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arXiv:1905.12649 · Physical review. B./Physical review. B

Local formula for the Z2 invariant of topological insulators

Zhi Li, Roger S. K. Mong

Published Nov 4, 2019• 14 citations
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Abstract

We proposed a formula for the ${\mathbb{Z}}_{2}$ invariant for topological insulators, which remains valid without translational invariance. Our formula is a local expression, in the sense that the contributions mainly come from quantities near a point. Using almost commute matrices, we proposed a method to approximate this invariant with local information. The validity of the formula and the approximation method is proved.

DOI
10.1103/physrevb.100.205101

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We proposed a formula for the Z2{\mathbb{Z}}_{2}Z2​ invariant for topological insulators, which remains valid without translational invariance. Our formula is a local expression, in the sense that the contributions mainly come from quantities near a point. Using almost commute matrices, we proposed a method to approximate this invariant with local information. The validity of the formula and the approximation method is proved.
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.
The ground state of translationally invariant insulators comprises bands which can assume topologically distinct structures. There are few known examples where this distinction is enforced by a point-group symmetry alone. In this paper we show that 1D and 2D insulators with the simplest point-group symmetry, inversion, have a ${\mathbb{Z}}^{\ensuremath{\ge}}$ classification. In 2D, we identify a relative winding number that is solely protected by inversion symmetry. By analysis of Berry phases, we show that this invariant has similarities with the first Chern class (of time-reversal breaking insulators), but is more closely analogous to the Z2{\mathbb{Z}}_{2}Z2​ invariant (of time-reversal invariant insulators). Implications of our work are discussed in holonomy, the geometric-phase theory of polarization, the theory of maximally localized Wannier functions, and in the entanglement spectrum.
We define a Z2-valued topological and gauge invariant associated with any one-dimensional, translation-invariant topological insulator that satisfies either particle–hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated with a suitable basis of Bloch functions that is compatible with the symmetries. We compute the invariant in the Su–Schrieffer–Heeger model for chiral symmetric insulators and in the Kitaev model for particle–hole symmetric insulators. We show that in both cases, the Z2 invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.
Topological quantum state described by the global invariant has been extensively studied in theory and experiment. In this Letter, we investigate the relationship between Zitterbewegung and the topology of systems that reflects the properties of the local and whole energy bands, respectively. We generalize the usual two-band effective Hamiltonian to characterize the topological phase transition of the spin-JJJ topological insulator. By studying Zitterbewegung dynamics before and after topological phase transition, we find that the direction of quasiparticles' oscillation can well reflect topological properties. Furthermore, we develop a quantitative calculation formula for the topological invariant in the spin-JJJ Chern insulator and give the selection rule of the corresponding dynamics. Finally, we demonstrate that our theory is valid in different topological systems. The topological invariant can be represented by local dynamical properties of the high-symmetry points in the first Brillouin zone, which provides a measurement method from the dynamical perspective.
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.