Classifying and measuring geometry of a quantum ground state manifold

By M. Kolodrubetz, V. Gritsev, A. Polkovnikov. 2013

134 citations recorded by the source corpus.

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: via integrating a properly weighted noise spectral function or by using leading-order responses of the work distribution to ramps and quenches in quantum many-body systems.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3dbe17e6f18b93867bd33a2a863180940f5617b7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. Bernardo Spagnolo
2. Davide Valenti
3. Angelo Carollo

In Statistical and Nonlinear Physics

1. Davide Valenti
2. Bernardo Spagnolo
3. Angelo Carollo

Citations in the corpus

• Geometry of quantum phase transitions (2019, methodology): “The resulting integrals leads to analytical expressions which differ in the two regions |h| < 1 and |h| > 1 [159, 166],”
• Geometry and Response of Lindbladians (2015, unknown): “We highlight this with a specific example.”