Valley Spin Sum Rule for Dirac Fermions: Topological Argument

TL;DR

This paper establishes a topological sum rule for valley spins in 2D bipartite lattices with Dirac fermions, revealing that the total valley spin sum vanishes regardless of symmetry constraints.

Contribution

It introduces a topological argument linking vortex and meron numbers to valley spins, deriving a sum rule applicable even without time-reversal or parity symmetry.

Findings

Total valley spin sum is zero in the system.

Topological numbers for vortices and merons are equivalent.

Similarity to Nielsen-Ninomiya no-go theorem in odd dimensions.

Abstract

We consider a two-dimensional bipartite lattice system. In such a system, the Bloch band spectrum can have some valley points, around which Dirac fermions appear as the low-energy excitations. Each valley point has a valley spin +1 or -1. In such a system, there are two topological numbers counting vortices and merons in the Brillouin zone, respectively. These numbers are equivalent, and this fact leads to a sum rule which states that the total sum of the valley spins is absent even in a system without time-reversal and parity symmetries. We can see some similarity between the valley spin and chirality in the Nielsen-Ninomiya no-go theorem in odd-spatial dimensions.