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 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 %0 Report %D 2010 %T The geometry emerging from the symmetries of a quantum system %A Giuseppe De Nittis %A Gianluca Panati %X We investigate the relation between the symmetries of a quantum system and its topological quantum numbers, in a general C*-algebraic framework. We prove that, under suitable assumptions on the symmetry algebra, there exists a generalization of the Bloch-Floquet transform which induces a direct-integral decomposition of the algebra of observables. Such generalized transform selects uniquely the set of \\\"continuous sections\\\" in the direct integral, thus yielding a Hilbert bundle. The emerging geometric structure provides some topological invariants of the quantum system. Two running examples provide an Ariadne\\\'s thread through the paper. For the sake of completeness, we review two related theorems by von Neumann and Maurin and compare them with our result. %G en_US %U http://hdl.handle.net/1963/3834 %1 493 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-01-26T16:38:34Z\\nNo. of bitstreams: 1\\n0911.5270v2.pdf: 578198 bytes, checksum: a06ef54ebf418d5d7b6d0a7c3410c054 (MD5) %0 Journal Article %D 2003 %T Effective dynamics for Bloch electrons: Peierls substitution and beyond %A Gianluca Panati %A Herbert Spohn %A Stefan Teufel %X We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $\\\\phi(\\\\epsi x)$, and vector potential $A(\\\\epsi x)$, with $x \\\\in \\\\R^d$ and $\\\\epsi \\\\ll 1$. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of $L^2(\\\\R^d)$ and an effective Hamiltonian governing the evolution inside this subspace to all orders in $\\\\epsi$. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics. %I Springer %G en_US %U http://hdl.handle.net/1963/3040 %1 1293 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-08T12:11:51Z\\nNo. of bitstreams: 1\\n0212041v2.pdf: 395908 bytes, checksum: 7b198d3311402d34945989b8f6edd8b8 (MD5) %0 Journal Article %J Adv. Theor. Math. Phys. 7 (2003) 145-204 %D 2003 %T Space-adiabatic perturbation theory %A Gianluca Panati %A Herbert Spohn %A Stefan Teufel %X We study approximate solutions to the Schr\\\\\\\"odinger equation $i\\\\epsi\\\\partial\\\\psi_t(x)/\\\\partial t = H(x,-i\\\\epsi\\\\nabla_x) \\\\psi_t(x)$ with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space $\\\\Hi_{\\\\rm f}$ of fast ``internal\\\'\\\' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \\\\cite{NS} we prove that interband transitions are suppressed to any order in $\\\\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\\\\mathbb{R}^d,\\\\Hi _{\\\\rm f})$ almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory. %B Adv. Theor. Math. Phys. 7 (2003) 145-204 %I International Press %G en_US %U http://hdl.handle.net/1963/3041 %1 1292 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-08T12:17:36Z\\nNo. of bitstreams: 1\\n0201055v3.pdf: 449361 bytes, checksum: a37ea04fc4a4f59a75d03e4b2ec3df16 (MD5) %0 Thesis %D 2002 %T Space-adiabatic Decoupling in Quantum Dynamics %A Gianluca Panati %I SISSA %G en %U http://hdl.handle.net/1963/6360 %1 6292 %2 Mathematics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2012-12-21T16:35:07Z\\nNo. of bitstreams: 1\\nPhD_Panati_Gianluca.pdf: 9088755 bytes, checksum: 75624cfaef0563f1f1184f2dd0f2f955 (MD5) %0 Journal Article %J Physical review letters. 2002 Jun; 88(25 Pt 1):250405 %D 2002 %T Space-adiabatic perturbation theory in quantum dynamics %A Gianluca Panati %A Herbert Spohn %A Stefan Teufel %X A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schrödinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders, and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications, we discuss the Born-Oppenheimer theory to second order and derive for the first time the nonperturbative definition of the g factor of the electron within nonrelativistic quantum electrodynamics. %B Physical review letters. 2002 Jun; 88(25 Pt 1):250405 %I American Physical Society %G en %U http://hdl.handle.net/1963/5985 %1 5841 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Marta Maurutto (maurutto@sissa.it) on 2012-07-15T14:04:06Z\\nNo. of bitstreams: 0 %R 10.1103/PhysRevLett.88.250405