Open agenda of the axion topological field theory of 3d time-reversal-invariant topological superconductors: beyond-weak-pairing validity, chiral-vortex realization and fermion-parity pumping, and measurable gravitational (thermal) response in real materials

Qi, Witten and Zhang (Phys. Rev. B 87, 134519 (2013)) derived an 'axion' topological field theory for three-dimensional time-reversal-invariant topological superconductors (symmetry class DIII), in which the superconducting phase fields $\theta_i$ of the individual Fermi surfaces (each carrying helicity Chern number $C_{1i}$) couple to the electromagnetic and gravitational Pontryagin densities, the bulk invariant being $N=\frac{1}{2}\sum_i C_{1i}e^{i\theta_i}$. The derivation is controlled only in the weak-pairing limit with weak inter-Fermi-surface Josephson coupling ($m^2\equiv 2J/\rho\ll|\Delta|^2$), and the paper left open: the theory's validity beyond this regime and under strong interactions; the confinement energetics and possible deconfinement of chiral vortex lines (binding one-dimensional chiral Majorana modes) and of the associated fermion-parity pumping at vortex crossings; the identification of a real bulk topological-superconductor material (then candidate: Cu-doped Bi2Se3); and the measurability of the gravitational (thermal) response. Later work has substantially narrowed the interaction question — the free-fermion integer classification reduces to $\mathbb{Z}_{16}$, and the axion framework itself reproduces this reduction — but no controlled beyond-weak-pairing theory, no realization of deconfined chiral vortex lines or fermion-parity pumping, no established bulk fully gapped 3d time-reversal-invariant topological superconductor, and no measured thermal/gravitational topological response is known. The problem — the source paper's own open agenda — remains open.

Background

Three-dimensional superconductors in symmetry class DIII — superconductors that preserve time-reversal symmetry ($\mathcal{T}^2=-1$ on electrons) and have no symmetries besides time reversal and the particle-hole redundancy of the Bogoliubov–de Gennes description (the standard mean-field framework of superconductivity, in which each quasiparticle excitation mixes electron and hole components) — can be topological: their fully gapped quasiparticle spectrum carries an integer invariant, and a nontrivial value forces gapless Majorana cones on any surface. (A Majorana fermion is a real superposition of an electron and a hole, i.e. its own antiparticle; Majorana modes bound inside the superconducting gap are the hallmark excitation of topological superconductors.) In the weak-pairing (BCS, Bardeen–Cooper–Schrieffer) limit the invariant is computed from the normal-state Fermi surfaces: each Fermi surface $i$ carries a first Chern number $C_{1i}$, the integer winding of its spin-helicity texture (the projection of the electron spin onto its direction of motion), and with the superconducting phase $\theta_i$ pinned by time reversal to $0$ or $\pi$ on each surface, the bulk index of the Qi–Witten–Zhang theory is

$$N=\frac{1}{2}\sum_i C_{1i}e^{i\theta_i},$$

which counts the net number of surface Majorana cones (time-reversal-related Fermi surfaces have opposite helicity Chern numbers and opposite pairing signs, so their contributions add to an integer).

Qi, Witten and Zhang derived the low-energy effective field theory of such superconductors by coupling the phase fields $\theta_i$ to the electromagnetic field. Its central term — called the axion coupling by analogy with axion electrodynamics of topological insulators, the superconducting phase playing the role of the axion — is

$$\mathcal{L}_{topo}=\frac{\sum_i C_{1i}\theta_i}{2\cdot 64\pi}\,\epsilon^{\mu\nu\sigma\tau}F_{\mu\nu}F_{\sigma\tau},$$

