Improved lattice measurement of the critical coupling in ${\varphi }_2^4$ theory

We use Monte Carlo simulations to obtain an improved lattice measurement of the critical coupling constant $[\lambda /{\mu }^2{]}_{\mathrm{crit}}$ for the continuum $(1+1)$-dimensional $(\lambda /4){\varphi }^4$ theory. We find that the critical coupling constant depends logarithmically on the lattice coupling, resulting in a continuum value of $[\lambda /{\mu }^2{]}_{\mathrm{crit}}={10.8}_{-0.05}^{+0.10}$, in considerable disagreement with the previously reported $[\lambda /{\mu }^2{]}_{\mathrm{crit}}={10.26}_{-0.04}^{+0.08}$. Although this logarithmic behavior was not observed in earlier lattice studies, it is consistent with them, and expected analytically.

Figure 1

The only divergent Feynman diagram in ${\varphi }_2^4$ theory.

Figure 2

Histograms of $\varphi$ for simulations with $N=32$ and $\widehat{\lambda }=0.05$, in the symmetric phase (left graph, ${\widehat{\mu }}_0^2=-0.075$) and the broken symmetry phase (right graph, ${\widehat{\mu }}_0^2=-0.11$).

Figure 3

Bimodality plotted against ${\widehat{\mu }}_0^2$ for simulations with $N=64$ and $\widehat{\lambda }=0.5$, before (left graph) and after (right graph) smoothing.

Figure 4

Critical coupling constant $\widehat{\lambda }/{\widehat{\mu }}_c^2$ plotted against $\widehat{\lambda }$.

Figure 5

Our data for $\widehat{\lambda }/{\widehat{\mu }}_c^2$ compared with the results of 3 (empty circles).