The 1/N expansion of two-dimensional spin models

Abstract

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.

Fighting topological freezing in the two-dimensional CPN−1 model

“…Finally we try to extract results for the continuum limit from our data and compare these with predictions obtained from the 1/N-expansion [31][32][33] …”

θ dependence of SU(N) gauge theories in the presence of a topological term

“…We consider the lattice action (8.17) which is more convenient for a large-N expansion [123,217,218]. One can easily prove that site-and link-reflection positivity holds for the lattice action (8.17).…”

“…An appealing feature of 2D CP N −1 models is the possibility of perfoming a systematic 1/N expansion, keeping g fixed, around the large-N saddle-point solution [122,123,166,548], unlike 4D SU (N ) gauge theories. This makes these models particularly interesting, because they allow us to also check general nonperturbative scenarios by analytic calculations, without necessarily resorting to numerical Monte Carlo methods of their lattice formulation.…”

“…This may be proved by performing an asymptotic expansion of G L (k) (at fixed kξ) in powers of ξ −2 , following the procedure outlined in Ref. [123].…”

The puzzle of apparent linear lattice artifacts in the 2d non-linear σ-model and Symanzik's solution

“…2-dimensional O(n) spin systems, including the Ising model and the large n limit, are reviewed in [13]. Various aspects (on the lattice and in the continuum) of the exactly solvable n → ∞ limit, using a number of different lattice actions, are reviewed in [14]. Symanzik's improvement program for the 2-dimensional O(n) models are studied in [15,16] and some exact results about the lattice artifacts in the large n limit are given in [12].…”