Hopf Semimetals

TL;DR

Hopf semimetals are four-dimensional topological phases featuring nodal lines, surface states with Hopf flux, and unique gapless surface and corner states, constructed via homotopy from the Brillouin zone to the sphere.

Contribution

This work introduces a new class of four-dimensional topological semimetals called Hopf semimetals, characterized by their unique topological and surface state properties.

Findings

Host nodal lines with Hopf flux in 4D Brillouin zone

Surface states include Fermi arcs, drumhead states, and Fermi surfaces

Presence of gapless corner states at intersections

Abstract

We construct two-band topological semimetals in four dimensions using the unstable homotopy of maps from the three-torus $T^3$ (Brillouin zone of a 3D crystal) to the two-sphere $S^2$. Dubbed ``Hopf semimetals'', these gapless phases generically host nodal lines, with a surface enclosing such a nodal line in the four-dimensional Brillouin zone carrying a Hopf flux. These semimetals show a unique class of surface states: while some three-dimensional surfaces host gapless Fermi-arc states {\em and} drumhead states, other surfaces have gapless Fermi surfaces. Gapless two-dimensional corner states are also present at the intersection of three-dimensional surfaces.