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arXiv:1103.4603 · Physical Review B

Manifestly gauge-independent formulations of the Z2 invariants

Emil V. Prodan

Published Jun 8, 2011• 30 citations
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Abstract

We use a ``monodromy'' argument to derive new expressions for the ${\mathbb{Z}}_{2}$ invariants of topological insulators with time-reversal symmetry in two and three dimensions. The derivations and the final expressions do not require any gauge choice and the calculation of the invariants is based entirely on the projectors onto the occupied states. Explicit numerical tests for tight-binding models with strongly broken inversion symmetry are presented in two and three dimensions.

DOI
10.1103/physrevb.83.235115

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We use a ``monodromy'' argument to derive new expressions for the Z2{\mathbb{Z}}_{2}Z2​ invariants of topological insulators with time-reversal symmetry in two and three dimensions. The derivations and the final expressions do not require any gauge choice and the calculation of the invariants is based entirely on the projectors onto the occupied states. Explicit numerical tests for tight-binding models with strongly broken inversion symmetry are presented in two and three dimensions.
For interacting Z2{Z}_{2}Z2​ topological insulators with inversion symmetry, we propose a simple topological invariant expressed in terms of the parity eigenvalues of the interacting Green's function at time-reversal invariant momenta and zero frequency. We derive this result from our previous formula involving the integral over the frequency-momenta space. This formula greatly simplifies the explicit calculation of Z2{Z}_{2}Z2​ topological invariants in inversion-symmetric insulators with strong interactions.
We show that the two-dimensional Z2{\mathbb{Z}}_{2}Z2​ invariant for time-reversal invariant insulators can be formulated in terms of the boundary-condition dependence of the ground state wave function for both noninteracting and strongly correlated insulators. By introducing a family of quasi-single-particle states associated to the many-body ground state of an insulator, we show that the Z2{\mathbb{Z}}_{2}Z2​ invariant can be expressed as the integral of a certain Berry connection over half the space of boundary conditions, providing an alternative expression to the formulations that appear in S.-S. Lee and S. Ryu [Phys. Rev. Lett. 100, 186807 (2008)]. We show the equivalence of the different many-body formulations of the invariant and show how they reduce to known band-theoretic results for Slater determinant ground states. Finally, we apply our results to analytically calculate the invariant for the Kane-Mele model with nonlocal (orbital) Hatsugai-Kohmoto (HK) interactions. This rigorously establishes the topological nontriviality of the Kane-Mele model with HK interactions and represents one of the few exact calculations of the Z2{\mathbb{Z}}_{2}Z2​ invariant for a strongly interacting system.
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Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Z. Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy CP\mathcal{CP}CP symmetry, leading to intuitive interpretations of the bulk invariant for a CP\mathcal{CP}CP-symmetric topological insulator upon putting the system on a Klein bottle. We also briefly discuss how these many-body invariants may be able to characterize models with only time-reversal symmetry.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.
We study three-dimensional Z2{Z}_{2}Z2​ topological phases enriched by Z2T{Z}_{2}^{T}Z2T​ time-reversal symmetry with bosonic bulk excitations. Some of these phases can be constructed by simply coupling the three-dimensional symmetry protected topological phases with ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T}$ symmetry to a deconfined Z2{Z}_{2}Z2​ gauge field. Besides these simple phases, we also construct two special root phases, whose boundary can have an extra Z2{Z}_{2}Z2​ topological order in addition to their bulk topological order, and the boundary anyon excitations can have fractional time-reversal transformation T\mathcal{T}T with ${\mathcal{T}}^{4}=\ensuremath{-}1$. In particular, the boundary eee and mmm anyons of one of the two root phases are interchanged under time-reversal transformation, and they must be degenerate and orthogonal with each other. Eventually we obtain ${({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$ classification (8 different phases) for Z2T{Z}_{2}^{T}Z2T​-enriched Z2{Z}_{2}Z2​ topological phases with bosonic bulk excitations in three dimensions.