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

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.

arXiv comment7+9 pages, 4+1 figures, published version

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• Extracting the Quantum Geometric Tensor of an Optical Raman Lattice by Bloch State Tomography
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• Direct Geometric Probe of Singularities in Band Structure
• Flat bands with band crossings enforced by symmetry representation
• Nonlinear photoconductivities and quantum geometry of chiral multifold fermions
• Topological properties of a non-Hermitian quasi-1D chain with a flat band
• Thermodynamic principle for quantum metrology
• Quantum metric on the Brillouin Zone in correlated electron systems and its relation to topology for Chern insulators
• Band geometry from position-momentum duality at topological band crossings
• Revisiting the magnetic responses of bilayer graphene from the perspective of the quantum distance
• Symmetry-enforced double Weyl points, multiband quantum geometry, and singular flat bands of doping-induced states at the Fermi level
• Pinch-point spectral singularity from the interference of topological loop states
• Pinch points and half moons encode Berry curvature
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Spinor ice correlation in flat-band electronic states on kagome and pyrochlore lattices with spin-orbit coupling