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 -