Efficient calculation of critical parameters in SU (2) gauge theory

Abstract

We show how the critical point and the ratio ~,/u of critical exponents of the finite temperature deconfinement transition of SU (2) gauge theory may be determined simply from the expectation value of the square of the Polyakov loop. In a similar way we estimate the ratio (a-1 )/~. The method is based on a consistent application of finite size scaling theory to results obtained with the density of states technique. It may also be used in other lattice theories at second order transitions.Finite size scaling (FSS) techniques have now become a well-established tool for the investigation of critical properties in SU(2) [1][2][3] and SU(3) [4][5][6] lattice gauge theories at finite temperature. Many ingenious methods have been devised using FSS theory to extract the infinite volume critical point and ratios of critical exponents from various variables. Most of these methods were invented in statistical physics and applied to high precision Monte Carlo data of Ising [7,8] and other comparatively simple models. Especially Binder's fourth-order cumulant [9] of the magnetization or the energy has become a favourite observable for the determination of the critical point. Besides that the peak positions of thermodynamic derivatives provide finite lattice transition points, which may be extrapolated by FSS formulas to the asymptotic critical point.Both the cumulant and the susceptibility, which are the most used observables for this purpose are quantities, which involve differences of powers of directly measured observables and require thus higher statistics to reach the same accuracy. Moreover, instead of the true susceptibility a pseudosusceptibility is commonly used, where the expectation value of the magWork supported by the Deutsche Forschungsgemeinschaft. Permanent address: Joint Institute for Nuclear Research, 101 000 Dubna, Russian Federation. netization is replaced by the one of the modulus of the magnetization.In this letter we want to show that the FSS behaviour of the expectation value of the square of the magnetization, though it is not peaked at the transition, allows to determine the asymptotic critical point and the ratio of the critical exponents 7 and u.We apply our idea to SU(2) gauge theory on N3XN~, N~=4 lattices using the standard Wilsonwhere Up is the product of link operators around a plaquette. The number of lattice points in the space (time) direction N~(~) and the lattice spacing a fix the volume and temperature asOn an infinite volume lattice the order parameter of magnetization for the deconfinement transition is the expectation value of the Polyakov loop

Critical behaviour of SU(2) lattice gauge theory. A complete analysis with the χ2-method

“…Proceeding in the same way as in ref. [1], we evaluated then the data in the very close vicinity ( 2.298 ≤ 4/g 2 ≤ 2.300) of the transition with the density of states method (DSM) [8]. The newly calculated MC data were in complete agreement with the former DSM interpolation and introduced only a tiny shift to the interpolation itself.…”

“…[5,7] and [1]. Since in the latter reference the critical point was determined to 4/g 2 c,∞ = 2.2988(1), we calculated in addition the still missing results at this point for N σ = 8, 12 and 18 and N τ = 4 with high statistics.…”

Scaling, asymptotic scaling, and Symanzik improvement: Deconfinement temperature in SU(2) pure gauge theory

“…Its application to the analysis of second order phase transitions permitted a detailed numerical test of theoretical predictions on critical exponents. The effectiveness in the study of SU(2) deconfining phase transition has been already demonstrated in a number of papers, where use of the Wilson's action was made [1,2,18].…”

Stability of Dirac sheet configurations

“…b = 4a and L s = aN s with a the lattice spacing). For N t = 4 the model is known to undergo the deconfinement phase transition at the critical coupling β c ≃ 2.299 [15]. In Ref.…”