We prove an adiabatic theorem for general densities of observables that are sums of local terms in finite systems of interacting fermions, without periodicity assumptions on the Hamiltonian and with error estimates that are uniform in the size of the system. Our result provides an adiabatic expansion to all orders, in particular, also for initial data that lie in eigenspaces of degenerate eigenvalues. Our proof is based on ideas from [6], where Bachmann et al. proved an adiabatic theorem for interacting spin systems.As one important application of this adiabatic theorem, we provide the first rigorous derivation of the adiabatic response formula for the current density induced by an adiabatic change of the Hamiltonian of a system of interacting fermions in a ground state, with error estimates uniform in the system size. We also discuss the application to quantum Hall systems.

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| → ∞, 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 localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is +∞. In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity.(2) Throughout this Section, we use Hartree atomic units, and moreover we reabsorb the reciprocal of the speed of light 1/c in the definition of the function A Γ .

Abstract. 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 Z2 topological phases.We review several proposals for Z2 indices that distinguish these topological phases, including the ones by Fu-Kane [FK], Prodan [Pr2], Graf-Porta [GP] and Fiorenza-Monaco-Panati [FMP2]. We show that all these formulations are equivalent. In particular, this allows to prove a geometric formula for the the Z2 invariant of 2-dimensional TRS topological insulators, originally indicated in [FK], which expresses it in terms of the Berry connection and the Berry curvature.

Abstract. We consider a real periodic Schrödinger operator and a physically relevant family of m ≥ 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 ≤ 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 composite Wannier functions which are real-valued and almost-exponentially localized.The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.

Abstract. We describe some applications of group-and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e. g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The rôle of additional Z 2 -symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same Z 2 -symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.

Abstract. We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i. e. the time-reversal operator squares to −1. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a Z 2 -valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four Z 2 invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones.

Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of |x| 2 diverges. Intermediate regimes are forbidden.Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane [15]. We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in [25], and finally we announce further generalizations to non-periodic models.Date: September 6, 2019. Extended version of the paper published in Rend. Mat. Appl. 39, 307-327 (2018). In comparison with the published version, we added some details and the whole Chapter 5.2010 Mathematics Subject Classification. 81Q70, 81V70, 47A56, 47A10.

Abstract. 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 Z 2 -valued complete invariants for these homotopy classes. When these invariants vanish, we provide an algorithm which constructs a "multi-step logarithm" that is employed to continuously deform the given family into a constant one, identically equal to the identity matrix. This algorithm leads to a constructive procedure to produce the composite Wannier bases mentioned above.