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Andrea Pelissetto

Publications and source records attributed to Andrea Pelissetto.

At least 217 records · Page 12Linked to original sources
arXiv1996

Corrections to finite-size scaling in two-dimensional O(N) sigma-models

We have considered the corrections to the finite-size-scaling functions for a general class of $O(N)$ $σ$-models with two-spin interactions in two dimensions for $N=\infty$. We have computed the leading corrections finding that they generically behave as $(f(z) \log L + g(z))/L^2$ where $z = m(L) L$ and $m(L)$ is a mass scale; $f(z)$ vanishes for Symanzik improved actions for which the inverse propagator behaves as $q^2 + O(q^6)$ for small $q$, but not for on-shell improved ones. We also discuss a model with four-spin interactions which shows a much more complicated behaviour.

hep-lat↗
arXiv1996

How to compute one-loop Feynman diagrams in lattice QCD with Wilson fermions

We describe an algebraic algorithm which allows to express every one-loop lattice integral with gluon or Wilson-fermion propagators in terms of a small number of basic constants which can be computed with arbitrary high precision. Although the presentation is restricted to four dimensions the technique can be generalized to every space dimension. We also give a method to express the lattice free propagator for Wilson fermions in coordinate space as a linear function of its values in eight points near the origin. This is an essential step in order to apply the recent methods of L\" uscher and Weisz to higher-loop integrals with fermions.

hep-lat↗
arXiv1996

Algebraic algorithm for the computation of one-loop Feynman diagrams in lattice QCD with Wilson fermions

We describe an algebraic algorithm which allows to express every one-loop lattice integral with gluon or Wilson-fermion propagators in terms of a small number of basic constants which can be computed with arbitrary high precision. Although the presentation is restricted to four dimensions the technique can be generalized to every space dimension. Various examples are given, including the one-loop self-energies of the quarks and gluons and the renormalization constants for some dimension-three and dimension-four lattice operators. We also give a method to express the lattice free propagator for Wilson fermions in coordinate space as a linear function of its values in eight points near the origin. This is an essential step in order to apply the recent methods of Lüscher and Weisz to higher-loop integrals with fermions.

hep-lat↗
arXiv1996

Multi-Grid Monte Carlo via $XY$ Embedding I. General Theory and Two-Dimensional $O(N)$-Symmetric Nonlinear $σ$-Models

We introduce a variant of the multi-grid Monte Carlo (MGMC) method, based on the embedding of an $XY$ model into the target model, and we study its mathematical properties for a variety of nonlinear $σ$-models. We then apply the method to the two-dimensional $O(N)$-symmetric nonlinear $σ$-models (also called $N$-vector models) with $N=3,4,8$ and study its dynamic critical behavior. Using lattices up to $256 \times 256$, we find dynamic critical exponents $z_{int,{\cal M}^2} \approx 0.70 \pm 0.08$, $0.60 \pm 0.07$, $0.52 \pm 0.10$ for $N=3,4,8$, respectively (subjective 68\% confidence intervals). Thus, for these asymptotically free models, critical slowing-down is greatly reduced compared to local algorithms, but not completely eliminated; and the dynamic critical exponent does apparently vary with $N$. We also analyze the static data for $N=8$ using a finite-size-scaling extrapolation method. The correlation length $ξ$ agrees with the four-loop asymptotic-freedom prediction to within $\approx 1\%$ over the interval $12 < ξ< 650$.

hep-lat↗
arXiv1996

A strong-coupling analysis of two-dimensional O(N) sigma models with N<=2 on square, triangular and honeycomb lattices

The critical behavior of two-dimensional ${\rm O}(N)$ $σ$ models with $N\leq 2$ on the square, triangular, and honeycomb lattices is investigated by an analysis of the strong-coupling expansion of the two-point fundamental Green's function $G(x)$, calculated up to 21st order on the square lattice, 15th order on the triangular lattice, and 30th order on the honeycomb lattice. For $N<2$ the critical behavior is of power-law type, and the exponents $γ$ and $ν$ extracted from our strong-coupling analysis confirm exact results derived assuming universality with solvable solid-on-solid models. At $N=2$, i.e., for the 2-$d$ XY model, the results from all lattices considered are consistent with the Kosterlitz-Thouless exponential approach to criticality, characterized by an exponent $σ=1/2$, and with universality. The value $σ=1/2$ is confirmed within an uncertainty of few per cent. The prediction $η=1/4$ is also roughly verified. For various values of $N\leq 2$, we determine some ratios of amplitudes concerning the two-point function $G(x)$ in the critical limit of the symmetric phase. This analysis shows that the low-momentum behavior of $G(x)$ in the critical region is essentially Gaussian at all values of $N\leq 2$. New exact results for the long-distance behavior of $G(x)$ when $N=1$ (Ising model in the strong-coupling phase) confirm this statement.

hep-lat↗
arXiv1996

Asymptotic scaling in the two-dimensional $SU(3)$ $σ$-model at correlation length $4 \times 10^5$

We carry out a high-precision simulation of the two-dimensional $SU(3)$ principal chiral model at correlation lengths $ξ$ up to $\approx\! 4 \times 10^5$, using a multi-grid Monte Carlo (MGMC) algorithm. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. For $ξ\gtapprox 10^3$ we observe good asymptotic scaling in the bare coupling; at $ξ\approx 4 \times 10^5$ the nonperturbative constant is within 2--3\% of its predicted limiting value.

