39 research outputs found
Systematic Speedup of Path Integrals of a Generic -fold Discretized Theory
We present and discuss a detailed derivation of a new analytical method that
systematically improves the convergence of path integrals of a generic -fold
discretized theory. We develop an explicit procedure for calculating a set of
effective actions , for which have the property that
they lead to the same continuum amplitudes as the starting action, but that
converge to that continuum limit ever faster. Discretized amplitudes calculated
using the level effective action differ from the continuum limit by a term
of order . We obtain explicit expressions for the effective actions for
levels . We end by analyzing the speedup of Monte Carlo simulations of
two different models: an anharmonic oscillator with quartic coupling and a
particle in a modified P\"oschl-Teller potential.Comment: 10 pages, 5 figures, biblio info correcte
Generalization of Euler's Summation Formula to Path Integrals
A recently developed analytical method for systematic improvement of the
convergence of path integrals is used to derive a generalization of Euler's
summation formula for path integrals. The first terms in this formula
improve convergence of path integrals to the continuum limit from 1/N to
, where is the coarseness of the discretization. Monte Carlo
simulations performed on several different models show that the analytically
derived speedup holds.Comment: 12 pages, 1 figure, uses elsart.cls, ref. [15] resolve
Jaggedness of Path Integral Trajectories
We define and investigate the properties of the jaggedness of path integral
trajectories. The new quantity is shown to be scale invariant and to satisfy a
self-averaging property. Jaggedness allows for a classification of path
integral trajectories according to their relevance. We show that in the
continuum limit the only paths that are not of measure zero are those with
jaggedness 1/2, i.e. belonging to the same equivalence class as random walks.
The set of relevant trajectories is thus narrowed down to a specific subset of
non-differentiable paths. For numerical calculations, we show that jaggedness
represents an important practical criterion for assessing the quality of
trajectory generating algorithms. We illustrate the obtained results with Monte
Carlo simulations of several different models.Comment: 11 pages, 7 figures, uses elsart.cl
Dipolar Bose-Einstein Condensates in Weak Anisotropic Disorder
Here we study properties of a homogeneous dipolar Bose-Einstein condensate in
a weak anisotropic random potential with Lorentzian correlation at zero
temperature. To this end we solve perturbatively the Gross-Pitaevskii equation
to second order in the random potential strength and obtain analytic results
for the disorder ensemble averages of both the condensate and the superfluid
depletion, the equation of state, and the sound velocity. For a pure contact
interaction and a vanishing correlation length, we reproduce the seminal
results of Huang and Meng, which were originally derived within a Bogoliubov
theory around a disorder-averaged background field. For dipolar interaction and
isotropic Lorentzian-correlated disorder, we obtain results which are
qualitatively similar to the case of an isotropic Gaussian-correlated disorder.
In the case of an anisotropic disorder, the physical observables show
characteristic anisotropies which arise from the formation of fragmented
dipolar condensates in the local minima of the disorder potential.Comment: 16 pages, 7 figure
Asymptotic Properties of Path Integral Ideals
We introduce and analyze a new quantity, the path integral ideal, governing
the flow of generic discrete theories to the continuum limit and greatly
increasing their convergence. The said flow is classified according to the
degree of divergence of the potential at spatial infinity. Studying the
asymptotic behavior of path integral ideals we isolate the dominant terms in
the effective potential that determine the behavior of a generic theory for
large discrete time steps.Comment: 4 pages, 1 figure, revte
Stability of quantum degenerate Fermi gases of tilted polar molecules
A recent experimental realization of quantum degenerate gas of
KRb molecules opens up prospects of exploring strong dipolar
Fermi gases and many-body phenomena arising in that regime. Here we derive a
mean-field variational approach based on the Wigner function for the
description of ground-state properties of such systems. We show that the
stability of dipolar fermions in a general harmonic trap is universal as it
only depends on the trap aspect ratios and the dipoles' orientation. We
calculate the species-independent stability diagram and the deformation of the
Fermi surface (FS) for polarized molecules, whose electric dipoles are oriented
along a preferential direction. Compared to atomic magnetic species, the
stability of a molecular electric system turns out to strongly depend on its
geometry and the FS deformation significantly increases.Comment: 6 + 5 pages, 5 + 3 figures; accepted for publication in Phys. Rev.
Researc
Energy Estimators and Calculation of Energy Expectation Values in the Path Integral Formalism
A recently developed method, introduced in Phys. Rev. Lett. 94 (2005) 180403,
Phys. Rev. B 72 (2005) 064302, Phys. Lett. A 344 (2005) 84, systematically
improved the convergence of generic path integrals for transition amplitudes.
This was achieved by analytically constructing a hierarchy of -fold
discretized effective actions labeled by a whole number and
starting at from the naively discretized action in the mid-point
prescription. The derivation guaranteed that the level effective actions
lead to discretized transition amplitudes differing from the continuum limit by
a term of order . Here we extend the applicability of the above method
to the calculation of energy expectation values. This is done by constructing
analytical expressions for energy estimators of a general theory for each level
. As a result of this energy expectation values converge to the continuum as
. Finally, we perform a series of Monte Carlo simulations of several
models, show explicitly the derived increase in convergence, and the ensuing
speedup in numerical calculation of energy expectation values of many orders of
magnitude.Comment: 12 pages, 3 figures, 1 appendix, uses elsart.cl
Improved Gaussian Approximation
In a recently developed approximation technique for quantum field theory the
standard one-loop result is used as a seed for a recursive formula that gives a
sequence of improved Gaussian approximations for the generating functional. In
this paper we work with the generic model in dimensions.
We compare the first, and simplest, approximation in the above sequence with
the one-loop and two-loop approximations, as well as the exact results
(calculated numericaly).Comment: 5 pages, Latex 2e, 7 figures, Lectures given at 11th Yugoslav
Symposium on Nuclear and Particle Physics, Studenica, September 199
Efficient Calculation of Energy Spectra Using Path Integrals
A newly developed method for systematically improving the convergence of path
integrals for transition amplitudes, introduced in Phys. Rev. Lett. 94 (2005)
180403, Phys. Rev. B 72 (2005) 064302, Phys. Lett. A 344 (2005) 84, and
expectation values, introduced in Phys. Lett. A 360 (2006) 217, is here applied
to the efficient calculation of energy spectra. We show how the derived
hierarchies of effective actions lead to substantial speedup of the standard
path integral Monte Carlo evaluation of energy levels. The general results and
the ensuing increase in efficiency of several orders of magnitude are shown
using explicit Monte Carlo simulations of several distinct models.Comment: 11 pages, 4 figures, 2 tables, uses elsart.cl
An Improved Gaussian Approximation for Quantum Field Theory
We present a new approximation technique for quantum field theory. The
standard one-loop result is used as a seed for a recursive formula that gives a
sequence of improved Gaussian approximations for the generating functional. In
a different setting, the basic idea of this recursive scheme is used in the
second part of the paper to substantialy speed up the standard Monte Carlo
algorithm.Comment: 9 pages, Latex 2e, 4 figure