Manifestly gauge-independent formulations of theZ2invariants

Abstract

We use a "monodromy" argument to derive new expressions for the 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.

Exactly solvable topological chiral spin liquid with random exchange

“…The most striking feature is the expansion of the topologically non-trivial phase region for intermediate to strong values of disorder strength, 1 < W < 2. Similar effects have also been observed in disordered topological insulators [11][12][13] and disordered Chern insulators 14 .…”

“…In the clean limit, this model is known to possess both Abelian and topological (finite Chern number) non-Abelian gapped chiral spin liquid phases [3][4][5] . In this work, we focus on how random exchange disorder affects the phase boundaries and show there are analogs to the recently studied topological Anderson insulators [11][12][13] and disordered Chern insulators 14 . Our main result is that disorder enlarges the parameter space of the topological phase and therefore can drive a transition into the topological phase.…”

Effect of strong disorder in a three-dimensional topological insulator: Phase diagram and maps of theZ2invariant

“…This robustness against disorder can be the key to many technological applications, [9][10][11] and because of that, a great deal of effort has been dedicated to understanding the behavior of the topological materials in the presence of disorder. [13][14][15][16][17][18][19][20][21][22][23][24][25][26] One important question, which is still opened for 3D topological insulators, is if the robustness against disorder extends into the strong disorder regime, particularly into the regime where the insulating gap is filled with dense localized spectrum. The theoretical argument based on the time-reversal symmetry is perturbative and therefore it breaks down in this regime.…”

“…29,36,[41][42][43] Here we will follow Ref. 29 and we will argue here that this new formulations bring certain numerical advantages which open the possibility of directly computing the Z 2 invariants for systems with extremely large unit cells, particularly for disordered samples (as opposed to indirectly inferring the Z 2 invariants from other type of calculations such as transport simulations of the surface states). We present a numerical analysis of the strong Z 2 invariant for a system without inversion symmetry, computed in the weak and strong disorder regimes via the twisted boundary conditions technique combined with the new formulation of the invariant.…”

“…The difficulty introduced by this requirement has been documented in our previous work. 29 For example, the original expressions of the Z 2 invariant, 3,7,30,31 require special smooth gauges and were computed only for analytically solvable band models. These expressions have been reformulated in an almost gauge invariant fashion by Fukui and his collaborators.…”