches in quantum many-body systems.

• The paper presents a detailed classification of quantum geometric singularities in the XY model using the quantum metric tensor and Gaussian curvature.
• It introduces experimental protocols based on non-adiabatic responses and noise spectral functions to measure quantum ground state properties.
• The findings highlight the universality and topological invariance of Gaussian curvature, providing robust tools for classifying phase transitions.

Classifying and Measuring the Geometry of the Quantum Ground State Manifold

Introduction

The paper "Classifying and measuring the geometry of the quantum ground state manifold" focuses on the quantum geometric tensor, with particular attention to its symmetric part, the quantum metric tensor. The study primarily utilizes the quantum XY chain in a transverse field to explore this geometry. The authors perform a detailed analysis of the Riemannian geometry of the ground state manifold, emphasizing the Gaussian curvature, its critical behavior near phase transitions, and its role as a geometric invariant.

Quantum Metric Tensor and Gaussian Curvature

The paper distinguishes itself by not only considering the antisymmetric Berry curvature but also exploring its symmetric counterpart—the quantum metric tensor. This tensor captures the fidelity susceptibility scaling near quantum phase transitions, providing a rich perspective on many-body ground states. Of particular interest is the Gaussian curvature, calculated from the metric tensor, which serves as a critical tool for characterizing topological properties of the quantum manifold.

Implementation of the XY Model Analysis

By examining the spin-1/2 XY chain, the authors illustrate the geometry induced by the quantum metric tensor. This model reveals a rich phase diagram characterized by paramagnetic and ferromagnetic phases. The study conducts a thorough classification of curvature singularities across phase transitions, identifying integrable, conical, and non-integrable singularities. The Gaussian curvature is shown to be protected against perturbations by critical scaling theory, highlighting its topological invariance within each phase.

Measurement Proposals

The authors propose several methodologies for empirically measuring the components of the quantum metric tensor. These techniques include mapping the metric tensor to measurable quantities such as noise spectral functions and examining non-adiabatic responses in quantum systems. They propose specific protocols using ramps, quenches, and work distribution analyses to extract the quantum metric in experimental settings like cold atoms or mesoscopic systems.

Singularities and Geometric Invariants

A major contribution of the paper is the classification of geometric singularities, which occur at the transition points of the model. Integrable singularities are characterized by finite curvatures that show jumps across transitions. Non-integrable or conical singularities, conversely, result from varying scaling dimensions of orthogonal directions and lead to non-integer Euler characteristics. This geometric framework provides a foundational understanding of singularities within quantum phase transitions.

Two-Dimensional Cuts and Visualization

The authors employ two-dimensional cuts through parameter space to visualize the geometry of the quantum manifold. By reconstructing three-dimensional equivalent surfaces, they provide an intuitive picture of the geometric properties of each phase. Such visualizations assert that the Gaussian curvature and Euler characteristic effectively classify quantum phases and transitions, offering robust tools for topological classifications.

Universality and Robustness

The robustness of the bulk Euler integral is demonstrated through critical scaling, emphasizing its universality across models within the same class of phase transitions. This universality suggests that the geometric characteristics identified are intrinsic features of quantum critical phenomena, largely independent of model-specific parameters.

Conclusions

The paper presents a comprehensive framework for understanding and measuring the quantum geometry of many-body ground states. By linking the quantum metric tensor to experimentally measurable quantities and providing geometric classifications of phase transitions, the study advances both theoretical insights and experimental methodologies in quantum physics. The findings emphasize the profound role of geometry in characterizing quantum phases, offering new avenues for exploring topological properties in diverse quantum systems.