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 - 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 -