Topological semimetals with Riemann surface states

TL;DR

This paper links the surface states of topological semimetals to Riemann surfaces generated by holomorphic functions, predicting new types of semimetals with complex surface state geometries and symmetry-protected degeneracies.

Contribution

It introduces a novel geometric framework connecting Riemann surfaces to topological semimetal surface states, predicting new semimetal types with complex Riemann surface structures.

Findings

Surface states in Weyl semimetals are represented by Riemann surfaces.

Predicted new topological semimetals with double- and quad-helicoid Riemann surface states.

Identified symmetry-protected degeneracies along high-symmetry lines.

Abstract

Riemann surfaces are geometric constructions in complex analysis that may represent multi-valued holomorphic functions using multiple sheets of the complex plane. We show that the energy dispersion of surface states in topological semimetals can be represented by Riemann surfaces generated by holomorphic functions in the two-dimensional momentum space, whose constant height contours correspond to Fermi arcs. This correspondence is demonstrated in the recently discovered Weyl semimetals and leads us to predict new types of topological semimetals, whose surface states are represented by double- and quad-helicoid Riemann surfaces. The intersection of multiple helicoids, or the branch cut of the generating function, appears on high-symmetry lines in the surface Brillouin zone, where surface states are guaranteed to be doubly degenerate by a glide reflection symmetry. We predict the heterostructure superlattice [(SrIrO$_3$)$_2$(CaIrO$_3$)$_2$] to be a topological semimetal with double-helicoid Riemann surface states.