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.

VL - 17 UR - https://doi.org/10.1007/s00023-015-0400-6 ER - TY - JOUR T1 - Z2 Invariants of Topological Insulators as Geometric Obstructions JF - Communications in Mathematical Physics Y1 - 2016 A1 - Domenico Fiorenza A1 - Domenico Monaco A1 - Gianluca Panati AB -We 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.

VL - 343 UR - https://doi.org/10.1007/s00220-015-2552-0 ER - TY - JOUR T1 - Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry JF - Acta Applicandae Mathematicae, vol. 137, Issue 1, 2015, pages: 185-203 Y1 - 2015 A1 - Domenico Monaco A1 - Gianluca Panati AB -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.

PB - Springer UR - http://urania.sissa.it/xmlui/handle/1963/34468 N1 - The article is composed of 23 pages and recorded in PDF format U1 - 34642 U2 - Mathematics U4 - 1 U5 - MAT/07 ER - TY - JOUR T1 - Topological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene JF - J. Stat. Phys 155 (2014) 1027-1071 Y1 - 2014 A1 - Domenico Monaco A1 - Gianluca Panati KW - Wannier functions, Bloch bundles, conical intersections, eigenspace vorticity, pseudospin winding number, graphene AB -We 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.

PB - Journal of Statistical Physics U1 - 7368 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - RPRT T1 - The geometry emerging from the symmetries of a quantum system Y1 - 2010 A1 - Giuseppe De Nittis A1 - Gianluca Panati AB - 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. UR - http://hdl.handle.net/1963/3834 U1 - 493 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Effective dynamics for Bloch electrons: Peierls substitution and beyond Y1 - 2003 A1 - Gianluca Panati A1 - Herbert Spohn A1 - Stefan Teufel AB - 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. PB - Springer UR - http://hdl.handle.net/1963/3040 U1 - 1293 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Space-adiabatic perturbation theory JF - Adv. Theor. Math. Phys. 7 (2003) 145-204 Y1 - 2003 A1 - Gianluca Panati A1 - Herbert Spohn A1 - Stefan Teufel AB - 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. PB - International Press UR - http://hdl.handle.net/1963/3041 U1 - 1292 U2 - Mathematics U3 - Mathematical Physics ER - TY - THES T1 - Space-adiabatic Decoupling in Quantum Dynamics Y1 - 2002 A1 - Gianluca Panati PB - SISSA UR - http://hdl.handle.net/1963/6360 U1 - 6292 U2 - Mathematics U4 - -1 ER - TY - JOUR T1 - Space-adiabatic perturbation theory in quantum dynamics JF - Physical review letters. 2002 Jun; 88(25 Pt 1):250405 Y1 - 2002 A1 - Gianluca Panati A1 - Herbert Spohn A1 - Stefan Teufel AB - 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. PB - American Physical Society UR - http://hdl.handle.net/1963/5985 U1 - 5841 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER -