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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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