Abstract
Continuum models for time-reversal (TR) invariant topological insulators (Tis) in d ≥ 3 dimensions are provided by harmonic oscillators coupled to certain SO(d) gauge fields. These models are equivalent to the presence of spin-orbit (SO) interaction in the oscillator Hamiltonians at a critical coupling strength (equivalent to the harmonic oscillator frequency) and leads to flat Landau Level (LL) spectra and therefore to infinite degeneracy of either the positive or the negative helicity states depending on the sign of the SO coupling. Generalizing the results of [1] to d ≥ 4, we construct vector operators commuting with these Hamiltonians and show that SO(d, 2) emerges as the non-compact extended dynamical symmetry. Focusing on the model in four dimensions, we demonstrate that the infinite degeneracy of the flat spectra can be fully explained in terms of the discrete unitary representations of SO(4,2), i.e. the doubletons. The degeneracy in the opposite helicity branch is finite, but can still be explained exploiting the complex conjugate doubleton representations. Subsequently, the analysis is generalized to d-dimensions, distinguishing the cases of odd and even d. We also determine the spectrum generating algebra in these models and briefly comment on the algebraic organization of the LL states w.r.t. an underlying “deformed” AdS geometry as well as on the organization of the surface states under open boundary conditions in view of our results.
Highlights
SU(2) ≃ SO(3) and SO(d), Aharanov-Casher type non-abelian gauge fields in three and d-dimensions, respectively
We show in full detail how the infinite degeneracy of the energy spectrum in the positive helicity branch can be explained in terms of the discrete unitary irreducible representation (UIR) of SO(4, 2), which are known as the doubletons [38,39,40]
The Hamiltonians of these models can be expressed as simple harmonic oscillator (SHO), with a spin-orbit terms, whose coupling strength is tuned to the SHO frequency
Summary
We may launch our discussion starting with the Hamiltonian of a four-dimensional (4D). Which may be expressed as the Hamiltonian for a simple harmonic oscillator (SHO) with the spin-orbit (SO) term at the coupling strength ω matching the SHO frequency as p2a 2m. In this expression Lab := rapb − rbpa , (a, b = 1, · · · , 4) are the orbital angular momentum operators, while Γab are proportional to the spin operator Sab in 4-dimensions, as will be explicitly defined in what follows. The Hamiltonian commutes with the total angular momentum operator Jab. Its spectrum and eigenfunctions are given in [15]. Trading the label l for l′, we can express the spectrum in (2.8) as
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