6,032 research outputs found
New Optimization Methods for Converging Perturbative Series with a Field Cutoff
We take advantage of the fact that in lambda phi ^4 problems a large field
cutoff phi_max makes perturbative series converge toward values exponentially
close to the exact values, to make optimal choices of phi_max. For perturbative
series terminated at even order, it is in principle possible to adjust phi_max
in order to obtain the exact result. For perturbative series terminated at odd
order, the error can only be minimized. It is however possible to introduce a
mass shift in order to obtain the exact result. We discuss weak and strong
coupling methods to determine the unknown parameters. The numerical
calculations in this article have been performed with a simple integral with
one variable. We give arguments indicating that the qualitative features
observed should extend to quantum mechanics and quantum field theory. We found
that optimization at even order is more efficient that at odd order. We compare
our methods with the linear delta-expansion (LDE) (combined with the principle
of minimal sensitivity) which provides an upper envelope of for the accuracy
curves of various Pade and Pade-Borel approximants. Our optimization method
performs better than the LDE at strong and intermediate coupling, but not at
weak coupling where it appears less robust and subject to further improvements.
We also show that it is possible to fix the arbitrary parameter appearing in
the LDE using the strong coupling expansion, in order to get accuracies
comparable to ours.Comment: 10 pages, 16 figures, uses revtex; minor typos corrected, refs. adde
Critical Behaviour of Structure Factors at a Quantum Phase Transition
We review the theoretical behaviour of the total and one-particle structure
factors at a quantum phase transition for temperature T=0. The predictions are
compared with exact or numerical results for the transverse Ising model, the
alternating Heisenberg chain, and the bilayer Heisenberg model. At the critical
wavevector, the results are generally in accord with theoretical expectations.
Away from the critical wavevector, however, different models display quite
different behaviours for the one-particle residues and structure factors.Comment: 17 pp, 10 figure
Large-n expansion for m-axial Lifshitz points
The large-n expansion is developed for the study of critical behaviour of
d-dimensional systems at m-axial Lifshitz points with an arbitrary number m of
modulation axes. The leading non-trivial contributions of O(1/n) are derived
for the two independent correlation exponents \eta_{L2} and \eta_{L4}, and the
related anisotropy index \theta. The series coefficients of these 1/n
corrections are given for general values of m and d with 0<m<d and
2+m/2<d<4+m/2 in the form of integrals. For special values of m and d such as
(m,d)=(1,4), they can be computed analytically, but in general their evaluation
requires numerical means. The 1/n corrections are shown to reduce in the
appropriate limits to those of known large-n expansions for the case of
d-dimensional isotropic Lifshitz points and critical points, respectively, and
to be in conformity with available dimensionality expansions about the upper
and lower critical dimensions. Numerical results for the 1/n coefficients of
\eta_{L2}, \eta_{L4} and \theta are presented for the physically interesting
case of a uniaxial Lifshitz point in three dimensions, as well as for some
other choices of m and d. A universal coefficient associated with the
energy-density pair correlation function is calculated to leading order in 1/n
for general values of m and d.Comment: 28 pages, 3 figures. Submitted to: J. Phys. C: Solid State Phys.,
special issue dedicated to Lothar Schaefer on the occasion of his 60th
birthday. V2: References added along with corresponding modifications in the
text, corrected figure 3, corrected typo
On the Divergence of Perturbation Theory. Steps Towards a Convergent Series
The mechanism underlying the divergence of perturbation theory is exposed.
This is done through a detailed study of the violation of the hypothesis of the
Dominated Convergence Theorem of Lebesgue using familiar techniques of Quantum
Field Theory. That theorem governs the validity (or lack of it) of the formal
manipulations done to generate the perturbative series in the functional
integral formalism. The aspects of the perturbative series that need to be
modified to obtain a convergent series are presented. Useful tools for a
practical implementation of these modifications are developed. Some resummation
methods are analyzed in the light of the above mentioned mechanism.Comment: 42 pages, Latex, 4 figure
Universality class of 3D site-diluted and bond-diluted Ising systems
We present a finite-size scaling analysis of high-statistics Monte Carlo
simulations of the three-dimensional randomly site-diluted and bond-diluted
Ising model. The critical behavior of these systems is affected by
slowly-decaying scaling corrections which make the accurate determination of
their universal asymptotic behavior quite hard, requiring an effective control
of the scaling corrections. For this purpose we exploit improved Hamiltonians,
for which the leading scaling corrections are suppressed for any thermodynamic
quantity, and improved observables, for which the leading scaling corrections
are suppressed for any model belonging to the same universality class.
