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Wannier functions and $$\mathbb{Z}_2$$ invariants in time-reversal symmetric topological insulators. (English) Zbl 1370.81081
The paper gives constructive proofs of the existence of exponentionally localized Wannier functions in gapped periodic quantum systems with a fermionic time-reversal symmetric (TRS) topological insulators in dimension $$d\leq 2$$. The paper also discusses the possibility of imposing a natural compatibility condition with the time-reversal operator and the relation of the latter with the $$\mathbb{Z}_2$$ invariants. There are proved that analytic and periodic frames can indeed be constructed in dimension $$d\leq 2$$ and encountered a topological obstruction in d = 2, when it is required also a TRS properties for the frame to hold. The 2d construction procedure can go through provided the Graf-Porta (GP) $$\mathbb{Z}_2$$ index vanishes: the latter is introduced as a “bulk invariant” for 2d TRS topological insulators. The topological obstruction is encoded in the $$\mathbb{Z}_2$$ Fiorenza-Monaco-Panati (FMP) invariant $$\delta$$. The GP index becomes manifestly a topological invariant of the quantum system, but the topological obstruction $$\delta$$ is brought into contact with the actual process of detection of topological phases in real experimental setups in view of the bulk-edge correspondence. These two quantities agree numerically. From its equality with the GP index, it is deduced an expression for invariant $$\delta$$, which depends explicitly on the family of projections, to which it is associated via its Berry connection and Berry curvature. As a result, after formulation of main results of the study, it is reformulated the problem of the construction of analytic, periodic and possibly TRS Bloch frames for a $$d$$-dimensional family of projectors. Their properties are modeled after the once of the eigen-projectors of the Hamiltonian of a gapped periodic quantum system with fermionic TRS in terms of an equivalent problem for particular families of unitary matrices $$\alpha$$. The latter emerge from the construction of a $$d$$-dimensional frame. The possibility to construct a periodic and TRS frame is then equivalent to the possibility of “rotating” this family $$\alpha$$ to the identity matrix by preserving its properties. Then the problem is specialized and solved in dimensions $$d=1$$ and 2. The problem is solved by constructing explicitly “good logarithms”. The topological obstruction to the existence of a “good logarithm” in $$d=2$$ is encoded in GP index, for which the authors prove several useful properties, including that it characterizes completely the homotopy class of a continuous, periodic and TRS family of unitary matrices. Due to the GP index is a topological obstruction, it allows discussion of several approaches to the formulation of $$\mathbb{Z}_2$$ invariants distinguishing the different topological phases in 2d TRS topological insulators. Equivalence between the GP and FMP invariants allows one to show that the $$\mathbb{Z}_2$$ invariant can be expressed in geometric terms as a function of the family of projectors.

##### MSC:
 81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory 35J10 Schrödinger operator, Schrödinger equation 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 82D25 Statistical mechanics of crystals
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