Monte Carlo determination of the critical coupling in $φ^4_2$ theory

TL;DR

The paper tackles the nonperturbative continuum limit of the two-dimensional φ^4 theory by determining the critical coupling ratio $f_0 = g/\mu^2$ using lattice regularisation and Monte Carlo simulations with the worm algorithm. A renormalisation scheme based on additive mass renormalisation, $\mu^2 = \mu_0^2 + 3 g A(\mu^2)$, is employed to obtain the continuum mass, and finite-volume extrapolations ($mL=z$) are used to locate the critical point before taking $g\to 0$. The authors report $f_0 = 11.15(6)(3)$, with an alternative $\,\eta = g/(g+1)\,$ parametrisation yielding $f_0 = 11.119(24)$, and they find general agreement with recent nonperturbative determinations while noting a $3\sigma$ discrepancy with a previous Monte Carlo result at very small $g$. This work advances precision in the nonperturbative determination of the continuum limit for $\phi^4_2$ and tests universality against Ising-like behavior in two dimensions.

Abstract

We use lattice formulation of $φ^4$ theory in order to investigate non--perturbative features of its continuum limit in two dimensions. In particular, by means of Monte Carlo calculations, we obtain the critical coupling constant $g/μ^2$ in the continuum, where $g$ is the {\em unrenormalised} coupling. Our final result is $g/μ^2=11.15(6)(3)$.

Figures (7)

• Figure 1: One--loop self--energy in $\phi^4$
• Figure 2: Linear interpolation ($\lambda = 0.25,\,L=256$) to obtain $\beta(z=4)$
• Figure 3: Linear extrapolation of $\beta$ to $a/L=0$ for $\lambda=0.25$.
• Figure 4: Extrapolation to $a/L=0$ with $z=4$ (blue steep curve) and $z=1$ (red curve) ($\lambda=1$).
• Figure 5: Final results for $f(g)$ in logarithmic scale. Error-bars, where not visible, are smaller than symbols size.