with an analogous coupling of $\sum_i C_{1i}\theta_i$ to the gravitational Pontryagin density (the curvature invariant $\epsilon RR$); this formalizes earlier proposals that 3d DIII superconductors possess a gravitational $\theta$-term with coefficient $0$ or $\pi$ mod $2\pi$, entailing a half-quantized surface thermal Hall response (the heat-current analog of the quantum Hall effect). The theory's physical content includes: (a) a vortex line of a single phase field $\theta_i$ — a chiral vortex — binds a one-dimensional chiral Majorana mode, a manifestation of the gravitational anomaly; (b) when chiral vortex lines cross or link, quasiparticle energy levels cross and the ground-state fermion parity (the conserved fermion number modulo 2) flips — an analog of the Witten $SU(2)$ anomaly (a classic instance of an 't Hooft anomaly, i.e. a symmetry signature of the low-energy theory that any microscopic realization must reproduce), and the proposed 'fermion-parity pump' readout; (c) in a two-Fermi-surface bulk, the inter-surface Josephson coupling $J$ (the energy cost of a relative phase between the two condensates) confines vortices of the individual $\theta_i$, but at an interface to a conventional $s$-wave superconductor a chiral vortex deconfines, and an $s$-wave Josephson trijunction on the superconductor surface with interface phases $\pm\pi/2$ was proposed as the geometry for creating and moving chiral vortices.

The derivation is controlled only in a restricted regime: energies below the superconducting gap, weak pairing, Josephson-mode masses satisfying $m^2\equiv 2J/\rho\ll|\Delta_{L,R}|^2$ (with $\rho$ the phase stiffness and $\Delta_{L,R}$ the superconducting pairing gaps on the two Fermi surfaces), and, for convenience of the derivation, a term lifting degeneracies between Fermi surfaces. The paper is explicitly framed as the response-theory answer to the question of whether the integer invariant of 3d time-reversal-invariant topological superconductors survives electron interactions — a question sharpened by the $\mathbb{Z}_2$ ambiguity of the gravitational $\theta$-coefficient — and it left open whether its predictions can be realized in any material; at the time, Cu-doped Bi2Se3, supported by a point-contact tunneling experiment (Sasaki et al. 2011), was the candidate.

Later work reshaped parts of this agenda. Wang and Senthil (Phys. Rev. B 89, 195124 (2014)) showed that interactions reduce the free-fermion $\mathbb{Z}$ classification of 3d DIII superconductors to $\mathbb{Z}_{16}$, with the $\nu=16$ state connectable to the trivial phase; Gu and Qi (arXiv:1512.04919) showed this reduction is internal to the axion framework itself (the vortex-crossing anomaly pumps charge $\nu/8$ in the $(4+1)$-dimensional parent theory, and charge pumping by 2 is trivial in a superconductor). On materials, Knight-shift measurements (the nuclear-magnetic-resonance probe of the spin susceptibility in the superconducting state) found spin-rotation symmetry breaking in CuxBi2Se3 (Matano et al. 2016), but thermodynamics revealed a nematic, two-component order parameter with point nodes and a small gap (Yonezawa et al. 2017), so it is not an established bulk, fully gapped DIII superconductor; interest has partly moved to spin-triplet candidates such as UTe2 (Ran et al. 2019), whose pairing structure remains debated.

Problem Statement

Following the source paper's own agenda, the problem is to establish the axion topological field theory of three-dimensional time-reversal-invariant (class DIII) topological superconductors as a validated description of real materials. This jointly requires: (1) a controlled description — a microscopic effective theory or equivalent derivation — of the axion (phase-field) electromagnetic and gravitational response outside the weak-pairing/weak-Josephson regime, i.e. when the relative-phase (Josephson) mode mass $m\equiv\sqrt{2J/\rho}$ is no longer small compared with the superconducting quasiparticle gap $\Delta$ (equivalently, $m^2\equiv 2J/\rho$ is comparable to $|\Delta|^2$), when the Fermi surfaces are degenerate, and for strongly interacting superconductors; (2) the realization and detection of the theory's distinctive vortex signatures in a concrete material — deconfined chiral vortex lines, the quasiparticle level crossings when chiral vortices cross or link, and the associated ground-state fermion-parity pumping; and (3) a determination of whether the gravitational Pontryagin coupling yields an accessible experimental signature, such as a quantized thermal response, in real materials. Items (2) and (3) presuppose a bulk, fully gapped three-dimensional time-reversal-invariant topological superconductor, of which no example is currently established. The common objective is to establish the axion topological field theory of a real topological superconductor through these linked results.

References

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