Wave-function geometry of band crossing points in two-dimensions

TL;DR

This paper explores the wave-function geometry of two-dimensional semimetals with band crossing points, revealing how the quantum metric governs Berry phases and linking geometric properties to topological features.

Contribution

It uncovers the relationship between quantum metric and Berry phase in 2D semimetals, providing new insights into the geometric origin of band crossing point properties.

Findings

Berry phase of linear BCPs is determined by quantum metric or maximum quantum distance.

Quadratic BCPs can have arbitrary Berry phases between 0 and 2π.

In lattice systems, quadratic BCPs with zero Berry curvature are paired to ensure zero total Berry phase.

Abstract

Geometry of the wave function is a central pillar of modern solid state physics. In this work, we unveil the wave-function geometry of two-dimensional semimetals with band crossing points (BCPs). We show that the Berry phase of BCPs are governed by the quantum metric describing the infinitesimal distance between quantum states. For generic linear BCPs, we show that the corresponding Berry phase is determined either by an angular integral of the quantum metric, or equivalently, by the maximum quantum distance of Bloch states. This naturally explains the origin of the $\pi$-Berry phase of a linear BCP. In the case of quadratic BCPs, the Berry phase can take an arbitrary value between 0 and $2\pi$. We find simple relations between the Berry phase, maximum quantum distance, and the quantum metric in two cases: (i) when one of the two crossing bands is flat; (ii) when the system has rotation and/or time-reversal symmetries. To demonstrate the implication of the continuum model analysis in lattice systems, we study tight-binding Hamiltonians describing quadratic BCPs. We show that, when the Berry curvature is absent, a quadratic BCP with an arbitrary Berry phase always accompanies another quadratic BCP so that the total Berry phase of the periodic system becomes zero. This work demonstrates that the quantum metric plays a critical role in understanding the geometric properties of topological semimetals.