Monte Carlo determination of the critical coupling in ${\varphi }_2^4$ theory

We use lattice formulation of ${\varphi }^4$ theory in order to investigate nonperturbative features of its continuum limit in two dimensions. In particular, by means of Monte Carlo calculations, we obtain the critical coupling constant $g/{\mu }^2$ in the continuum, where $g$ is the unrenormalized coupling. Our final result is $g/{\mu }^2=11.15\pm 0.06_{\text{stat}}\pm 0.03_{\text{syst}}$.

Figure 1

One–loop self–energy in ${\varphi }^4$.

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 symbol size.

Figure 6

Extrapolation of ${\mu }^2$ at $g\to \infty$, as described in text.

Figure 7

Final plot of $f(\eta )$ with our results.