10,716 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\'
Index Information Algorithm with Local Tuning for Solving Multidimensional Global Optimization Problems with Multiextremal Constraints
Multidimensional optimization problems where the objective function and the
constraints are multiextremal non-differentiable Lipschitz functions (with
unknown Lipschitz constants) and the feasible region is a finite collection of
robust nonconvex subregions are considered. Both the objective function and the
constraints may be partially defined. To solve such problems an algorithm is
proposed, that uses Peano space-filling curves and the index scheme to reduce
the original problem to a H\"{o}lder one-dimensional one. Local tuning on the
behaviour of the objective function and constraints is used during the work of
the global optimization procedure in order to accelerate the search. The method
neither uses penalty coefficients nor additional variables. Convergence
conditions are established. Numerical experiments confirm the good performance
of the technique.Comment: 29 pages, 5 figure
Testing a quintessence model with CMBR peaks locations
We show that a model of quintessence with exponential potential,
which allows to obtain general exact solutions, can generate location of
CMBR peaks which are fully compatible with present observational data
Testing a quintessence model with CMBR peaks location
We show that a model of quintessence with exponential potential, which allows
to obtain general exact solutions, can generate locations of CMBR peaks which
are fully compatible with present observational dataComment: 7 pages, no figure
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
Phase behavior of polydisperse sticky hard spheres: analytical solutions and perturbation theory
We discuss phase coexistence of polydisperse colloidal suspensions in the
presence of adhesion forces. The combined effect of polydispersity and Baxter's
sticky-hard-sphere (SHS) potential, describing hard spheres interacting via
strong and very short-ranged attractive forces, give rise, within the
Percus-Yevick (PY) approximation, to a system of coupled quadratic equations
which, in general, cannot be solved either analytically or numerically. We
review and compare two recent alternative proposals, which we have attempted to
by-pass this difficulty. In the first one, truncating the density expansion of
the direct correlation functions, we have considered approximations simpler
than the PY one. These approximations can be systematically improved.
We have been able to provide a complete analytical description of polydisperse
SHS fluids by using the simplest two orders and , respectively.
Such a simplification comes at the price of a lower accuracy in the phase
diagram, but has the advantage of providing an analytical description of
various new phenomena associated with the onset of polydispersity in phase
equilibria (e.g. fractionation). The second approach is based on a perturbative
expansion of the polydisperse PY solution around its monodisperse counterpart.
This approach provides a sound approximation to the real phase behavior, at the
cost of considering only weak polydispersity. Although a final seattlement on
the soundness of the latter method would require numerical simulations for the
polydisperse Baxter model, we argue that this approach is expected to keep
correctly into account the effects of polydispersity, at least qualitatively.Comment: 12 pages, 4 figures, to appear in Molec. Phys. special issue Liblice
200
On the compressibility equation of state for multicomponent adhesive hard sphere fluids
The compressibility equation of state for a multicomponent fluid of particles
interacting via an infinitely narrow and deep potential, is considered within
the mean spherical approximation (MSA). It is shown that for a class of models
leading to a particular form of the Baxter functions containing
density-independent stickiness coefficient, the compressibility EOS does not
exist, unlike the one-component case. The reason for this is that a direct
integration of the compressibility at fixed composition, cannot be carried out
due to the lack of a reciprocity relation on the second order partial
derivatives of the pressure with respect to two different densities. This is,
in turn, related to the inadequacy of the MSA. A way out to this drawback is
presented in a particular example, leading to a consistent compressibility
pressure, and a possible generalization of this result is discussed.Comment: 13 pages, no figures, accepted for publication Molec. Physics (2002
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