201 research outputs found
Optimally localized Wannier functions for quasi one-dimensional nonperiodic insulators
It is proved that for general, not necessarily periodic quasi one dimensional
systems, the band position operator corresponding to an isolated part of the
energy spectrum has discrete spectrum and its eigenfunctions have the same
spatial localization as the corresponding spectral projection.
As a consequence, an eigenbasis of the band position operator provides a
basis of optimally localized (generalized) Wannier functions for quasi one
dimensional systems, thus proving the "strong conjecture" of Marzari and
Vanderbilt. If the system has some translation symmetries (e.g. usual
translations, screw transformations), they are "inherited" by the Wannier
basis.Comment: 15 pages, final version. Accepted for publication in J.Phys.
On essential self-adjointness for magnetic Schroedinger and Pauli operators on the unit disc in R^2
We study the question of magnetic confinement of quantum particles on the
unit disk \ID in \IR^2, i.e. we wish to achieve confinement solely by means
of the growth of the magnetic field near the boundary of the disk.
In the spinless case we show that , for close to 1, insures the confinement provided we
assume that the non-radially symmetric part of the magnetic field is not very
singular near the boundary. Both constants and
are optimal. This answers, in this context, an open
question from Y. Colin de Verdi\`ere and F. Truc. We also derive growth
conditions for radially symmetric magnetic fields which lead to confinement of
spin 1/2 particles.Comment: 18 pages; the main theorem has been expanded and generalize
The Faraday effect revisited: Thermodynamic limit
This paper is the second in a series revisiting the (effect of) Faraday
rotation. We formulate and prove the thermodynamic limit for the transverse
electric conductivity of Bloch electrons, as well as for the Verdet constant.
The main mathematical tool is a regularized magnetic and geometric
perturbation theory combined with elliptic regularity and Agmon-Combes-Thomas
uniform exponential decay estimates.Comment: 35 pages, accepted for publication in Journal of Functional Analysi
The Faraday effect revisited: General theory
This paper is the first in a series revisiting the Faraday effect, or more
generally, the theory of electronic quantum transport/optical response in bulk
media in the presence of a constant magnetic field. The independent electron
approximation is assumed. At zero temperature and zero frequency, if the Fermi
energy lies in a spectral gap, we rigorously prove the Widom-Streda formula.
For free electrons, the transverse conductivity can be explicitly computed and
coincides with the classical result. In the general case, using magnetic
perturbation theory, the conductivity tensor is expanded in powers of the
strength of the magnetic field . Then the linear term in of this
expansion is written down in terms of the zero magnetic field Green function
and the zero field current operator. In the periodic case, the linear term in
of the conductivity tensor is expressed in terms of zero magnetic field
Bloch functions and energies. No derivatives with respect to the quasi-momentum
appear and thereby all ambiguities are removed, in contrast to earlier work.Comment: Final version, accepted for publication in J. Math. Phy
Instability of a Pseudo-Relativistic Model of Matter with Self-Generated Magnetic Field
For a pseudo-relativistic model of matter, based on the no-pair Hamiltonian,
we prove that the inclusion of the interaction with the self-generated magnetic
field leads to instability for all positive values of the fine structure
constant. This is true no matter whether this interaction is accounted for by
the Breit potential, by an external magnetic field which is chosen to minimize
the energy, or by the quantized radiation field.Comment: 13 pages, AMS-LaTe
Wick Theorem for General Initial States
We present a compact and simplified proof of a generalized Wick theorem to
calculate the Green's function of bosonic and fermionic systems in an arbitrary
initial state. It is shown that the decomposition of the non-interacting
-particle Green's function is equivalent to solving a boundary problem for
the Martin-Schwinger hierarchy; for non-correlated initial states a one-line
proof of the standard Wick theorem is given. Our result leads to new
self-energy diagrams and an elegant relation with those of the imaginary-time
formalism is derived. The theorem is easy to use and can be combined with any
ground-state numerical technique to calculate time-dependent properties.Comment: 9 pages, 5 figure; extended version published in Phys. Rev.
Faraday effect revisited: sum rules and convergence issues
This is the third paper of a series revisiting the Faraday effect. The
question of the absolute convergence of the sums over the band indices entering
the Verdet constant is considered. In general, sum rules and traces per unit
volume play an important role in solid state physics, and they give rise to
certain convergence problems widely ignored by physicists. We give a complete
answer in the case of smooth potentials and formulate an open problem related
to less regular perturbations.Comment: Dedicated to the memory of our late friend Pierre Duclos. Accepted
for publication in Journal of Physics A: Mathematical and Theoretical
General Adiabatic Evolution with a Gap Condition
We consider the adiabatic regime of two parameters evolution semigroups
generated by linear operators that are analytic in time and satisfy the
following gap condition for all times: the spectrum of the generator consists
in finitely many isolated eigenvalues of finite algebraic multiplicity, away
from the rest of the spectrum. The restriction of the generator to the spectral
subspace corresponding to the distinguished eigenvalues is not assumed to be
diagonalizable. The presence of eigenilpotents in the spectral decomposition of
the generator forbids the evolution to follow the instantaneous eigenprojectors
of the generator in the adiabatic limit. Making use of superadiabatic
renormalization, we construct a different set of time-dependent projectors,
close to the instantaneous eigeprojectors of the generator in the adiabatic
limit, and an approximation of the evolution semigroup which intertwines
exactly between the values of these projectors at the initial and final times.
Hence, the evolution semigroup follows the constructed set of projectors in the
adiabatic regime, modulo error terms we control
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