Topology in time-reversal symmetric crystals

The discovery of topological insulators has reformed modern materials science, promising to be a platform for tabletop relativistic physics, electronic transport without scattering, and stable quantum computation. Topological invariants are used to label distinct types of topological insulators. But, it is not generally known how many or which invariants can exist in any given crystalline material. Using a new and efficient counting algorithm, we study the topological invariants that arise in time-reversal symmetric crystals. This results in a unified picture that explains the relations between all known topological invariants in these systems. It also predicts new topological phases and one entirely new topological invariant. We present explicitly the classification of all two-dimensional crystalline fermionic materials, and give a straightforward procedure for finding the analogous result in any three-dimensional structure. Our study represents a single, intuitive physical picture applicable to all topological invariants in real materials, with crystal symmetries.

Figure 1

(a) The typical band structure of a Kramers' pair close to a high-symmetry point. Two bands related by the time-reversal operation necessarily come together into a degenerate Kramers pair at the time-reversal invariant momentum in the center. Also shown schematically is a band inversion which brings together states at points away from the high-symmetry momentum. This results in the formation of vortices in the Berry connection, indicated here by yellow and orange arrows. (b) A more schematic representation of two bands containing states $|\psi \rangle$ and $T|\psi \rangle$, which form Kramers pairs at two time-reversal invariant momenta, chosen here to be $\mathrm{\Gamma }$ and $M$. (c) Vortices in the Berry connection, depicted by $+$ and $-$, can be moved throughout the Brillouin zone without annihilating. The color indicates the band to which the vortices belong. (d) An even number of vortices can be created by a band inversion within a set of states related by TRS. (e) Vortices can hop between partner bands using a band inversion to create two vortex-antivortex pairs.

Figure 2

(a) Topologically nontrivial vortex configurations with $p4$ symmetry in class AII. (b) A band inversion involving a second, trivial, Kramers pair connects the configurations with a single vortex at $\mathrm{\Gamma }$ to one with an ${\mathrm{FKM}}_2$-trivial band with vortices at both $\mathrm{\Gamma }$ and $M$, and one ${\mathrm{FKM}}_2$-nontrivial band with only a vortex at $M$. Notice, however, that the final situation cannot be deformed into a band with a single vortex at $M$ and no vortices in the second band. That would require a change in the value of the new torsion invariant described in Sec. 6. (c) Vortex configuration with $p4$ symmetry in class AII in which the ${\mathrm{FKM}}_2$ invariant is trivial, but the new invariant of Sec. 6 is not.

Figure 3

(a) Sketch of the Berry connection projected onto the highest-energy state within a Kramers pair. A vortex-antivortex pair can arise in two topologically distinct ways within a Berry connection vector field that is continuous on the Brillouin zone torus. (b) Sketch of vortex lines extending across the bulk of a three-dimensional Brillouin zone with trivial ${\mathrm{FKM}}_3$ invariant. (c) Sketch of a vortex line extending into the bulk of a three-dimensional Brillouin zone, and closing onto itself. This situation is described by a nontrivial ${\mathrm{FKM}}_3$ invariant.

Figure 4

The BZ divided into two patches, $A$ and $B$, with a transition function $U\left(\varphi \right)$ in-between. The coordinate $\varphi$ parametrizes the overlapping circle between $A$ and $B$.