623 research outputs found
Evaluation of the BCS Approximation for the Attractive Hubbard Model in One Dimension
The ground state energy and energy gap to the first excited state are
calculated for the attractive Hubbard model in one dimension using both the
Bethe Ansatz equations and the variational BCS wavefunction. Comparisons are
provided as a function of coupling strength and electron density. While the
ground state energies are always in very good agreement, the BCS energy gap is
sometimes incorrect by an order of magnitude, particularly at half-filling.
Finite size effects are also briefly discussed for cases where an exact
solution in the thermodynamic limit is not possible. In general, the BCS result
for the energy gap is poor compared to the exact result.Comment: 25 pages, 5 Postscript figure
Analytical results on quantum interference and magnetoconductance for strongly localized electrons in a magnetic field: Exact summation of forward-scattering paths
We study quantum interference effects on the transition strength for strongly
localized electrons hopping on 2D square and 3D cubic lattices in the presence
of a magnetic field B. These effects arise from the interference between phase
factors associated with different electron paths connecting two distinct sites.
For electrons confined on a square lattice, with and without disorder, we
obtain closed-form expressions for the tunneling probability, which determines
the conductivity, between two arbitrary sites by exactly summing the
corresponding phase factors of all forward-scattering paths connecting them. An
analytic field-dependent expression, valid in any dimension, for the
magnetoconductance (MC) is derived. A positive MC is clearly observed when
turning on the magnetic field. In 2D, when the strength of B reaches a certain
value, which is inversely proportional to twice the hopping length, the MC is
increased by a factor of two compared to that at zero field. We also
investigate transport on the much less-studied and experimentally important 3D
cubic lattice case, where it is shown how the interference patterns and the
small-field behavior of the MC vary according to the orientation of B. The
effect on the low-flux MC due to the randomness of the angles between the
hopping direction and the orientation of B is also examined analytically.Comment: 24 pages, RevTeX, 8 figures include
Role of Umklapp Processes in Conductivity of Doped Two-Leg Ladders
Recent conductivity measurements performed on the hole-doped two-leg ladder
material reveal an approximately linear
power law regime in the c-axis DC resistivity as a function of temperature for
. In this work, we employ a bosonic model to argue that umklapp processes
are responsible for this feature and for the high spectral weight in the
optical conductivity which occurs beyond the finite frequency Drude-like peak.
Including quenched disorder in our model allows us to reproduce experimental
conductivity and resistivity curves over a wide range of energies. We also
point out the differences between the effect of umklapp processes in a single
chain and in the two-leg ladder.Comment: 10 pages, 2 figure
Quantum phase transitions and collapse of the Mott gap in the dimensional half-filled Hubbard model
We study the low-energy asymptotics of the half-filled Hubbard model with a
circular Fermi surface in continuous dimensions, based on the
one-loop renormalization-group (RG) method. Peculiarity of the
dimensions is incorporated through the mathematica structure of the elementary
particle-partcile (PP) and particle-hole (PH) loops: infrared logarithmic
singularity of the PH loop is smeared for . The RG flows indicate
that a quantum phase transition (QPT) from a metallic phase to the Mott
insulator phase occurs at a finite on-site Coulomb repulsion for
. We also discuss effects of randomness.Comment: 12 pages, 10 eps figure
Analytical solutions for two heteronuclear atoms in a ring trap
We consider two heteronuclear atoms interacting with a short-range
potential and confined in a ring trap. By taking the Bethe-ansatz-type
wavefunction and considering the periodic boundary condition properly, we
derive analytical solutions for the heteronuclear system. The eigen-energies
represented in terms of quasi-momentums can then be determined by solving a set
of coupled equations. We present a number of results, which display different
features from the case of identical atoms. Our result can be reduced to the
well-known Lieb-Liniger solution when two interacting atoms have the same
masses.Comment: 6 pages, 6 figure
A new variational approach to the stability of gravitational systems
We consider the three dimensional gravitational Vlasov Poisson system which
describes the mechanical state of a stellar system subject to its own gravity.
A well-known conjecture in astrophysics is that the steady state solutions
which are nonincreasing functions of their microscopic energy are nonlinearly
stable by the flow. This was proved at the linear level by several authors
based on the pioneering work by Antonov in 1961. Since then, standard
variational techniques based on concentration compactness methods as introduced
by P.-L. Lions in 1983 have led to the nonlinear stability of subclasses of
stationary solutions of ground state type.
In this paper, inspired by pioneering works from the physics litterature
(Lynden-Bell 94, Wiechen-Ziegler-Schindler MNRAS 88, Aly MNRAS 89), we use the
monotonicity of the Hamiltonian under generalized symmetric rearrangement
transformations to prove that non increasing steady solutions are local
minimizer of the Hamiltonian under equimeasurable constraints, and extract
compactness from suitable minimizing sequences. This implies the nonlinear
stability of nonincreasing anisotropic steady states under radially symmetric
perturbations
Uniqueness and Nondegeneracy of Ground States for in
We prove uniqueness of ground state solutions for the
nonlinear equation in , where
and for and for . Here denotes the fractional Laplacian
in one dimension. In particular, we generalize (by completely different
techniques) the specific uniqueness result obtained by Amick and Toland for
and in [Acta Math., \textbf{167} (1991), 107--126]. As a
technical key result in this paper, we show that the associated linearized
operator is nondegenerate;
i.\,e., its kernel satisfies .
This result about proves a spectral assumption, which plays a central
role for the stability of solitary waves and blowup analysis for nonlinear
dispersive PDEs with fractional Laplacians, such as the generalized
Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page
Nonlinear Dynamical Stability of Newtonian Rotating White Dwarfs and Supermassive Stars
We prove general nonlinear stability and existence theorems for rotating star
solutions which are axi-symmetric steady-state solutions of the compressible
isentropic Euler-Poisson equations in 3 spatial dimensions. We apply our
results to rotating and non-rotating white dwarf, and rotating high density
supermassive (extreme relativistic) stars, stars which are in convective
equilibrium and have uniform chemical composition. This paper is a continuation
of our earlier work ([28])
Existence and Nonlinear Stability of Rotating Star Solutions of the Compressible Euler-Poisson Equations
We prove existence of rotating star solutions which are steady-state
solutions of the compressible isentropic Euler-Poisson (EP) equations in 3
spatial dimensions, with prescribed angular momentum and total mass. This
problem can be formulated as a variational problem of finding a minimizer of an
energy functional in a broader class of functions having less symmetry than
those functions considered in the classical Auchmuty-Beals paper. We prove the
nonlinear dynamical stability of these solutions with perturbations having the
same total mass and symmetry as the rotating star solution. We also prove local
in time stability of W^{1, \infty}(\RR^3) solutions where the perturbations
are entropy-weak solutions of the EP equations. Finally, we give a uniform (in
time) a-priori estimate for entropy-weak solutions of the EP equations
Hybrid Stars in a Strong Magnetic Field
We study the effects of high magnetic fields on the particle population and
equation of state of hybrid stars using an extended hadronic and quark SU(3)
non-linear realization of the sigma model. In this model the degrees of freedom
change naturally from hadrons to quarks as the density and/or temperature
increases. The effects of high magnetic fields and anomalous magnetic moment
are visible in the macroscopic properties of the star, such as mass, adiabatic
index, moment of inertia, and cooling curves. Moreover, at the same time that
the magnetic fields become high enough to modify those properties, they make
the star anisotropic.Comment: Revised version with updated reference
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