Chern-Simons theory and Wilson loops in the Brillouin zone

Berry connection is conventionally defined as a static gauge field in the Brillouin zone. Here we show that for three-dimensional (3D) time-reversal invariant superconductors, a generalized Berry gauge field behaves as a fluctuating field of a Chern-Simons gauge theory. The gapless nodal lines in the momentum space play the role of Wilson loop observables, while their linking and knot invariants modify the gravitational theta angle. This angle induces a topological gravitomagnetoelectric effect where a temperature gradient induces a rotational energy flow. We also show how topological strings may be realized in the six-dimensional phase space, where the physical space defects play the role of topological D-branes.

Figure 1

(a) Illustration of local crossing of two nodal lines, which leads to a $\pi /2$ change in the $\theta \left[a\right]$ angle. (b) This shows how a nodal line ${\mathcal{L}}_b$ can cross itself, after which its self-linking number changes by 2. (c) The generic skein relation for knot polynomials, which gives the rules for the local surgery of a knot configuration.

Figure 2

(a) Two linked nodal lines in the same band can be realized on a torus Fermi surface. (b) A nodal line can be created at point $A$ and annihilated at point $B$, whose trajectory is a closed surface $\mathcal{M}$. (c),(d) The sign of pairing ${\mathrm{\Delta }}_{\alpha }$ on a Fermi surface of a gapped TRI superconductor can be reversed by creating nodal lines and making them sweep around the Fermi surface, which leads to a phase transition between topological superconductors if the Fermi-surface Chern number $C_1\left(\alpha \right)$ is nonzero.