Quantitative picture of the scaling behaviour of lattice nonlinear σ-models from the 1/Nexpansion (original) (raw)

1/N contribution to physical quantities in the lattice O(N) σ-model

Nuclear Physics B, 1985

We evaluate the 1/N contribution to the mass gap and to the magnetic susceptibility for the two-dimensional O(N) nonlinear o-model with standard lattice action. The same quantities are also obtained through a strong coupling analysis and found to be in good agreement with the direct evaluation. The relations with the results for different choices of the action are discussed.

Lattice O(N) nonlinear sigma model: Scaling in high temperature expansion

Nuclear Physics B, 1988

In the two-dimensional lattice O(N) nonlinear sigma model (also called the classical N-vector model), we compute for various physical quantities the high temperature expansion coefficients as functions of N up to if1. The calculation is performed by a new computer program which solves iteratively the Schwinger-Dyson equations of the model. Here we describe the algorithm which is similar to the one recently introduced for the N = 2 case and used to extend the fl-expansion. As a first application we use these new results to study the N-dependence of the singularities of the susceptibility in the complex B/N plane. We can study in this way how the location of the "critical singularity" depends on N. For N ~< 2 the critical singularity stays on the real positive fl axis increasing with N, and for N > 2 it splits into two complex conjugate unphysical singularities which move into the complex plane as N grows large, in agreement with the fact that the theory becomes asymptotically free. We suggest that the difficulties with the asymptotic scaling laws encountered in high temperature studies and in Monte Carlo simulations for N = 3 could be explained by the fact that these complex singularities sit near the values of fl which are investigated in these studies.

Magnetic susceptibility of O(N) σ-models in 2D. Weak coupling results from expansion

Physics Letters B, 1991

The magnetic susceptibilities of the non-linear a-models in 2D are given to three leading orders in 1/N as functions of the inverse bare coupling fl and up to correlation lengths 150. Within the systematic error introduced by truncating the 1/N expansion, our results agree with Monte Carlo results for N>~ 3. We argue that the 1/N series is convergent with a fl-dependent radius of convergence approaching ½ at weak coupling, and use this to predict the value for the magnetic susceptibility at asymptotically weak coupling. Our calculation features "Fourier accelerated" numerical evaluation of Feynman diagrams, and extrapolation of finite volume results to infinite volume by phenomenological scaling.

1/N expansion of two-dimensional models in the scaling region

Nuclear Physics B - Proceedings Supplements, 1993

The main technical and conceptual features of the lattice 1/N expansion in the scaling region are discussed in the context of a two-parameter two-dimensional spin model interpolating between CP N−1 and O(2N) σ models, with standard and improved lattice actions. We show how to perform the asymptotic expansion of effective propagators for small values of the mass gap and how to employ this result in the evaluation of physical quantities in the scaling regime. The lattice renormalization group β function is constructed explicitly and exactly to O(1/N).

Extended scaling relations for planar lattice models

It is widely believed that the critical properties of several planar lattice models, like the Eight Vertex or the Ashkin-Teller models, are well described by an effective Quantum Field Theory obtained as formal scaling limit. On the basis of this assumption several extended scaling relations among their indices were conjectured. We prove the validity of some of them, among which the ones by Kadanoff, , and by Luther and Peschel, .

Perturbation theory predictions and Monte Carlo simulations for the 2D O(n) non-linear σ-models

Nuclear Physics B, 1997

By using the results of a high-statistics (O(10 7 ) measurements) Monte Carlo simulation we test several predictions of perturbation theory on the O(n) non-linear σ-model in 2 dimensions. We study the O(3) and O(8) models on large enough lattices to have a good control on finite-size effects. The magnetic susceptibility and three different definitions of the correlation length are measured. We check our results with large-n expansions as well as with standard formulae for asymptotic freedom up to 4 loops in the standard and effective schemes. For this purpose the weak coupling expansions of the energy up to 4 loops for the standard action and up to 3 loops for the Symanzik action are calculated. For the O(3) model we have used two different effective schemes and checked that they lead to compatible results. A great improvement in the results is obtained by using the effective scheme based on the energy at 3 and 4 loops. We find that the O(8) model follows very nicely (within few per mille) the perturbative predictions. For the O(3) model an acceptable agreement (within few per cent) is found.

The 1/N expansion of two-dimensional spin models

Nuclear Physics B - Proceedings Supplements, 1994

A short review of all available results (perturbative, nonperturbative, and exact) on d-dimensional spin models is presented in order to introduce the discussion of their 1/N expansion at d = 2, where the models are asymptotically free. A general two-dimensional spin model with U(N) invariance, interpolating between CP N −1 and O(2N) models, is studied in detail in order to illustrate both the general features of the 1/N expansion on the lattice and the specific techniques devised to extract scaling (field-theoretical) behavior. The continuum version of the model is carefully analyzed deriving quantitative O(1/N) physical predictions in order to establish a benchmark for lattice computations. The 1/N expansion on the lattice, including second-nearest-neighbor interactions, is set up by constructing explicitly effective propagators and vertices, and exhibiting a number of exact results and integral representations that allow a substantial reduction of the numerical effort. The technique of asymptotic expansion of the lattice propagators, basic to the derivation of analytical results in the scaling domain, is presented in full detail and applied to the model. Physical quantities, like the free energy and different definitions of correlation length, are evaluated. The lattice renormalization-group trajectories are identified and universality among different lattice (and continuum) schemes in the scaling region is explicitly proven. As a byproduct, representations of the O(1/N) contribution to the Λ-parameter ratios and to the lattice β functions are obtained. A review of other developments based on the lattice 1/N expansion (finite size scaling, higher orders, fermionic models) is presented.

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

Nuclear Physics B-proceedings Supplements, 1996

We report progress in the computation and analysis of strong-coupling series of two- and three-dimensional rmO(N){\rm O}(N)rmO(N) sigma\sigmasigma 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.

Scaling and topology in the 2-d 0(3) σ-model on the lattice with the fixed point action

Nuclear Physics B, 1995

We study scaling properties and topological aspects of the 2-d 0(3) non-linear o'-model on the lattice with the fixed point action recently found by P. Hasenfratz and F. Niedermayer. The behavior of the mass gap confirms the good properties of scaling of the fixed point action. Concerning the topology, lattice classical solutions are proved to be very stable under local minimization of the action; this outcome ensures the reliability of the cooling method for the computation of the topological susceptibility, which indeed reproduces the results of the field theoretical approach. Disagreement is instead observed with a different approach in which the fixed point topological charge operator is used: we argue that the discrepancy is related to the ultraviolet dominated nature of the model. * Work supported in part by Fondazione "A. Delia Riccia" (Italy). t On leave from the