Axion topological field theory of topological superconductors (1206.1407v2)

Abstract: Topological superconductors are gapped superconductors with gapless and topologically robust quasiparticles propagating on the boundary. In this paper, we present a topological field theory description of three-dimensional time-reversal invariant topological superconductors. In our theory the topological superconductor is characterized by a topological coupling between the electromagnetic field and the superconducting phase fluctuation, which has the same form as the coupling of "axions" with an Abelian gauge field. As a physical consequence of our theory, we predict the level crossing induced by the crossing of special "chiral" vortex lines, which can be realized by considering s-wave superconductors in proximity with the topological superconductor. Our theory can also be generalized to the coupling with a gravitational field.

Axion Topological Field Theory of Topological Superconductors

In this paper, Qi, Witten, and Zhang present a meaningful theoretical framework for understanding three-dimensional (3D) time-reversal invariant (TRI) topological superconductors (TSCs). Such superconductors are characterized by a bulk energy gap and the presence of gapless, topologically protected quasiparticle states on their boundaries. The principal innovation in this work is the development of a topological field theory for these materials that connects electromagnetic fields with the fluctuations of the superconducting phase. The authors characterize topological superconductors via a coupling analogous to the axion coupling with Abelian gauge fields from high-energy physics.

Summary and Numerical Results

The paper explores the properties of 3D TSCs using a field theory approach that includes both electromagnetic fields and superconducting phase fluctuations. The core result of this theory is a topological term akin to an axion field theory, described as:

$$L_{\text{topo}} = \frac{θ}{64π^2} i\epsilon^{\mu\nu\sigma\tau}F_{\mu\nu}F_{\sigma\tau},$$

where $\theta$ is dynamical and represents the superconducting phase on each Fermi surface, and $F_{\mu\nu}$ denotes the electromagnetic field strength tensor. The authors highlight that this topological term induces anomalies and level crossings when special chiral vortex lines intersect, leading to changes in ground state fermion number parity.

One of the crucial results presented is the topological invariant description based on the winding numbers derived from the Chern numbers of Fermi surfaces, leading to a classification aligned with the $\mathbb{Z}_2$ invariant in TRI TSCs, as opposed to the integer classification in non-interacting systems. They further discuss the implications of such an invariant: the presence of an axion field-like behavior that influences surface states and responses to external fields.

Implications and Future Directions

The authors propose that the theoretical framework they have developed has significant implications for understanding the topological magneto-electric effect and other observable consequences in materials proposed to be topological superconductors. It also opens a pathway for experimental verification, primarily through the realization of chiral vortex lines via proximity effects on interfaces between trivial and topological superconductors. Experiments involving tunneling and Josephson junctions could probe these phenomena further.

In terms of future research directions, the presented field theory serves as a foundational framework for exploring gravitational effects and charge-parity (CP) invariant models where the superconducting phase encompasses gravitational axionic coupling. Such extensions can potentially unravel novel topological states and offer deeper insights into the behavior of quasiparticles in exotic superconducting materials.

The theoretical advancements made in this work thus represent a solid step toward a unified description of superconductors in higher dimensions, influencing future experimental investigations into materials like copper-doped bismuth selenide. The development and prediction of new topological phenomena founded on fermion parity and coupling dynamics enrich the theoretical landscape and beckon empirical validation.