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 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.

%B Annales Henri Poincaré %V 17 %P 63–97 %8 Jan %G eng %U https://doi.org/10.1007/s00023-015-0400-6 %R 10.1007/s00023-015-0400-6 %0 Journal Article %J Applied Categorical Structures %D 2016 %T t-Structures are Normal Torsion Theories %A Domenico Fiorenza %A Fosco Loregian %XWe characterize $t$-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure $\mathcal{t}$ on a stable $\infty$-category $\mathbb{C}$ is equivalent to a normal torsion theory $\mathbf{F}$ on $\mathbb{C}$, i.e. to a factorization system $\mathbf{F} = (\mathcal{\epsilon}, \mathcal{M})$ where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.

%B Applied Categorical Structures %V 24 %P 181–208 %8 Apr %G eng %U https://doi.org/10.1007/s10485-015-9393-z %R 10.1007/s10485-015-9393-z %0 Journal Article %J Communications in Mathematical Physics %D 2016 %T Z2 Invariants of Topological Insulators as Geometric Obstructions %A Domenico Fiorenza %A Domenico Monaco %A Gianluca Panati %XWe consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to $-\mathbb{1}$. 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 $\mathbb{Z}_2$-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 $\mathbb{Z}_2$ 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.

%B Communications in Mathematical Physics %V 343 %P 1115–1157 %8 May %G eng %U https://doi.org/10.1007/s00220-015-2552-0 %R 10.1007/s00220-015-2552-0