Wilsonʼs momentum shell renormalization group from Fourier Monte Carlo simulations

Abstract

Previous attempts to accurately compute critical exponents from Wilsonʼs momentum shell renormalization prescription suffered from the difficulties posed by the presence of an infinite number of irrelevant couplings. Taking the example of the 1d long-ranged Ising model, we calculate the momentum shell renormalization flow in the plane spanned by the coupling constants ( u 0 , r 0 ) for different values of the momentum shell thickness parameter b by simulation using our recently developed Fourier Monte Carlo algorithm. We report strong anomalies in the b-dependence of the fixed point couplings and the resulting exponents y τ and ω in the vicinity of a shell parameter b ⁎ < 1 characterizing a thin but finite momentum shell. Evaluation of the exponents for this value of b yields a dramatic improvement of their numerical accuracy, indicating a strong damping of the influence of irrelevant couplings for b = b ⁎ .