CPN-1 Models with a   Term and Fixed Point Action

Abstract

The topological charge distribution P (Q) is calculated for lattice CP N−1 models. In order to suppress lattice cut-off effects, we employ a fixed point (FP) action. Through transformation of P (Q), we calculate the free energy F (θ) as a function of the θ parameter. For N=4, scaling behavior is observed for P (Q) and F (θ), as well as the correlation lengths ξ(Q). For N=2, however, scaling behavior is not observed, as expected. For comparison, we also make a calculation for the CP 3 model with a standard action. We furthermore pay special attention to the behavior of P (Q) in order to investigate the dynamics of instantons. For this purpose, we carefully consider the behavior of γ eff , which is an effective power of P (Q) (∼ exp(−CQ γ eff )), and reflects the local behavior of P (Q) as a function of Q. We study γ eff for two cases, the dilute gas approximation based on the Poisson distribution of instantons and the Debye-Hückel approximation of instanton quarks. In both cases, we find behavior similar to that observed in numerical simulations. §1. IntroductionIt is interesting to study the phase structure of asymptotic free theories such as QCD and the CP N −1 model. Non-perturbative studies of the phase structure of such theories are necessary in order to understand why effects of the topological term (θ term) are suppressed in Nature. The θ term affects the dynamics at low energy and is expected to lead to rich phase structures. 1) Actually, in the Z(N) gauge model, it has been shown by use of free energy arguments that oblique confinement phases could emerge and that an interesting phase structure may be realized. 2) In this paper we are concerned with the dynamics of the θ vacuum of CP N −1 models with a topological term, which have several dynamical properties in common with QCD. We believe that study of the two-dimensional model will be useful in acquiring information about realistic physics.From the numerical point of view, the topological term introduces a complex Boltzmann weight in the Euclidean lattice path integral formalism. The complex nature of the weight prevents one from straightforwardly applying the standard algorithm used for Monte Carlo simulations. This problem can be circumvented * )

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

“…Refs. [38,54,56,59,70,89,114,350,437,450,451,475]. Like 4D SU (N ) gauge theories, the Euclidean action with the θ term is complex, which impedes a Monte Carlo simulation of the theory using its straightforward discretization, i.e.…”

“…[29,55,70,75,77,95,113,114,124,125,193,201,202,299,331,338,350,374,393,450,451,458,475,533,535,554]. A wide range of values of N has been considered, both small and large, in order to test large-N calculations.…”

Topological lattice actions

“…Taking advantage of universality, and following Symanzik's improvement program [3,4], one can systematically construct improved lattice actions [5,6] which eliminate lattice artifacts up to a given order in the lattice spacing. At a fixed point of the renormalization group, so-called classically perfect lattice actions have been constructed, which are free of lattice artifacts at the classical level [7][8][9][10]. In particular, in asymptotically free theories, including QCD and the 2-d O(3) model, a classically perfect fixed point action has been constructed by solving a minimization problem.…”

Efficient cluster algorithm for models

“…However, due to a very severe complex action problem, no efficient algorithm has ever been found for non-zero vacuum angle. Consequently, in the Wilson formulation numerical studies of θ-vacua [11,12] have been limited to moderate volumes, or are based on additional assumptions [13,14,15,16]. D-theory is an alternative formulation of field theory in which continuous classical fields arise dynamically as collective excitations by the dimensional reduction of discrete variables such as quantum spins [17,18,19].…”