Valley Spin Sum Rule for Dirac Fermions: Topological Argument

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.

5 pages, 1 figure, some comments are added/revised, accepted for publication in J. Phys. Soc. Jpn

References in corpus (4)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Topological Insulators with Inversion Symmetry
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topological aspects of graphene: Dirac fermions and the bulk-edge correspondence in magnetic fields