Fourier Monte Carlo Renormalization Group Approach to Crystalline Membranes

TL;DR

This paper introduces a novel Fourier Monte Carlo renormalization group method to accurately compute the critical exponent eta for crystalline membranes, addressing previous discrepancies in numerical results.

Contribution

It combines Wilson's momentum shell RG with Fourier Monte Carlo simulations, enabling improved estimates of eta and the Wegner exponent omega.

Findings

Results agree with functional renormalization group data.

First estimate of the Wegner exponent omega for this system.

Refined numerical value for eta from correction-to-scaling analysis.

Abstract

The computation of the critical exponent eta characterizing the universal elastic behavior of crystalline membranes in the flat phase continues to represent challenges to theorists as well as computer simulators that manifest themselves in a considerable spread of numerical results for eta published in the literature. We present new insight to this problem that results from combining Wilson's momentum shell renormalization group method with the power of modern computer simulations based on the Fourier Monte Carlo algorithm. After discussing the ideas and difficulties underlying this combined scheme, we present a calculation of the renormalization group flow of the effective 2d Young modulus for momentum shells of different thickness. Extrapolation to infinite shell thickness allows to produce results in reasonable agreement with those obtained by functional renormalization group or by Fourier Monte Carlo simulations in combination with finite size scaling. Moreover, our new method allows for the first time to obtain a decent estimate for the value of the Wegner exponent omega that determines the leading correction to scaling, which in turn allows to refine our numerical estimate for eta previously obtained from precise finite size scaling data.