3,367 research outputs found
Stability of Relativistic Matter With Magnetic Fields
Stability of matter with Coulomb forces has been proved for non-relativistic
dynamics, including arbitrarily large magnetic fields, and for relativistic
dynamics without magnetic fields. In both cases stability requires that the
fine structure constant alpha be not too large. It was unclear what would
happen for both relativistic dynamics and magnetic fields, or even how to
formulate the problem clearly. We show that the use of the Dirac operator
allows both effects, provided the filled negative energy `sea' is defined
properly. The use of the free Dirac operator to define the negative levels
leads to catastrophe for any alpha, but the use of the Dirac operator with
magnetic field leads to stability.Comment: This is an announcement of the work in cond-mat/9610195 (LaTeX
Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields
The stability of matter composed of electrons and static nuclei is
investigated for a relativistic dynamics for the electrons given by a suitably
projected Dirac operator and with Coulomb interactions. In addition there is an
arbitrary classical magnetic field of finite energy. Despite the previously
known facts that ordinary nonrelativistic matter with magnetic fields, or
relativistic matter without magnetic fields is already unstable when the fine
structure constant, is too large it is noteworthy that the combination of the
two is still stable provided the projection onto the positive energy states of
the Dirac operator, which defines the electron, is chosen properly. A good
choice is to include the magnetic field in the definition. A bad choice, which
always leads to instability, is the usual one in which the positive energy
states are defined by the free Dirac operator. Both assertions are proved here.Comment: LaTeX fil
The TF Limit for Rapidly Rotating Bose Gases in Anharmonic Traps
Starting from the full many body Hamiltonian we derive the leading order
energy and density asymptotics for the ground state of a dilute, rotating Bose
gas in an anharmonic trap in the ` Thomas Fermi' (TF) limit when the
Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to
infinity. Although the many-body wave function is expected to have a
complicated phase, the leading order contribution to the energy can be computed
by minimizing a simple functional of the density alone
The Flux-Phase of the Half-Filled Band
The conjecture is verified that the optimum, energy minimizing magnetic flux
for a half-filled band of electrons hopping on a planar, bipartite graph is
per square plaquette. We require {\it only} that the graph has
periodicity in one direction and the result includes the hexagonal lattice
(with flux 0 per hexagon) as a special case. The theorem goes beyond previous
conjectures in several ways: (1) It does not assume, a-priori, that all
plaquettes have the same flux (as in Hofstadter's model); (2) A Hubbard type
on-site interaction of any sign, as well as certain longer range interactions,
can be included; (3) The conclusion holds for positive temperature as well as
the ground state; (4) The results hold in dimensions if there is
periodicity in directions (e.g., the cubic lattice has the lowest energy
if there is flux in each square face).Comment: 9 pages, EHL14/Aug/9
Improved Lieb-Oxford exchange-correlation inequality with gradient correction
We prove a Lieb-Oxford-type inequality on the indirect part of the Coulomb
energy of a general many-particle quantum state, with a lower constant than the
original statement but involving an additional gradient correction. The result
is similar to a recent inequality of Benguria, Bley and Loss, except that the
correction term is purely local, which is more usual in density functional
theory. In an appendix, we discuss the connection between the indirect energy
and the classical Jellium energy for constant densities. We show that they
differ by an explicit shift due to the long range of the Coulomb potential.Comment: Final version to appear in Physical Review A. Compared to the very
first version, this one contains an appendix discussing the link with the
Jellium proble
Ground State Energy of the Low Density Bose Gas
Now that the properties of low temperature Bose gases at low density, ,
can be examined experimentally it is appropriate to revisit some of the
formulas deduced by many authors 4-5 decades ago. One of these is that the
leading term in the energy/particle is , where is
the scattering length. Owing to the delicate and peculiar nature of bosonic
correlations, four decades of research have failed to establish this plausible
formula rigorously. The only known lower bound for the energy was found by
Dyson in 1957, but it was 14 times too small. The correct bound is proved here.Comment: 4 pages, Revtex, reference 12 change
The ground state of a general electron-phonon Hamiltonian is a spin singlet
The many-body ground state of a very general class of electron-phonon
Hamiltonians is proven to contain a spin singlet (for an even number of
electrons on a finite lattice). The phonons interact with the electronic system
in two different ways---there is an interaction with the local electronic
charge and there is a functional dependence of the electronic hopping
Hamiltonian on the phonon coordinates. The phonon potential energy may include
anharmonic terms, and the electron-phonon couplings and the hopping matrix
elements may be nonlinear functions of the phonon coordinates. If the hopping
Hamiltonian is assumed to have no phonon coordinate dependence, then the ground
state is also shown to be unique, implying that there are no ground-state level
crossings, and that the ground-state energy is an analytic function of the
parameters in the Hamiltonian. In particular, in a finite system any
self-trapping transition is a smooth crossover not accompanied by a
nonanalytical change in the ground state. The spin-singlet theorem applies to
the Su-Schrieffer-Heeger model and both the spin-singlet and uniqueness
theorems apply to the Holstein and attractive Hubbard models as special cases.
These results hold in all dimensions --- even on a general graph without
periodic lattice structure.Comment: 25 pages, no figures, plainte
Improved Mean-Field Scheme for the Hubbard Model
Ground state energies and on-site density-density correlations are calculated
for the 1-D Hubbard model using a linear combination of the Hubbard projection
operators. The mean-field coefficients in the resulting linearized Equations of
Motion (EOM) depend on both one-particle static expectation values as well as
static two-particle correlations. To test the model, the one particle
expectation values are determined self-consistently while using Lanczos
determined values for the two particle correlation terms. Ground state energies
and on-site density-density correlations are then compared as a function of
to the corresponding Lanczos values on a 12 site Hubbard chain for 1/2 and 5/12
fillings. To further demonstrate the validity of the technique, the static
correlation functions are also calculated using a similar EOM approach, which
ignores the effective vertex corrections for this problem, and compares those
results as well for a 1/2 filled chain. These results show marked improvement
over standard mean-field techniques.Comment: 10 pages, 3 figures, text and figures as one postscript file -- does
not need to be "TeX-ed". LA-UR-94-294
The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive
-function interaction can be rigorously derived via a scaling limit
from a dilute three-dimensional Bose gas with arbitrary repulsive interaction
potential of finite scattering length. For this purpose, we prove bounds on
both the eigenvalues and corresponding eigenfunctions of three-dimensional
bosons in strongly elongated traps and relate them to the corresponding
quantities in the Lieb-Liniger model. In particular, if both the scattering
length and the radius of the cylindrical trap go to zero, the
Lieb-Liniger model with coupling constant is derived. Our bounds
are uniform in in the whole parameter range , and apply
to the Hamiltonian for three-dimensional bosons in a spectral window of size
above the ground state energy.Comment: LaTeX2e, 19 page
Charged and spin-excitation gaps in half-filled strongly correlated electron systems: A rigorous result
By exploiting the particle-hole symmetries of the Hubbard model, the periodic
Anderson model and the Kondo lattice model at half-filling and applying a
generalized version of Lieb's spin-reflection positivity method, we show that
the charged gaps of these models are always larger than their spin excitation
gaps. This theorem confirms the previous results derived by either the
variational approach or the density renormalization group approach.Comment: 20 pages, no figur
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