The results of the finite-size scaling analysis provide strong numerical
evidence that phase transitions in three-dimensional randomly site-diluted and
bond-diluted Ising models belong to the same randomly dilute Ising universality
class. We obtain accurate estimates of the critical exponents, ,
, , , ,
, and of the leading and next-to-leading correction-to-scaling
exponents, and .Comment: 45 pages, 22 figs, revised estimate of n
Critical Casimir effect in films for generic non-symmetry-breaking boundary conditions
Systems described by an O(n) symmetrical Hamiltonian are considered
in a -dimensional film geometry at their bulk critical points. A detailed
renormalization-group (RG) study of the critical Casimir forces induced between
the film's boundary planes by thermal fluctuations is presented for the case
where the O(n) symmetry remains unbroken by the surfaces. The boundary planes
are assumed to cause short-ranged disturbances of the interactions that can be
modelled by standard surface contributions corresponding
to subcritical or critical enhancement of the surface interactions. This
translates into mesoscopic boundary conditions of the generic
symmetry-preserving Robin type .
RG-improved perturbation theory and Abel-Plana techniques are used to compute
the -dependent part of the reduced excess free energy per
film area to two-loop order. When , it takes the scaling
form as
, where are scaling fields associated with the
surface-enhancement variables , while is a standard
surface crossover exponent. The scaling function
and its analogue for the Casimir force
are determined via expansion in and extrapolated to
dimensions. In the special case , the expansion
becomes fractional. Consistency with the known fractional expansions of D(0,0)
and to order is achieved by appropriate
reorganisation of RG-improved perturbation theory. For appropriate choices of
and , the Casimir forces can have either sign. Furthermore,
crossovers from attraction to repulsion and vice versa may occur as
increases.Comment: Latex source file, 40 pages, 9 figure
On the Dominance of Trivial Knots among SAPs on a Cubic Lattice
The knotting probability is defined by the probability with which an -step
self-avoiding polygon (SAP) with a fixed type of knot appears in the
configuration space. We evaluate these probabilities for some knot types on a
simple cubic lattice. For the trivial knot, we find that the knotting
probability decays much slower for the SAP on the cubic lattice than for
continuum models of the SAP as a function of . In particular the
characteristic length of the trivial knot that corresponds to a `half-life' of
the knotting probability is estimated to be on the cubic
lattice.Comment: LaTeX2e, 21 pages, 8 figur
Absence of a Spin Liquid Phase in the Hubbard Model on the Honeycomb Lattice
A spin liquid is a novel quantum state of matter with no conventional order
parameter where a finite charge gap exists even though the band theory would
predict metallic behavior. Finding a stable spin liquid in two or higher
spatial dimensions is one of the most challenging and debated issues in
condensed matter physics. Very recently, it has been reported that a model of
graphene, i.e., the Hubbard model on the honeycomb lattice, can show a spin
liquid ground state in a wide region of the phase diagram, between a semi-metal
(SM) and an antiferromagnetic insulator (AFMI). Here, by performing numerically
exact quantum Monte Carlo simulations, we extend the previous study to much
larger clusters (containing up to 2592 sites), and find, if any, a very weak
evidence of this spin liquid region. Instead, our calculations strongly
indicate a direct and continuous quantum phase transition between SM and AFMI.Comment: 15 pages with 7 figures and 9 tables including supplementary
information, accepted for publication in Scientific Report
Critical adsorption near edges
Symmetry breaking surface fields give rise to nontrivial and long-ranged
order parameter profiles for critical systems such as fluids, alloys or magnets
confined to wedges. We discuss the properties of the corresponding universal
scaling functions of the order parameter profile and the two-point correlation
function and determine the critical exponents eta_parallel and
eta_perpendicular for the so-called normal transition.Comment: 22 pages, 5 figures, accepted for publication in PR
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