Critical coupling in $φ_2^4$ theory

TL;DR

The paper addresses the nonperturbative critical coupling in the two-dimensional $\phi^4$ theory by lattice Monte Carlo simulations. It determines the finite-volume critical bare mass $\hat{\mu}_{0\mathrm{c}}^2(\hat{\lambda})$, extrapolates to infinite volume, and renormalizes to obtain $\hat{\mu}_c^2(\hat{\lambda})$, from which the continuum ratio $f_c$ is defined as $f_c=\lim_{\hat{\lambda}\to0} \hat{\lambda}/\hat{\mu}_c^2(\hat{\lambda})$. A robust statistical workflow—including finite-volume extrapolation, a multi-parameter fit across $\hat{\lambda}$ with logarithmic corrections, and AIC-based model selection with pruning and CDF weighting—yields $f_c=11.1097(22)$. This result provides a precise universal constant for the symmetry-breaking transition in 2D $\phi^4$ theory and engages in critical comparisons with prior work, discussing potential sources of discrepancies and the role of logarithmic terms in the continuum extrapolation.

Abstract

We consider $φ^4$ theory with $φ(x)\in\mathbb{R}$ in two Euclidean dimensions. We determine for a variety of self-couplings $\hatλ$ the (negative) critical bare mass $\hatμ_{0\mathrm{c}}^2(\hatλ)$ where the lattice-regularized system changes from the symmetric to the broken phase. Based on these data, the transition to infinite volume and a universal scheme with the renormalized parameter $\hatμ_\mathrm{c}^2(\hatλ)$ is made. Finally, $f_\mathrm{c}=\lim_{\hatλ\to0}\hatλ/\hatμ_\mathrm{c}^2(\hatλ)$ is determined, with a judicious choice of the parameterizations considered. Our final result reads $f_\mathrm{c}=11.1097(20)_\mathrm{stat}(09)_\mathrm{sys}=11.1097(22)_\mathrm{tot}$.

Figures (6)

• Figure 1: Left: Field susceptibility $\chi$ versus $\hat{\mu}_0^2$ at $\hat{\lambda}=0.01$ from $21$ independent simulations on $7168\times7168$ lattices. The peak position (with uncertainty) is shown by the orange vertical band. The closest simulation (marked by a cross) is reweighted to give a second determination (with uncertainty as indicated by the green vertical band). Right: Histogram of the field variable $\Phi$, as defined in (\ref{['def_susc']}), in the simulation that was marked with a green cross in the left panel.
• Figure 2: Left: Extrapolation of the peak positions $\hat{\mu}_{0\mathrm{c}}^2(\hat{\lambda},N_x)$ at $\hat{\lambda}=0.01$ to $N_x\to\infty$ for $N_x\geq2048$, along with the resulting pulls (in standard deviations) of the respective data point. The data with $N_x=512,768,1024$ are not included in the fit and out of scale. Right: $f_\mathrm{c}$ at $\hat{\lambda}=0.01$ (after loop-correction) versus the minimal $N_x$ included in the fit. The red diamonds indicate the resulting $p$-values.
• Figure 3: Phase diagrams of $\phi_2^4$ theory in terms of the bare $\hat{\mu}_{0\mathrm{c}}^2$ (left) and the renormalized $\hat{\mu}_{\mathrm{c}}^2$ (right). In either panel cubic splines connect the points (whose error-bars are dwarfed by the symbol size), and the broken phase is beneath the line. The desired quantity $f_\mathrm{c}$ is the inverse of the asymptotic slope (near the origin) of the transition line in the right panel.
• Figure 4: Left: Example of a continuum fit with $\hat{\lambda}_\mathrm{max}=1.0$. Right: Cumulative distribution function (CDF) of all statistically acceptable fits with $\hat{\lambda}_\mathrm{max}=1.0$, weighted with their AIC scores. At each step of the red dashed curve, the horizontal band indicates the statistical uncertainty of the fit. The blue curve shows a CDF obtained by combining the AIC weights with the Gaussian distributions corresponding to the statistical uncertainties.
• Figure 5: Left: Values of $f_\mathrm{c}$ obtained with the procedure shown in the right panel of Fig. \ref{['fig:fc_lambdamax_1']}, separately for $\hat{\lambda}_\mathrm{max} \in \{1.0, 0.7, 0.5, 0.35, 0.25\}$. Right: CDF of all statistically acceptable fits from the left panel except for $\lambda_\mathrm{max}=0.25$, merging results from the other four choices of $\hat{\lambda}_\mathrm{max}$ in the same manner as for a single $\hat{\lambda}_\mathrm{max}$ in Fig. \ref{['fig:fc_lambdamax_1']}. From this combined analysis, we obtain our final result $f_\mathrm{c}=11.1097(20)_\mathrm{stat}(09)_\mathrm{sys}=11.1097(22)_\mathrm{tot}$.