171 research outputs found
A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory
The goal of this article is to provide a practical method to calculate, in a
scalar theory, accurate numerical values of the renormalized quantities which
could be used to test any kind of approximate calculation. We use finite
truncations of the Fourier transform of the recursion formula for Dyson's
hierarchical model in the symmetric phase to perform high-precision
calculations of the unsubtracted Green's functions at zero momentum in
dimension 3, 4, and 5. We use the well-known correspondence between statistical
mechanics and field theory in which the large cut-off limit is obtained by
letting beta reach a critical value beta_c (with up to 16 significant digits in
our actual calculations). We show that the round-off errors on the magnetic
susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the
systematic errors (finite truncations and volume) can be controlled with an
exponential precision and reduced to a level lower than the numerical errors.
We justify the use of the truncation for calculations of the high-temperature
expansion. We calculate the dimensionless renormalized coupling constant
corresponding to the 4-point function and show that when beta -> beta_c, this
quantity tends to a fixed value which can be determined accurately when D=3
(hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.Comment: Uses revtex with psfig, 31 pages including 15 figure
The Oscillatory Behavior of the High-Temperature Expansion of Dyson's Hierarchical Model: A Renormalization Group Analysis
We calculate 800 coefficients of the high-temperature expansion of the
magnetic susceptibility of Dyson's hierarchical model with a Landau-Ginzburg
measure. Log-periodic corrections to the scaling laws appear as in the case of
a Ising measure. The period of oscillation appears to be a universal quantity
given in good approximation by the logarithm of the largest eigenvalue of the
linearized RG transformation, in agreement with a possibility suggested by K.
Wilson and developed by Niemeijer and van Leeuwen. We estimate to be
1.300 (with a systematic error of the order of 0.002) in good agreement with
the results obtained with other methods such as the -expansion. We
briefly discuss the relationship between the oscillations and the zeros of the
partition function near the critical point in the complex temperature plane.Comment: 21 pages, 10 Postcript figures, latex file, uses revte
High-Accuracy Calculations of the Critical Exponents of Dyson's Hierarchical Model
We calculate the critical exponent gamma of Dyson's hierarchical model by
direct fits of the zero momentum two-point function, calculated with an Ising
and a Landau-Ginzburg measure, and by linearization about the Koch-Wittwer
fixed point. We find gamma= 1.299140730159 plus or minus 10^(-12). We extract
three types of subleading corrections (in other words, a parametrization of the
way the two-point function depends on the cutoff) from the fits and check the
value of the first subleading exponent from the linearized procedure. We
suggest that all the non-universal quantities entering the subleading
corrections can be calculated systematically from the non-linear contributions
about the fixed point and that this procedure would provide an alternative way
to introduce the bare parameters in a field theory model.Comment: 15 pages, 9 figures, uses revte
Non-Gaussian numerical errors versus mass hierarchy
We probe the numerical errors made in renormalization group calculations by
varying slightly the rescaling factor of the fields and rescaling back in order
to get the same (if there were no round-off errors) zero momentum 2-point
function (magnetic susceptibility). The actual calculations were performed with
Dyson's hierarchical model and a simplified version of it. We compare the
distributions of numerical values obtained from a large sample of rescaling
factors with the (Gaussian by design) distribution of a random number generator
and find significant departures from the Gaussian behavior. In addition, the
average value differ (robustly) from the exact answer by a quantity which is of
the same order as the standard deviation. We provide a simple model in which
the errors made at shorter distance have a larger weight than those made at
larger distance. This model explains in part the non-Gaussian features and why
the central-limit theorem does not apply.Comment: 26 pages, 7 figures, uses Revte
Evidence for Complex Subleading Exponents from the High-Temperature Expansion of the Hierarchical Ising Model
Using a renormalization group method, we calculate 800 high-temperature
coefficients of the magnetic susceptibility of the hierarchical Ising model.
The conventional quantities obtained from differences of ratios of coefficients
show unexpected smooth oscillations with a period growing logarithmically and
can be fitted assuming corrections to the scaling laws with complex exponents.Comment: 10 pages, Latex , uses revtex. 2 figures not included (hard copies
available on request
A Two-Parameter Recursion Formula For Scalar Field Theory
We present a two-parameter family of recursion formulas for scalar field
theory. The first parameter is the dimension . The second parameter
() allows one to continuously extrapolate between Wilson's approximate
recursion formula and the recursion formula of Dyson's hierarchical model. We
show numerically that at fixed , the critical exponent depends
continuously on . We suggest the use of the independence as a
guide to construct improved recursion formulas.Comment: 7 pages, uses Revtex, one Postcript figur
A Numerical Study of the Hierarchical Ising Model: High Temperature Versus Epsilon Expansion
We study numerically the magnetic susceptibility of the hierarchical model
with Ising spins () above the critical temperature and for two
values of the epsilon parameter. The integrations are performed exactly, using
recursive methods which exploit the symmetries of the model. Lattices with up
to sites have been used. Surprisingly, the numerical data can be fitted
very well with a simple power law of the form for the {\it whole} temperature range. The numerical values for
agree within a few percent with the values calculated with a high-temperature
expansion but show significant discrepancies with the epsilon-expansion. We
would appreciate comments about these results.Comment: 15 Pages, 12 Figures not included (hard copies available on request),
uses phyzzx.te
Unity of Supersymmetry Breaking Models
We examine the models with gauge group U(1)^{k-1}\times\prod_{i=1}^k SU(n_i),
which are obtained from decomposing the supersymmetry breaking model of
Affleck, Dine and Seiberg containing an antisymmetric tensor field. We note
that all of these models are distinct vacua of a single SU(N) gauge theory with
an adjoint superfield. The dynamics of this model may be analyzed using the
duality of Kutasov and Schwimmer and the deconfinement trick of Berkooz. This
analysis leads to a simple picture for supersymmetry breaking for k=2,
complementing that of previous work. We examine the flat directions of these
models, and give straightforward criteria for lifting them, explaining the
requisite peculiar form of the superpotential. For all cases with k>2, the
duality argument fails to give supersymmetry breaking dynamics, and we identify
a class of problematic flat directions, which we term 2m-baryons. We study in
some detail the requirements for lifting these directions, and uncover some
surprising facts regarding the relationship between R-symmetry and
supersymmetry breaking in models with several gauge groups.Comment: harvmac, 40 page
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