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 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 %0 Thesis %D 2015 %T Geometric phases in graphene and topological insulators %A Domenico Monaco %K Geometric phases, graphene, topological insulators, Wannier functions, Bloch frames %X This thesis collects three of the publications that the candidate produced during his Ph.D. studies. They all focus on geometric phases in solid state physics. We first study topological phases of 2-dimensional periodic quantum systems, in absence of a spectral gap, like e.g. (multilayer) graphene. A topological invariant n_v in Z, baptized eigenspace vorticity, is attached to any intersection of the energy bands, and characterizes the local topology of the eigenprojectors around that intersection. With the help of explicit models, each associated to a value of n_v in Z, we are able to extract the decay at infinity of the single-band Wannier function w in mono- and bilayer graphene, obtaining |w(x)| <= const |x|^{-2} as |x| tends to infinity. Next, we investigate gapped periodic quantum systems, in presence of time-reversal symmetry. When the time-reversal operator Theta is of bosonic type, i.e. it satisfies Theta^2 = 1, we provide an explicit algorithm to construct a frame of smooth, periodic and time-reversal symmetric (quasi-)Bloch functions, or equivalently a frame of almost-exponentially localized, real-valued (composite) Wannier functions, in dimension d <= 3. In the case instead of a fermionic time-reversal operator, satisfying Theta^2 = -1, we show that the existence of such a Bloch frame is in general topologically obstructed in dimension d=2 and d=3. This obstruction is encoded in Z_2-valued topological invariants, which agree with the ones proposed in the solid state literature by Fu, Kane and Mele. %I SISSA %G en %1 34702 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Domenico Monaco (dmonaco@sissa.it) on 2015-09-13T00:09:51Z No. of bitstreams: 1 Monaco_PhDThesis.pdf: 1801696 bytes, checksum: 83cdd0dee2cc65cb246735d92f21c74a (MD5) %0 Report %D 2015 %T Stability of closed gaps for the alternating Kronig-Penney Hamiltonian %A Alessandro Michelangeli %A Domenico Monaco %X We consider the Kronig-Penney model for a quantum crystal with equispaced periodic delta-interactions of alternating strength. For this model all spectral gaps at the centre of the Brillouin zone are known to vanish, although so far this noticeable property has only been proved through a very delicate analysis of the discriminant of the corresponding ODE and the associated monodromy matrix. We provide a new, alternative proof by showing that this model can be approximated, in the norm resolvent sense, by a model of regular periodic interactions with finite range for which all gaps at the centre of the Brillouin zone are still vanishing. In particular this shows that the vanishing gap property is stable in the sense that it is present also for the "physical" approximants and is not only a feature of the idealised model of zero-range interactions. %I SISSA %G en %U http://urania.sissa.it/xmlui/handle/1963/34460 %1 34629 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-04-01T14:57:03Z No. of bitstreams: 1 SISSA_preprint_16-2015-MATE.pdf: 349807 bytes, checksum: f5e6dacd85fb41c96420795cc37cc12c (MD5) %0 Journal Article %J Acta Applicandae Mathematicae, vol. 137, Issue 1, 2015, pages: 185-203 %D 2015 %T Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry %A Domenico Monaco %A Gianluca Panati %XWe 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.

%B Acta Applicandae Mathematicae, vol. 137, Issue 1, 2015, pages: 185-203 %I Springer %G en %U http://urania.sissa.it/xmlui/handle/1963/34468 %1 34642 %2 Mathematics %4 1 %# MAT/07 %$ Submitted by Domenico Monaco (dmonaco@sissa.it) on 2015-05-15T10:12:11Z No. of bitstreams: 1 CalaGonone.pdf: 500680 bytes, checksum: 00aa2839c4d737f45da3d828e5cc0a53 (MD5) %R 10.1007/s10440-014-9995-8 %0 Journal Article %J J. Stat. Phys 155 (2014) 1027-1071 %D 2014 %T Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene %A Domenico Monaco %A Gianluca Panati %K Wannier functions, Bloch bundles, conical intersections, eigenspace vorticity, pseudospin winding number, graphene %XWe investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value n∈Z of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function w satisfies |w(x)|≤const |x|^{−2} as |x|→∞, both in monolayer and bilayer graphene.

%B J. Stat. Phys 155 (2014) 1027-1071 %I Journal of Statistical Physics %G en %1 7368 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Domenico Monaco (dmonaco@sissa.it) on 2014-05-25T14:32:44Z No. of bitstreams: 1 WannierGraphene_arXiv_v3.pdf: 618532 bytes, checksum: 975235799a61f551f3c4889727353676 (MD5) %R 10.1007/s10955-014-0918-x