Phase Diagram of the Attractive Kane-Mele-Hubbard Model at Half Filling

Zlatko Koinov

doi:10.3390/atoms8030058

Abstract

Motivated by recent developments in the experimental study of ultracold atoms in graphene-like honeycomb optical lattices, we investigate superconductivity of the attractive Kane-Mele-Habbard (KMH) model with the next-nearest-neighbor (NNN) hoping at half filling. The mean-field approximation is used to study the phase diagram which interpolates the trivial and the non-trivial topological states. It is shown that: (a) when the NNN hoping is taken into account, one has to introduce two mean-field gap equations for the two sublattices, instead of a single gap when the NNN hopping is neglected, and (b) in the non-trivial topological region the phase diagram with the NNN hopping is significantly different compared to the phase diagram calculated previously, but without the NNN term. We also discuss the superconducting instability of the attractive KMH model that is driven by condensation of Cooperons.

Highlights

The present-day experiments with ultracold atoms in optical lattices allow for us to simulate both the Haldane’s model [1] and the situation [2,3] considered by Kane and Mele (KM)
The Haldane’s model [4,5] is a tight-binding representation of electron motion on a honeycomb lattice in the presence of a magnetic field, where vector potential has the full symmetry of the lattice and generates a magnetic field with zero total flux through the unit cell
We use the mean-field approximation to numerically calculate the phase diagram of the attractive KMH model with NNN hoping at half filling, which interpolates the trivial and the non-trivial topological states

Summary

Introduction

The present-day experiments with ultracold atoms in optical lattices allow for us to simulate both the Haldane’s model [1] and the situation [2,3] considered by Kane and Mele (KM). Because of the zero magnetic flux through each unit cell, the phase accumulated through a nearest neighbor hopping vanishes, whereas the phase accumulated through NNN hopping is nonzero This extra phase breaks the time-reversal symmetry. Our tight-binding Hamiltonian H = HKM + HNNN + HU includes the KM terms, as well as the NNN hopping and the onsite attractive Hubbard interaction, where. We apply the mean-field approximation to obtain the matrix elements of the single-particle Green’s functions of the KMH model with NNN hopping term at half filling. If the NNN term is taken into account, the matrix elements of the mean-field single-particle Green’s function in the momentum space, proportional to < Ψ Ak↓ Ψ A−k↑ >.

Single-Particle Dispersion in the Mean-Field Approximation

Discussion