Cooperon condensation and intra-valley pairing states in honeycomb Dirac systems

TL;DR

This paper investigates intra-valley spin-triplet pairing in honeycomb Dirac systems, revealing a helical valley-triplet state with pair-density wave characteristics driven by Cooperon condensation and intrinsic spin-orbit coupling.

Contribution

It introduces a novel intra-valley spin-triplet pairing mechanism in the Kane-Mele model, combining Cooperon condensation and mean-field analysis, highlighting a helical valley-triplet state with spatial modulation.

Findings

Intra-valley spin-triplet pairing arises from NN attraction and spin-orbit coupling.

Cooperons with antiparallel spins condense at K and K' points.

The pairing state exhibits a pair-density wave with a p-Kekulé pattern.

Abstract

Motivated by recent developments in the experimental study of superconducting graphene and transition metal dichalcogenides, we investigate superconductivity of the Kane-Mele (KM) model with short-range attractive interactions on the two-dimensional honeycomb lattice. We show that intra-valley spin-triplet pairing arises from nearest-neighbor (NN) attractive interaction and the intrinsic spin-orbit coupling. We demonstrate this in two independent approaches: We study superconducting instability driven by condensation of Cooperons, which are in-gap bound states of two conduction electrons, within the $T$-matrix approximation and also study the superconducting ground state within the mean-field theory. We find that Cooperons with antiparallel spins condense at the $K$ and $K'$ points. This leads to the emergence of an intra-valley spin-triplet pairing state belonging to the irreducible representation A$_1$ of the point group $C_{6v}$. The fact that this pairing state has opposite chirality for $K$ and $K'$ identifies this state as a "helical" valley-triplet state, the valley-analog to the $^3$He-B phase in two dimension. Because of the finite center of mass momentum of Cooper pairs, the pair amplitude in NN bonds exhibits spatial modulation on the length scale of lattice constant, such that this pairing state may be viewed as a pair-density wave state. We find that the pair amplitude spontaneously breaks the translational symmetry and exhibits a $p$-Kekul\'e pattern. We also discuss the selection rule for pairing states focusing the characteristic band structure of the KM model and the Berry phase effects to the emergence of the intra-valley pairing state.