Pseudo-Riemannian metric: a new perspective on the quantum realm

TL;DR

This paper introduces a pseudo-Riemannian quantum geometry framework that reveals new topological phases, such as the Pauli semimetal and Pauli Chern insulator, expanding the understanding of quantum materials beyond traditional Riemannian metrics.

Contribution

It proposes a novel pseudo-Riemannian quantum geometric tensor incorporating spin, leading to the discovery of new topological phases characterized by the Pauli Chern number.

Findings

Discovery of the Pauli semimetal phase in PT-symmetric systems.

Introduction of the Pauli Chern number for classifying topological insulators.

Identification of helical edge states in the Pauli Chern insulator.

Abstract

As a fundamental concept in condensed matter physics, quantum geometry within the Riemannian metric elucidates various exotic phenomena, including the Hall effects driven by Berry curvature and quantum metric. In this work, we propose novel quantum geometries within a pseudo-Riemannian framework to explore unique characteristic of quantum matter. By defining distinct distances on pseudo-Riemannian manifolds and incorporating spin degree of freedom, we introduce the Pauli quantum geometric tensor. The imaginary part of this tensor corresponds to the Pauli Berry curvature, leading to the discovery a novel quantum phase: Pauli semimetal in PT-symmetric systems. This phase, characterized by the topological Pauli Chern number, manifests as a two-dimensional Pauli Chern insulator with helical edge states. These topological phases, uniquely revealed by the Pauli-Riemannian metric, go beyond the familiar Riemannian metric, where Berry curvature vanishes due to PT-symmetry. Pauli Chern number can classify helical topological insulator with or without time reversal symmetry. Pseudo-Riemannian metrics offer new insights into quantum materials and extend the scope of quantum geometry.