An improved lattice measurement of the critical coupling in phi^4_2 theory

TL;DR

The paper addresses the problem of determining the continuum critical coupling $f_c = [\lambda/\mu^2]_{crit}$ in the 2D $\phi^4$ theory by performing lattice Monte Carlo simulations with a mass renormalization scheme $\hat{\mu}^2 = \hat{\mu}_0^2 + 3\hat{\lambda} A_{\mu^2}$ and mapping the critical line $\hat{\mu}_{0c}^2(\hat{\lambda})$. It combines three indicators—susceptibility, bimodality, and Binder cumulant—to locate criticality at finite $N$, followed by finite-size scaling and a continuum extrapolation that reveals a nonlinear, logarithmic dependence on the lattice coupling, yielding $f_c = 10.78(3)$ (with broader fits giving $f_c \approx 10.9$ when including higher-order terms). The authors demonstrate that linear extrapolations are insufficient and that a logarithmic term in $\hat{\lambda}$ is required, which is also supported by analytic expectations for super-renormalizable theories. This work resolves discrepancies with earlier Monte Carlo results, provides a precise benchmark for analytic methods, and reinforces the role of logarithmic corrections in the continuum limit of $\phi_2^4$ theory.

Abstract

We use Monte Carlo simulations to obtain an improved lattice measurement of the critical coupling constant [lambda / mu^2]_crit for the continuum (1 + 1)-dimensional (lambda / 4) phi^4 theory. We find that the critical coupling constant depends logarithmically on the lattice coupling, resulting in a continuum value of [lambda / mu^2]_crit = 10.8(1), in considerable disagreement with the previously reported [lambda / mu^2]_crit = 10.26(8). Although this logarithmic behavior was not observed in earlier lattice studies, it is consistent with them, and expected analytically.

Figures (5)

• Figure 1: The only divergent Feynman diagram in $\phi^4_2$ theory.
• Figure 2: Histograms of $\phi$ for simulations with $N = 32$ and $\hat{\lambda} = 0.05$, in the symmetric phase (left, $\hat{\mu}_0^2 = -0.075$) and broken symmetry phase (right, $\hat{\mu}_0^2 = -0.11$).
• Figure 3: Bimodality plotted against $\hat{\mu}_0^2$ for simulations with $N = 64$ and $\hat{\lambda} = 0.5$, before (left) and after (right) smoothing.
• Figure 4: Critical coupling constant $\hat{\lambda} / \hat{\mu}_c^2$ plotted against $\hat{\lambda}$.
• Figure 5: Our data for $\hat{\lambda} / \hat{\mu}_c^2$ compared with the results of loinaz (empty circles).