20 research outputs found
Re-examining Bogoliubov's theory of an interacting Bose gas
As is well-known, in the conventional formulation of Bogoliubov's theory of
an interacting Bose gas, the Hamiltonian is written as a decoupled
sum of contributions from different momenta of the form . Then, each of the single-mode Hamiltonians is
diagonalized separately, and the resulting ground state wavefunction of the
total Hamiltonian is written as a simple product of the ground state
wavefunctions of each of the single-mode Hamiltonians . We argue
that, from a number-conserving perspective, this diagonalization method may not
be adequate since the true Hilbert spaces where the Hamiltonians
should be diagonalized all have the state in common,
and hence the ground state wavefunction of the total Hamiltonian may
{not} be written as a simple product of the ground state wavefunctions of the
's. In this paper, we give a thorough review of Bogoliubov's
method, and discuss a variational and number-conserving formulation of this
theory in which the state is restored to the Hilbert space of the
interacting gas, and where, instead of diagonalizing the Hamiltonians
separately, we diagonalize the total Hamiltonian as a
whole. When this is done, we find that the ground state energy is lowered below
the Bogoliubov result, and the depletion of bosons is significantly reduced
with respect to the one obtained in the number non-conserving treatment. We
also find that the spectrum of the usual excitations of
Bogoliubov's method changes from a gapless one, as predicted by the standard,
number non-conserving formulation of this theory, to one which exhibits a
finite gap in the limit.Comment: Published version, 81 pages, 16 figure
Anisotropic states of two-dimensional electrons in high magnetic fields
We study the collective states formed by two-dimensional electrons in Landau
levels of index near half-filling. By numerically solving the
self-consistent Hartree-Fock (HF) equations for a set of oblique
two-dimensional lattices, we find that the stripe state is an anisotropic
Wigner crystal (AWC), and determine its precise structure for varying values of
the filling factor. Calculating the elastic energy, we find that the shear
modulus of the AWC is small but finite (nonzero) within the HF approximation.
This implies, in particular, that the long-wavelength magnetophonon mode in the
stripe state vanishes like as in an ordinary Wigner crystal, and not
like as was found in previous studies where the energy of shear
deformations was neglected.Comment: minor corrections; 5 pages, 4 figures; version to be published in
Physical Review Letter
"Soft" Anharmonic Vortex Glass in Ferromagnetic Superconductors
Ferromagnetic order in superconductors can induce a {\em spontaneous} vortex
(SV) state. For external field , rotational symmetry guarantees a
vanishing tilt modulus of the SV solid, leading to drastically different
behavior than that of a conventional, external-field-induced vortex solid. We
show that quenched disorder and anharmonic effects lead to elastic moduli that
are wavevector-dependent out to arbitrarily long length scales, and non-Hookean
elasticity. The latter implies that for weak external fields , the magnetic
induction scales {\em universally} like , with
. For weak disorder, we predict the SV solid is a
topologically ordered vortex glass, in the ``columnar elastic glass''
universality class.Comment: minor corrections; version published in PR
Elasticity, fluctuations and vortex pinning in ferromagnetic superconductors: A "columnar elastic glass"
We study the elasticity, fluctuations and pinning of a putative spontaneous
vortex solid in ferromagnetic superconductors. Using a rigorous thermodynamic
argument, we show that in the idealized case of vanishing crystalline pinning
anisotropy the long-wavelength tilt modulus of such a vortex solid vanishes
identically, as guaranteed by the underlying rotational invariance. The
vanishing of the tilt modulus means that, to lowest order, the associated
tension elasticity is replaced by the softer, curvature elasticity. The effect
of this is to make the spontaneous vortex solid qualitatively more susceptible
to the disordering effects of thermal fluctuations and random pinning. We study
these effects, taking into account the nonlinear elasticity, that, in three
dimensions, is important at sufficiently long length scales, and showing that a
``columnar elastic glass'' phase of vortices results. This phase is controlled
by a previously unstudied zero-temperature fixed point and it is characterized
by elastic moduli that have universal strong wave-vector dependence out to
arbitrarily long length scales, leading to non-Hookean elasticity. We argue
that, although translationally disordered for weak disorder, the columnar
elastic glass is stable against the proliferation of dislocations and is
therefore a topologically ordered {\em elastic} glass. As a result, the
phenomenology of the spontaneous vortex state of isotropic magnetic
superconductors differs qualitatively from a conventional,
external-field-induced mixed state. For example, for weak external fields ,
the magnetic induction scales {\em universally} like , with .Comment: Minor editorial changes, version to be published in PRB, 39 pages, 7
figure
Static and dynamic properties of crystalline phases of two-dimensional electrons in a strong magnetic field
We study the cohesive energy and elastic properties as well as normal modes
of the Wigner and bubble crystals of the two-dimensional electron system (2DES)
in higher Landau levels. Using a simple Hartree-Fock approach, we show that the
shear moduli ('s) of these electronic crystals show a non-monotonic
behavior as a function of the partial filling factor at any given
Landau level, with increasing for small values of , before
reaching a maximum at some intermediate filling factor , and
monotonically decreasing for . We also go beyond previous
treatments, and study how the phase diagram and elastic properties of electron
solids are changed by the effects of screening by electrons in lower Landau
levels, and by a finite thickness of the experimental sample. The implications
of these results on microwave resonance experiments are briefly discussed.Comment: Discussion updated - 16 pages, 10 figures; version accepted for
publication in Phys. Rev.
Dynamical matrix of two-dimensional electron crystals
In a quantizing magnetic field, the two-dimensional electron (2DEG) gas has a
rich phase diagram with broken translational symmetry phases such as Wigner,
bubble, and stripe crystals. In this paper, we derive a method to get the
dynamical matrix of these crystals from a calculation of the density response
function performed in the Generalized Random Phase Approximation (GRPA). We
discuss the validity of our method by comparing the dynamical matrix calculated
from the GRPA with that obtained from standard elasticity theory with the
elastic coefficients obtained from a calculation of the deformation energy of
the crystal.Comment: Revised version published in Phys. Rev. B. 12 pages with 11
postscripts figure
Variational theory of flux-line liquids
We formulate a variational (Hartree like) description of flux line liquids
which improves on the theory we developed in an earlier paper [A.M. Ettouhami,
Phys. Rev. B 65, 134504 (2002)]. We derive, in particular, how the massive term
confining the fluctuations of flux lines varies with temperature and show that
this term vanishes at high enough temperatures where the vortices behave as
freely fluctuating elastic lines.Comment: 10 pages, 1 postscript figur