Spherical topological insulator

Imura, Ken‐Ichiro; Yoshimura, Yukinori; Takane, Yositake; Fukui, Tsuguya

doi:10.1103/physrevb.86.235119

Cited by 80 publications

(145 citation statements)

References 43 publications

(140 reference statements)

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“…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. The finite size of the spherical surface yields a problem with a discrete energy spectrum and well-defined eigenstates that is well-suited to numerical analysis, whether the objective be to study real systems with spherical geometry (i.e.…”

Section: Introductionmentioning

confidence: 99%

“…The finite size of the spherical surface yields a problem with a discrete energy spectrum and well-defined eigenstates that is well-suited to numerical analysis, whether the objective be to study real systems with spherical geometry (i.e. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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] .…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Durst

2016

Phys. Rev. B

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We consider a topological insulator (TI) of spherical geometry and numerically investigate the influence of disorder on the density of surface states. The energy spectrum of the spherical TI surface is discrete, for a sphere of finite radius, and can be truncated by imposing a high-energy cutoff at the scale of the bulk band gap. To this clean system we add a surface disorder potential of the most general hermitian form, V = V 0 (θ, φ)1 1 + V(θ, φ) · σ, where V 0 describes the spin-independent part of the disorder and the three components of V describe the spin-dependent part. We expand these four disorder functions in spherical harmonics and draw the expansion coefficients randomly from a four-dimensional, zero-mean gaussian distribution. Different strengths and classes of disorder are realized by specifying the 4 × 4 covariance matrix. For each instantiation of the disorder, we solve for the energy spectrum via exact diagonalization. Then we compute the disorder-averaged density of states, ρ(E), by averaging over 200,000 different instantiations. Disorder broadens the Landau-level delta functions of the clean density of states into peaks that decay and merge together. If the spin-dependent term is dominant, these peaks split due to the breaking of the degeneracy between time-reversed partner states. Increasing disorder strength pushes states closer and closer to zero energy (the Dirac point), resulting in a low-energy density of states that becomes nonzero for sufficient disorder, typically approaching an energy-independent saturation value, for most classes of disorder. But for purely spin-dependent disorder with V either entirely out-of-surface or entirely in-surface, we identify intriguing disorder-induced features in the vicinity of the Dirac point. In the out-of-surface case, a new peak emerges at zero energy. In the in-surface case, we see a symmetryprotected zero at zero energy, with ρ(E) increasing linearly toward nonzero-energy peaks. These striking features are explained in terms of the breaking (or not) of two chiral symmetries of the clean Hamiltonian.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Durst

2016

Phys. Rev. B

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“…While a consensus regarding possible states of matter consistent with the constraints given by the 3DTI setup is emerging, little is known which of these states might actually be realized for which kind of interaction profile. For this purpose, we investigate the surface states of a 3DTI on a sphere [29], the geometry where these states have the maximally attainable symmetry but no boundary. The single Dirac cone at the surface is intimately related to the Landau level quantization on the sphere [30], with an individual magnetic monopole of unit (anti-)charge for spin up (down) electrons at the center of the sphere, so as to respect TRS.…”

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confidence: 99%

Interacting Surface States of Three-Dimensional Topological Insulators

Neupert¹,

Rachel²,

Thomale³

et al. 2015

Phys. Rev. Lett.

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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

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