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

D’Elia, Massimo; Farchioni, F.; Papa, Alessandro

doi:10.1016/0550-3213(98)00492-1

Cited by 13 publications

(12 citation statements)

References 57 publications

(60 reference statements)

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“…In principle, the questions we are asking and the methods by which we answer them are similar to the problems associated with topology in two-dimensional spin models [10,11,12]. Because we deal with four dimensional gauge theories we face additional obstacles in numerical tests due to computer speed and memory limitations.…”

Section: Introductionmentioning

confidence: 99%

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Instantons and fixed-point actions in SU(2) gauge theory

1996

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We describe the properties of instantons in lattice gauge theory when the action is a fixed point action of some renormalization group transformation. We present a theoretically consistent method for measuring topological charge using an inverse renormalization group transformation. We show that, using a fixed point action, the action of smooth configurations with non-zero topological charge is greater than or equal to its continuum value 8π 2 /g 2 .

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Section: Introductionmentioning

confidence: 99%

“…The algebraic definition could be improved by using a FP instanton charge as described by Ref. [12] -if it could be done nonperturbatively. We elect not to pursue the algebraic method in this paper due to the complication of the Z(β) factor.…”

Section: Formal Considerations For Ideal Instantonsmentioning

confidence: 99%

Instantons and fixed-point actions in SU(2) gauge theory

1996

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“…[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.…”

Section: Lattice Calculations At θ =mentioning

confidence: 99%

“…[75,77,95,124,201,202,205,243,393], have been dedicated to the N = 2 case, which also corresponds to the O(3) nonlinear σ model. The most recent simulations using the so-called [193] (N = 10, 15, 21, obtained by the geometrical method), by a circle from Ref.…”

Section: Lattice Calculations At θ =mentioning

confidence: 99%

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

2009

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We review results concerning the θ dependence of 4D SU (N ) gauge theories and QCD, where θ is the coefficient of the CP-violating topological term in the Lagrangian. In particular, we discuss θ dependence in the large-N limit.Most results have been obtained within the lattice formulation of the theory via numerical simulations. We review results at zero and finite temperature. We show that the results support the scenario obtained by general large-N scaling arguments, and in particular the Witten-Veneziano mechanism to explain the U (1) A problem. We also compare with results obtained by other approaches, especially in the large-N limit, where the issue has been also addressed using, for example, the AdS/CFT correspondence.We discuss issues related to theta dependence in full QCD: the neutron electric dipole moment, the dependence of the topological susceptibility on the quark masses, the U (1) A symmetry breaking at finite temperature.We also review results in the 2D CP N −1 model, which is an interesting theoretical laboratory to study issues related to topology.Finally, we discuss the main features of the two-point correlation function of the topological charge density.

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“…The distribution P (Q) is given in terms of the real Boltzmann weight as 6) where [dzdz] Q is the constrained measure on which the value of the topological charge given in Eq. (2 .…”

Section: Definition and Algorithmmentioning

confidence: 99%

CPN-1 Models with a Term and Fixed Point Action

Burkhalter

Imachi

Shinno

et al. 2001

Progress of Theoretical Physics

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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 * )

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

Cited by 13 publications

References 57 publications

Instantons and fixed-point actions in SU(2) gauge theory

Instantons and fixed-point actions in SU(2) gauge theory

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

CPN-1 Models with a Term and Fixed Point Action

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