We provide a constructive proof of exponentially localized Wannier functions and related Bloch frames in 1- and 2-dimensional time-reversal symmetric (TRS) topological insulators. The construction is formulated in terms of periodic TRS families of projectors (corresponding, in applications, to the eigenprojectors on an arbitrary number of relevant energy bands), and is thus model-independent. The possibility to enforce also a TRS constraint on the frame is investigated. This leads to a topological obstruction in dimension 2, related to $\mathbb{Z}_2$ topological phases.
We review several proposals for $\mathbb{Z}_2$ indices that distinguish these topological phases, including the ones by Fu--Kane [Phys. Rev. B 74 (2006), 195312], Prodan [Phys. Rev. B 83 (2011), 235115], Graf--Porta [Commun. Math. Phys. 324 (2013), 851] and Fiorenza--Monaco--Panati [Commun. Math. Phys., in press]. We show that all these formulations are equivalent. In particular, this allows to prove a geometric formula for the the $\mathbb{Z}_2$ invariant of 2-dimensional TRS topological insulators, originally indicated in [Phys. Rev. B 74 (2006), 195312], which expresses it in terms of the Berry connection and the Berry curvature.
We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e.g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as $|x| \rightarrow \infty$, of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially l…
We consider a real periodic Schrödinger operator and a physically relevant family of $m \geq 1$ Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension $d \leq 3$ there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of com…
For a large class of physically relevant operators on a manifold with discrete group action, we prove general results on the (non-)existence of a basis of smooth well-localised Wannier functions for their spectral subspaces. This turns out to be equivalent to the freeness of a certain Hilbert module over the group $C^*$-algebra canonically associated to the spectral subspace. This brings into play $K$-theoretic methods and justifies their importance as invariants of topological insulators in physics.
We investigate the possibility of constructing exponentially localized composite Wannier bases, or equivalently smooth periodic Bloch frames, for 3-dimensional time-reversal symmetric topological insulators, both of bosonic and of fermionic type, so that the bases in question are also compatible with time-reversal symmetry. This problem is translated in the study, of independent interest, of homotopy classes of continuous, periodic, and time-reversal symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued complete invariants for these homotopy classes. When these invari…
Being Wannierizable is not the end of the story for topological insulators. We introduce a family of topological insulators that would be considered trivial in the paradigm set by the tenfold way, topological quantum chemistry, and the method of symmetry-based indicators. Despite having a symmetric, exponentially-localized Wannier representation, each Wannier function cannot be completely localized to a single primitive unit cell in the bulk. Such multicellular topology is shown to be neither stable, nor fragile, but delicate, i.e., the topology can be nullified by adding trivial bands to eith…
Localized bases play an important role in understanding electronic structure. In periodic insulators, a natural choice of localized basis is given by the Wannier functions which depend a choice of unitary transform known as a gauge transformation. Over the past few decades, there have been many works which have focused on optimizing the choice of gauge so that the corresponding Wannier functions are maximally localized or reflect some symmetry of the underlying system. In this work, we consider fully non-periodic materials where the usual Wannier functions are not well defined and gauge optimi…
We study spinful non-interacting electrons moving in two-dimensional materials which exhibit a spectral gap about the Fermi energy as well as time-reversal invariance. Using Fredholm theory we revisit the (known) bulk topological invariant, define a new one for the edge, and show their equivalence (the bulk-edge correspondence) via homotopy.
We provide a mathematically proven, simple and efficient algorithm to build localised Wannier functions, with the only requirement that the system has vanishing Chern numbers. Our algorithm is able to build localised Wannier for topological insulators such as the Kane-Mele model. It is based on an explicit and constructive proof of homotopies for the unitary group. We provide numerical tests validating the methods for several systems, including the Kane-Mele model.
The existence and construction of exponentially localised Wannier functions for insulators is a well-studied problem. In comparison, the case of metallic systems has been much less explored, even though localised Wannier functions constitute an important and widely used tool for the numerical band interpolation of metallic condensed matter systems. In this paper we prove that, under generic conditions, $N$ energy bands of a metal can be exactly represented by $N+1$ Wannier functions decaying faster than any polynomial. We also show that, in general, the lack of a spectral gap does not allow fo…
The Feynman amplitudes with the two-dimensional Wess-Zumino action functional have a geometric interpretation as bundle gerbe holonomy. We present details of the construction of a distinguished square root of such holonomy and of a related 3d-index and briefly recall the application of those to the building of topological invariants for time-reversal-symmetric two- and three-dimensional crystals, both static and periodically forced.
We study disordered topological insulators with time reversal symmetry. Relying on the noncommutative index theorem which relates the Chern number to the projection onto the Fermi sea and the magnetic flux operator, we give a precise definition of the $Z_2$ index which is a noncommutative analogue of the Atiyah-Singer $Z_2$ index. We prove that the noncommutative $Z_2$ index is robust against any time-reversal symmetric perturbation including disorder potentials as long as the spectral gap at the Fermi level does not close.
In this work, we investigate conditions which ensure the existence of an exponentially localized Wannier basis for a given periodic hamiltonian. We extend previous results in [Pan07] to include periodic zero flux magnetic fields which is the setting also investigated in [Kuc09]. The new notion of magnetic symmetry plays a crucial role; to a large class of symmetries for a non-magnetic system, one can associate "magnetic" symmetries of the related magnetic system. Observing that the existence of an exponentially localized Wannier basis is equivalent to the triviality of the so-called Bloch bund…
For models of non-interacting fermions moving within sites arranged on a surface in three dimensional space, there can be obstructions to finding localized Wannier functions. We show that such obstructions are $K$-theoretic obstructions to approximating almost commuting, complex-valued matrices by commuting matrices, and we demonstrate numerically the presence of this obstruction for a lattice model of the quantum Hall effect in a spherical geometry. The numerical calculation of the obstruction is straightforward, and does not require translational invariance or introducing a flux torus. We fu…
We introduce a web of strongly correlated interacting 3+1D topological superconductors/insulators of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and their new SU(N) generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism. We complete the list of field theories of bulk symmetry-protected topological invariants (SPT invariants/partition functions that exhibit boundary 't Hooft anomalies) via cobordism …
