6,085 research outputs found
An exchange-correlation energy for a two-dimensional electron gas in a magnetic field
We present the results of a variational Monte Carlo calculation of the
exchange-correlation energy for a spin-polarized two-dimensional electron gas
in a perpendicular magnetic field. These energies are a necessary input to the
recently developed current-density functional theory. Landau-level mixing is
included in a variational manner, which gives the energy at finite density at
finite field, in contrast to previous approaches. Results are presented for the
exchange-correlation energy and excited-state gap at 1/7, 1/5, 1/3, 1,
and 2. We parameterize the results as a function of and in a form
convenient for current-density functional calculations.Comment: 36 pages, including 6 postscript figure
Minimization of a fractional perimeter-Dirichlet integral functional
We consider a minimization problem that combines the Dirichlet energy with
the nonlocal perimeter of a level set, namely \int_\Om |\nabla
u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big), with . We obtain
regularity results for the minimizers and for their free boundaries \p
\{u>0\} using blow-up analysis. We will also give related results about
density estimates, monotonicity formulas, Euler-Lagrange equations and
extension problems
Stable ground states and self-similar blow-up solutions for the gravitational Vlasov-Manev system
In this work, we study the orbital stability of steady states and the
existence of blow-up self-similar solutions to the so-called Vlasov-Manev (VM)
system. This system is a kinetic model which has a similar Vlasov structure as
the classical Vlasov-Poisson system, but is coupled to a potential in (Manev potential) instead of the usual gravitational potential in
, and in particular the potential field does not satisfy a Poisson
equation but a fractional-Laplacian equation. We first prove the orbital
stability of the ground states type solutions which are constructed as
minimizers of the Hamiltonian, following the classical strategy: compactness of
the minimizing sequences and the rigidity of the flow. However, in driving this
analysis, there are two mathematical obstacles: the first one is related to the
possible blow-up of solutions to the VM system, which we overcome by imposing a
sub-critical condition on the constraints of the variational problem. The
second difficulty (and the most important) is related to the nature of the
Euler-Lagrange equations (fractional-Laplacian equations) to which classical
results for the Poisson equation do not extend. We overcome this difficulty by
proving the uniqueness of the minimizer under equimeasurabilty constraints,
using only the regularity of the potential and not the fractional-Laplacian
Euler-Lagrange equations itself. In the second part of this work, we prove the
existence of exact self-similar blow-up solutions to the Vlasov-Manev equation,
with initial data arbitrarily close to ground states. This construction is
based on a suitable variational problem with equimeasurability constraint
Interactions and the Theta Term in One-Dimensional Gapped Systems
We study how the \theta -term is affected by interactions in certain
one-dimensional gapped systems that preserve charge-conjugation, parity, and
time-reversal invariance. We exploit the relation between the chiral anomaly of
a fermionic system and the classical shift symmetry of its bosonized dual. The
vacuum expectation value of the dual boson is identified with the value of the
\theta -term for the corresponding fermionic system. Two (related) examples
illustrate the identification. We first consider the massive Luttinger liquid
and find the \theta -term to be insensitive to the strength of the interaction.
Next, we study the continuum limit of the Heisenberg XXZ spin-1/2 chain,
perturbed by a second nearest-neighbor spin interaction. For a certain range of
the XXZ anisotropy, we find that we can tune between two distinct sets of
topological phases by varying the second nearest-neighbor coupling. In the
first, we find the standard vacua at \theta = 0, \pi, while the second contains
vacua that spontaneously break charge-conjugation and parity with fractional
\theta / \pi = 1/ 2, 3/2. We also study quantized pumping in both examples
following recent work.Comment: 17 pages, harvmac; v.2 typo corrected and slight re-wording
On Perturbation theory improved by Strong coupling expansion
In theoretical physics, we sometimes have two perturbative expansions of
physical quantity around different two points in parameter space. In terms of
the two perturbative expansions, we introduce a new type of smooth
interpolating function consistent with the both expansions, which includes the
standard Pad\'e approximant and fractional power of polynomial method
constructed by Sen as special cases. We point out that we can construct
enormous number of such interpolating functions in principle while the "best"
approximation for the exact answer of the physical quantity should be unique
among the interpolating functions. We propose a criterion to determine the
"best" interpolating function, which is applicable except some situations even
if we do not know the exact answer. It turns out that our criterion works for
various examples including specific heat in two-dimensional Ising model,
average plaquette in four-dimensional SU(3) pure Yang-Mills theory on lattice
and free energy in c=1 string theory at self-dual radius. We also mention
possible applications of the interpolating functions to system with phase
transition.Comment: 31+11 pages, 15 figures, 9 tables, 1 Mathematica file; v3: minor
correction
An operator splitting scheme for the fractional kinetic Fokker-Planck equation
In this paper, we develop an operator splitting scheme for the fractional
kinetic Fokker-Planck equation (FKFPE). The scheme consists of two phases: a
fractional diffusion phase and a kinetic transport phase. The first phase is
solved exactly using the convolution operator while the second one is solved
approximately using a variational scheme that minimizes an energy functional
with respect to a certain Kantorovich optimal transport cost functional. We
prove the convergence of the scheme to a weak solution to FKFPE. As a
by-product of our analysis, we also establish a variational formulation for a
kinetic transport equation that is relevant in the second phase. Finally, we
discuss some extensions of our analysis to more complex systems
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