6,194 research outputs found
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
Spectral functions and time evolution from the Chebyshev recursion
We link linear prediction of Chebyshev and Fourier expansions to analytic
continuation. We push the resolution in the Chebyshev-based computation of
many-body spectral functions to a much higher precision by deriving a
modified Chebyshev series expansion that allows to reduce the expansion order
by a factor . We show that in a certain limit the Chebyshev
technique becomes equivalent to computing spectral functions via time evolution
and subsequent Fourier transform. This introduces a novel recursive time
evolution algorithm that instead of the group operator only involves
the action of the generator . For quantum impurity problems, we introduce an
adapted discretization scheme for the bath spectral function. We discuss the
relevance of these results for matrix product state (MPS) based DMRG-type
algorithms, and their use within dynamical mean-field theory (DMFT). We present
strong evidence that the Chebyshev recursion extracts less spectral information
from than time evolution algorithms when fixing a given amount of created
entanglement.Comment: 12 pages + 6 pages appendix, 11 figure
Introspective Pushdown Analysis of Higher-Order Programs
In the static analysis of functional programs, pushdown flow analysis and
abstract garbage collection skirt just inside the boundaries of soundness and
decidability. Alone, each method reduces analysis times and boosts precision by
orders of magnitude. This work illuminates and conquers the theoretical
challenges that stand in the way of combining the power of these techniques.
The challenge in marrying these techniques is not subtle: computing the
reachable control states of a pushdown system relies on limiting access during
transition to the top of the stack; abstract garbage collection, on the other
hand, needs full access to the entire stack to compute a root set, just as
concrete collection does. \emph{Introspective} pushdown systems resolve this
conflict. Introspective pushdown systems provide enough access to the stack to
allow abstract garbage collection, but they remain restricted enough to compute
control-state reachability, thereby enabling the sound and precise product of
pushdown analysis and abstract garbage collection. Experiments reveal
synergistic interplay between the techniques, and the fusion demonstrates
"better-than-both-worlds" precision.Comment: Proceedings of the 17th ACM SIGPLAN International Conference on
Functional Programming, 2012, AC
A Check of a D=4 Field-Theoretical Calculation Using the High-Temperature Expansion for Dyson's Hierarchical Model
We calculate the high-temperature expansion of the 2-point function up to
order 800 in beta. We show that estimations of the critical exponent gamma
based on asymptotic analysis are not very accurate in presence of confluent
logarithmic singularities. Using a direct comparison between the actual series
and the series obtained from a parametrization of the form (beta_c
-beta)^(-gamma) (Ln(beta_c -beta))^p +r), we show that the errors are minimized
for gamma =0.9997 and p=0.3351, in very good agreement with field-theoretical
calculations. We briefly discuss the related questions of triviality and
hyperscalingComment: Uses Revtex, 27 pages including 13 figure
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
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