88 research outputs found
On the construction of Wannier functions in topological insulators: the 3D case
We investigate the possibility of constructing exponentially localized
composite Wannier bases, or equivalently smooth periodic Bloch frames, for
3-dimensional time-reversal symmetric topological insulators, both of bosonic
and of fermionic type, so that the bases in question are also compatible with
time-reversal symmetry. This problem is translated in the study, of independent
interest, of homotopy classes of continuous, periodic, and time-reversal
symmetric families of unitary matrices. We identify three -valued
complete invariants for these homotopy classes. When these invariants vanish,
we provide an algorithm which constructs a "multi-step" logarithm that is
employed to continuously deform the given family into a constant one,
identically equal to the identity matrix. This algorithm leads to a
constructive procedure to produce the composite Wannier bases mentioned above.Comment: 29 pages. Version 2: minor corrections of misprints, corrected proofs
of Theorems 2.4 and 2.9, added references. Accepted for publication in
Annales Henri Poicar\'
Spectral edge regularity of magnetic Hamiltonians
We analyse the spectral edge regularity of a large class of magnetic
Hamiltonians when the perturbation is generated by a globally bounded magnetic
field. We can prove Lipschitz regularity of spectral edges if the magnetic
field perturbation is either constant or slowly variable. We also recover an
older result by G. Nenciu who proved Lipschitz regularity up to a logarithmic
factor for general globally bounded magnetic field perturbations.Comment: 18 pages, submitte
On the Verdet constant and Faraday rotation for graphene-like materials
We present a rigorous and rather self-contained analysis of the Verdet
constant in graphene- like materials. We apply the gauge-invariant magnetic
perturbation theory to a nearest- neighbour tight-binding model and obtain a
relatively simple and exactly computable formula for the Verdet constant, at
all temperatures and all frequencies of sufficiently large absolute value.
Moreover, for the standard nearest neighbour tight-binding model of graphene we
show that the transverse component of the conductivity tensor has an asymptotic
Taylor expansion in the external magnetic field where all the coefficients of
even powers are zero.Comment: 23 pages, 4 figures, revised versio
On the skeleton method and an application to a quantum scissor
In the spectral analysis of few one dimensional quantum particles interacting
through delta potentials it is well known that one can recast the problem into
the spectral analysis of an integral operator (the skeleton) living on the
submanifold which supports the delta interactions. We shall present several
tools which allow direct insight into the spectral structure of this skeleton.
We shall illustrate the method on a model of a two dimensional quantum particle
interacting with two infinitely long straight wires which cross one another at
a certain angle : the quantum scissor.Comment: Submitte
Metastable states when the Fermi Golden Rule constant vanishes
Resonances appearing by perturbation of embedded non-degenerate eigenvalues
are studied in the case when the Fermi Golden Rule constant vanishes. Under
appropriate smoothness properties for the resolvent of the unperturbed
Hamiltonian, it is proved that the first order Rayleigh-Schr\"odinger expansion
exists. The corresponding metastable states are constructed using this
truncated expansion. We show that their exponential decay law has both the
decay rate and the error term of order , where is
the perturbation strength.Comment: To appear in Commun. Math. Phy
Parseval frames of exponentially localized magnetic Wannier functions
Motivated by the analysis of gapped periodic quantum systems in presence of a
uniform magnetic field in dimension , we study the possibility to
construct spanning sets of exponentially localized (generalized) Wannier
functions for the space of occupied states. When the magnetic flux per unit
cell satisfies a certain rationality condition, by going to the momentum-space
description one can model occupied energy bands by a real-analytic and
-periodic family of
orthogonal projections of rank . A moving orthonormal basis of consisting of real-analytic and -periodic Bloch
vectors can be constructed if and only if the first Chern number(s) of
vanish(es). Here we are mainly interested in the topologically obstructed case.
First, by dropping the generating condition, we show how to algorithmically
construct a collection of orthonormal, real-analytic, and periodic Bloch
vectors. Second, by dropping the linear independence condition, we construct a
Parseval frame of real-analytic and periodic Bloch vectors which generate
. Both algorithms are based on a two-step logarithm
method which produces a moving orthonormal basis in the topologically trivial
case. A moving Parseval frame of analytic, periodic Bloch vectors corresponds
to a Parseval frame of exponentially localized composite Wannier functions. We
extend this construction to the case of magnetic Hamiltonians with an
irrational magnetic flux per unit cell and show how to produce Parseval frames
of exponentially localized generalized Wannier functions also in this setting.
Our results are illustrated in crystalline insulators modelled by discrete
Hofstadter-like Hamiltonians, but apply to certain continuous models of
magnetic Schr\"{o}dinger operators as well.Comment: 40 pages. Improved exposition and minor corrections. Final version
matches published paper on Commun. Math. Phy
Beyond Diophantine Wannier diagrams: Gap labelling for Bloch-Landau Hamiltonians
It is well known that, given a purely magnetic Landau Hamiltonian with a
constant magnetic field which generates a magnetic flux per unit
area, then any spectral island consisting of infinitely
degenerate Landau levels carries an integrated density of states
. Wannier later discovered a similar Diophantine
relation expressing the integrated density of states of a gapped group of bands
of the Hofstadter Hamiltonian as a linear function of the magnetic field flux
with integer slope.
We extend this result to a gap labelling theorem for any Bloch-Landau
operator which also has a bounded -periodic electric
potential. Assume that has a spectral island which remains
isolated from the rest of the spectrum as long as lies in a compact
interval . Then on such
intervals, where the constant while .
The integer is the Chern marker of the spectral projection onto the
spectral island . This result also implies that the Fermi projection
on , albeit continuous in in the strong topology, is nowhere
continuous in the norm topology if either or and is
rational.
Our proofs, otherwise elementary, do not use non-commutative geometry but are
based on gauge covariant magnetic perturbation theory which we briefly review
for the sake of the reader. Moreover, our method allows us to extend the
analysis to certain non-covariant systems having slowly varying magnetic
fields.Comment: 20 pages, no figures. Appendix C added. Final version accepted for
publication in Journal of the European Mathematical Societ
A rigorous proof of the Landau-Peierls formula and much more
We present a rigorous mathematical treatment of the zero-field orbital
magnetic susceptibility of a non-interacting Bloch electron gas, at fixed
temperature and density, for both metals and semiconductors/insulators. In
particular, we obtain the Landau-Peierls formula in the low temperature and
density limit as conjectured by T. Kjeldaas and W. Kohn in 1957.Comment: 30 pages - Accepted for publication in A.H.
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