Interacting Surface States of Three-Dimensional Topological Insulators

Abstract

We numerically investigate the surface states of a strong topological insulator in the presence of strong electron-electron interactions. We choose a spherical topological insulator geometry to make the surface amenable to a finite size analysis. The single-particle problem maps to that of Landau orbitals on the sphere with a magnetic monopole at the center that has unit strength and opposite sign for electrons with opposite spin. Assuming density-density contact interactions, we find superconducting and anomalous (quantum) Hall phases for attractive and repulsive interactions, respectively, as well as chiral fermion and chiral Majorana fermion boundary modes between different phases. Our setup is preeminently adapted to the search for topologically ordered surface terminations that could be microscopically stabilized by tailored surface interaction profiles.Introduction.-Three-dimensional topological insulators (3DTIs) [1][2][3][4][5][6], since their prediction in 2007, realize a quantum state of matter which triggered enormous interest in condensed matter physics, and have been subsequently discovered in various material classes [7][8][9][10][11].When viewed as a symmetry-protected topological phase, 3DTIs exhibit a gapped bulk with twodimensional gapless edge states protected by U(1) electron number conservation and time reversal symmetry (TRS), forbidding any adiabatic deformation into a trivial insulator. The effective electromagnetic field theory characterizing the 3DTI contains a topological axion term, with quantized coefficient θ = π for fermionic 3DTIs as compared to θ = 0 for TRS trivial insulators [12]. On the surface of a 3DTI, where θ changes from 0 to π, the gauge invariance of this topological field theory demands the presence of a half-integer Chern-Simons term at the surface. In the absence of symmetry breaking, this Chern-Simons term is precisely what allows a single Dirac cone surface state to be gauge invariant and hence consistently defined on the 3DTI surface, whereas a single massless Dirac field in odd space-time dimensions would usually exhibit a parity anomaly [13][14][15].When the protecting U(1) particle number symmetry is broken, such as by a superconducting proximity effect, the 3DTI surface yields an unconventional gapped s-wave superconductor with Majorana modes in its vortex cores [16]. Upon breaking TRS, such as by a magnetic coating on the surface, the single surface Dirac cone gaps out, and the Chern-Simons boundary term of the axion bulk action manifests itself as a ν = 1/2 quantum Hall effect [12] without fractionalized excitations. The axion term implies the Witten effect [17] by which a odd-half integer charge binds to magnetic monopoles in the bulk of a 3DTI (see also e.g. Ref. 18).All aforementioned properties of 3DTIs do not involve interactions in the bulk or at the surface. Assuming that the gapped 3DTI bulk is negligibly renormalized by interactions, it remains to be investigated how interactions

Disorder-induced density of states on the surface of a spherical topological insulator

“…An elegant solution to this problem, understood in terms of Landau levels on the sphere spanned by two mutually commuting SU(2) algebras (for the cyclotron momentum S and the guiding center momentum L) was obtained by Neupert et al 16 via the formalism developed in Refs. 18,19.…”

“…Recent work by Neupert et al 16 highlighted the utility of adopting a spherical geometry for numerical studies of the TI surface. As shown by Imura et al 17 , the massless Dirac Hamiltonian of the flat TI surface can be mapped to a spherical TI surface with the introduction of a fictitious magnetic monopole at the center of the sphere that has opposite sign for electrons of opposite spin.…”

“…TI nanoparticles 17 ) or to gain insight regarding the flat TI surface that is recovered in the large radius limit. Neupert et al 16 adopt the latter point of view, making use of spherical geometry to study the effect of electronelectron interactions on TI surface states.…”

“…In the present work, we make use of the same spherical geometry [16][17][18][19] to perform a numerical disorder analysis, investigating the effects of disorder on the density of states of the TI surface. Of course, the study of disorder effects in two-dimensional massless Dirac systems has a long history that predates the discovery of topological insulators and was primarily focused on the quasiparticle density of states of so-called dirty d-wave superconductors [20][21][22][23][24][25] .…”

“…II A by reviewing the solution of the clean, single-particle, spherical TI problem, as solved via Refs. [16][17][18][19], and noting the eigenvalues and eigenstates of the clean Hamiltonian. In Sec.…”