Abstract
We study a tight-binding model on the honeycomb lattice of chiral $d$-wave superconductivity that breaks time-reversal symmetry. Due to its nontrivial sublattice structure, we show that it is possible to construct a gauge-invariant time-reversal-odd bilinear of the pairing potential. The existence of this bilinear reflects the sublattice polarization of the pairing state. We show that it generates persistent loop current correlations around each lattice site and opens a topological mass gap at the Dirac points, resembling Haldane's model of the anomalous quantum Hall effect. In addition to the usual chiral $d$-wave edge states, there also exist electron-like edge resonances due to the topological mass gap. We show that the presence of loop-current correlations directly leads to a nonzero intrinsic ac Hall conductivity, which produces the polar Kerr effect without an external magnetic field. Similar results also hold for the nearest-neighbor chiral $p$-wave pairing. We briefly discuss the relevance of our results to superconductivity in twisted bilayer graphene.
Highlights
Chiral superconductors, which possess order parameters that break time-reversal symmetry, are currently the subject of much attention due to their nontrivial topological properties [1,2]
We show that the loop current correlations imply a nonzero anomalous Hall conductivity, connecting the polar Kerr effect in superconductors with the time-reversal-odd bilinear (TROB) product of the pairing potentials
In this paper we have examined the appearance of the polar Kerr effect in a minimal model of time-reversal symmetry-breaking chiral d-wave superconductivity on the honeycomb lattice
Summary
Chiral superconductors, which possess order parameters that break time-reversal symmetry, are currently the subject of much attention due to their nontrivial topological properties [1,2]. We discuss the relationship between our model and these proposals in more detail near the end of the paper Using this minimal model as an example, we show how to construct a gauge-invariant time-reversal-odd term by taking the product of the pairing potential and its Hermitian conjugate. The bilinear arises from the varying participation of the two sublattices in the pairing across the Brillouin zone and describes spontaneous breaking of the discrete Z2 time-reversal symmetry The presence of this term results in the opening of a topological mass gap at the Dirac points and the emergence of persistent loop current correlations, in a striking analogy to Haldane’s model of the anomalous Hall insulator [63]. We show that the loop current correlations imply a nonzero anomalous Hall conductivity, connecting the polar Kerr effect in superconductors with the time-reversal-odd bilinear (TROB) product of the pairing potentials. The high-frequency small-gap limit of the ac Hall conductivity is derived in Appendix D
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