Lattice model for the surface states of a topological insulator with applications to magnetic and exciton instabilities

TL;DR

This paper introduces a lattice model for the surface states of strong topological insulators that captures their unique Dirac fermion characteristics and allows for the study of magnetic and exciton instabilities due to interactions.

Contribution

A novel lattice model for a pair of STI surfaces with an odd number of Dirac states, enabling detailed analysis of surface phenomena and instabilities.

Findings

Model accurately describes surface Dirac states

Reveals magnetic instability driven by interactions

Shows exciton instability consistent with continuum models

Abstract

A surface of a strong topological insulator (STI) is characterized by an odd number of linearly dispersing gapless electronic surface states. It is well known that such a surface cannot be described by an effective two-dimensional lattice model (without breaking the time-reversal symmetry), which often hampers theoretical efforts to quantitatively understand some of the properties of such surfaces, including the effect of strong disorder, interactions and various symmetry-breaking instabilities. Here we formulate a lattice model that can be used to describe a {\em pair} of STI surfaces and has an odd number of Dirac fermion states with wavefunctions localized on each surface. The Hamiltonian consists of two planar tight-binding models with spin-orbit coupling, representing the two surfaces, weakly coupled by terms that remove the extra Dirac points from the low-energy spectrum. We illustrate the utility of this model by studying the magnetic and exciton instabilities of the STI surface state driven by short-range repulsive interactions and show that this leads to results that are consistent with calculations based on the continuum model as well as three-dimensional lattice models. We expect the model introduced in this work to be widely applicable to studies of surface phenomena in STIs.