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.

1 aFiorenza, Domenico1 aMonaco, Domenico1 aPanati, Gianluca uhttps://doi.org/10.1007/s00023-015-0400-601400nas a2200169 4500008004100000022001400041245007000055210006900125260000800194300001600202490000800218520089300226100002301119700002101142700002101163856004601184 2016 eng d a1432-091600aZ2 Invariants of Topological Insulators as Geometric Obstructions0 aZ2 Invariants of Topological Insulators as Geometric Obstruction cMay a1115–11570 v3433 aWe 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.

1 aFiorenza, Domenico1 aMonaco, Domenico1 aPanati, Gianluca uhttps://doi.org/10.1007/s00220-015-2552-001557nas a2200121 4500008004100000245012000041210006900161260001300230520109900243100002101342700002101363856005101384 2015 en d00aSymmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry0 aSymmetry and localization in periodic crystals triviality of Blo bSpringer3 aWe 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.

1 aMonaco, Domenico1 aPanati, Gianluca uhttp://urania.sissa.it/xmlui/handle/1963/3446801418nas a2200133 4500008004100000245010100041210006900142260003500211520074700246653011900993100002101112700002101133856013001154 2014 en d00aTopological Invariants of Eigenvalue Intersections and Decrease of Wannier Functions in Graphene0 aTopological Invariants of Eigenvalue Intersections and Decrease bJournal of Statistical Physics3 aWe 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.

10aWannier functions, Bloch bundles, conical intersections, eigenspace vorticity, pseudospin winding number, graphene1 aMonaco, Domenico1 aPanati, Gianluca uhttps://www.math.sissa.it/publication/topological-invariants-eigenvalue-intersections-and-decrease-wannier-functions-graphene01140nas a2200109 4500008004300000245006600043210006200109520077800171100002400949700002100973856003600994 2010 en_Ud 00aThe geometry emerging from the symmetries of a quantum system0 ageometry emerging from the symmetries of a quantum system3 aWe 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.1 aDe Nittis, Giuseppe1 aPanati, Gianluca uhttp://hdl.handle.net/1963/383401206nas a2200133 4500008004300000245007600043210006900119260001300188520077600201100002100977700001900998700001901017856003601036 2003 en_Ud 00aEffective dynamics for Bloch electrons: Peierls substitution and beyond0 aEffective dynamics for Bloch electrons Peierls substitution and bSpringer3 aWe 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.1 aPanati, Gianluca1 aSpohn, Herbert1 aTeufel, Stefan uhttp://hdl.handle.net/1963/304001403nas a2200133 4500008004300000245004000043210003900083260002400122520102800146100002101174700001901195700001901214856003601233 2003 en_Ud 00aSpace-adiabatic perturbation theory0 aSpaceadiabatic perturbation theory bInternational Press3 aWe 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.1 aPanati, Gianluca1 aSpohn, Herbert1 aTeufel, Stefan uhttp://hdl.handle.net/1963/304100307nas a2200097 4500008004100000245005100041210005000092260001000142100002100152856003600173 2002 en d00aSpace-adiabatic Decoupling in Quantum Dynamics0 aSpaceadiabatic Decoupling in Quantum Dynamics bSISSA1 aPanati, Gianluca uhttp://hdl.handle.net/1963/636001131nas a2200133 4500008004100000245006000041210005900101260003000160520071200190100002100902700001900923700001900942856003600961 2002 en d00aSpace-adiabatic perturbation theory in quantum dynamics0 aSpaceadiabatic perturbation theory in quantum dynamics bAmerican Physical Society3 aA 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.1 aPanati, Gianluca1 aSpohn, Herbert1 aTeufel, Stefan uhttp://hdl.handle.net/1963/5985