hep-lat↗
arXiv1996

A strong-coupling analysis of two-dimensional O(N) sigma models with $N\geq 3$ on square, triangular and honeycomb lattices

Recently-generated long strong-coupling series for the two-point Green's functions of asymptotically free ${\rm O}(N)$ lattice $σ$ models are analyzed, focusing on the evaluation of dimensionless renormalization-group invariant ratios of physical quantities and applying resummation techniques to series in the inverse temperature $β$ and in the energy $E$. Square, triangular, and honeycomb lattices are considered, as a test of universality and in order to estimate systematic errors. Large-$N$ solutions are carefully studied in order to establish benchmarks for series coefficients and resummations. Scaling and universality are verified. All invariant ratios related to the large-distance properties of the two-point functions vary monotonically with $N$, departing from their large-$N$ values only by a few per mille even down to $N=3$.

hep-lat↗
arXiv1995

Four-Loop Perturbative Expansion for the Lattice $N$-Vector Model

We compute the four-loop contributions to the $β$-function and the anomalous dimension of the field for the $O(N)$-invariant $N$-vector model. These results are used to compute the second analytic corrections to the correlation length and the general spin-$n$ susceptibility.

hep-lat↗
arXiv1995

Strong-coupling expansion of lattice O(N) sigma models

We report progress in the computation and analysis of strong-coupling series of two- and three-dimensional ${\rm O}(N)$ $σ$ models. We show that, through a combination of long strong-coupling series and judicious choice of observables, one can compute continuum quantities reliably and with a precision at least comparable with the best available Monte Carlo data.

hep-lat↗
arXiv1995

Application of the $O(N)$-Hyperspherical Harmonics to the Study of the Continuum Limits of One-Dimensional $σ$-Models and to the Generation of High-Temperature Expansions in Higher Dimensions

In this talk we present the exact solution of the most general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbour interactions, and we discuss the possible continuum limits. All these results are obtained using a high-temperature expansion in terms of hyperspherical harmonics. Applications in higher dimensions of the same technique are then discussed.

hep-lat↗
arXiv1995

Comparison between Theoretical Four-Loop Predictions and Monte Carlo Calculations in the Two-Dimensional $N$-Vector Model for $N=3,4,8$

We have computed the four-loop contribution to the beta-function and to the anomalous dimension of the field for the two-dimensional lattice $N$-vector model. This allows the determination of the second perturbative correction to various long-distance quantities like the correlation lengths and the susceptibilities. We compare these predictions with new Monte Carlo data for $N = 3,4,8$. From these data we also extract the values of various universal nonperturbative constants, which we compare with the predictions of the $1/N$ expansion.

hep-lat↗
arXiv1995

Dynamic Critical Behavior of Multi-Grid Monte Carlo for Two-Dimensional Nonlinear $σ$-Models

We introduce a new and very convenient approach to multi-grid Monte Carlo (MGMC) algorithms for general nonlinear $σ$-models: it is based on embedding an $XY$ model into the given $σ$-model, and then updating the induced $XY$ model using a standard $XY$-model MGMC code. We study the dynamic critical behavior of this algorithm for the two-dimensional $O(N)$ $σ$-models with $N = 3,4,8$ and for the $SU(3)$ principal chiral model. We find that the dynamic critical exponent $z$ varies systematically between these different asymptotically free models: it is approximately 0.70 for $O(3)$, 0.60 for $O(4)$, 0.50 for $O(8)$, and 0.45 for $SU(3)$. It goes without saying that we have no theoretical explanation of this behavior.

hep-lat↗
arXiv1995

Continuum Limits and Exact Finite-Size-Scaling Functions for One-Dimensional $O(N)$-Invariant Spin Models

We solve exactly the general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbor interactions, in finite volume with periodic boundary conditions, by an expansion in hyperspherical harmonics. The possible continuum limits are discussed for a general one-parameter family of interactions, and an infinite number of universality classes is found. For these classes we compute the finite-size-scaling functions and the leading corrections to finite-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated, and no additional universality classes appear. In the appendices we give new formulae for the Clebsch-Gordan coefficients and 6--$j$ symbols of the $O(N)$ group, and some new generalizations of the Poisson summation formula; these may be of independent interest.

hep-lat↗
arXiv1994

New Method for the Extrapolation of Finite-Size Data to Infinite Volume

We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional $O(3)$ and $O(\infty)$ $σ$-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.

hep-lat↗
arXiv1994

Two-Dimensional $O(3)$ $σ$-Model up to Correlation Length $10^5$

We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $σ$-model at correlation lengths $ξ$ up to $\sim 10^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $ξ\sim 10^2$, decreases to $\approx 4\%$ at $ξ\sim 10^5$.

hep-lat↗
arXiv1994

Asymptotic Scaling in the Two-Dimensional $O(3)$ $σ$-Model at Correlation Length $10^5$

We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $σ$-model at correlation lengths $ξ$ up to $\sim 10^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\%$ at $ξ\sim 10^2$, decreases to $\approx 4\%$ at $ξ\sim 10^5$.

hep-lat↗
arXiv1994

Finite-Size Scaling at $ξ/L \gg 1$

We present a simple and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully its systematic and statistical errors, and we illustrate it using three examples: the two-dimensional three-state Potts antiferromagnet on the square lattice, and the two-dimensional $O(3)$ and $O(\infty)$ $σ$-models. In favorable cases it is possible to obtain reliable extrapolations (errors of a few percent) even when the correlation length is 1000 times larger than the lattice.

hep-lat↗