${\varphi }_3^4$ lattice field theory viewed from the high-temperature side

We analyze high-temperature series expansions of the two- and four-point correlation functions in the three-dimensional Euclidean lattice scalar field theory with quartic self-coupling, which have been recently extended through 25th order for the simple-cubic and body-centered-cubic lattices. We conclude that the length of the present series is sufficient for a fairly accurate description of the critical behavior of the model and confirm the validity of universality, scaling, and hyperscaling. In the case of the body-centered-cubic lattice, we determine the value of the quartic self-coupling for which the leading corrections to scaling approximately vanish and correspondingly the universal critical parameters can be determined with high accuracy. In particular, for the susceptibility and the correlation-length exponents we find $\gamma =1.2373\left(2\right)$ and $\nu =0.6301\left(2\right)$. For the four-point renormalized coupling we find $g=23.56\left(3\right)$. In the case of the simple-cubic lattice our results are consistent with earlier estimates.

Figure 1

Phase diagram of ${\varphi }_3^4$ on the sc lattice. The critical line $K=K\left(g_0\right)$ is plotted vs. $G1=g_0∕(g_0+1)$.

Figure 2

Phase diagram of ${\varphi }_3^4$ on the bcc lattice. The critical line $K=K\left(g_0\right)$ is plotted vs. $G1=g_0∕(g_0+1)$.

Figure 3

The susceptibility exponent $\gamma \left(g_0\right)$ as a function of $G1=g_0∕(g_0+1)$ for the sc lattice. The continuous curves represent the estimates obtained by the MRAs of orders 17, 21, and 25, respectively. The dashed curve indicates the central value of our estimate (Ref. 5) of $\gamma$ for the Ising universality class. The dotted curve shows the results of an extrapolation of the MRA sequences by a fit to the asymptotic expansion (10) supplemented by a single additional higher order correction $O(1∕n^4)$. The triangles show estimates of $\gamma \left(g_0\right)$ by unbiased first-order DAs.

Figure 4

The same as in Fig. 3, but for the bcc lattice.

Figure 5

The amplitude $a\left(g_0\right)$ of the leading correction to scaling in (4) vs. $G1=g_0∕(g_0+1)$, as obtained from a fit to the asymptotic expansion (9) supplemented by a single higher-order correction $O(1∕n^{6+\theta })$ in the bcc lattice case.

Figure 6

The renormalized coupling $g({\xi }^2,g_0)$ as a function of $X4={\xi }^2∕({\xi }^2+4)$ for several fixed values of the ${\varphi }^4$ self-coupling $g_0$ on the sc lattice. The values of $g_0$ increase from the lowest curve up. The dashed line represents the central value of our estimate $g=23.52\left(5\right)$, obtained in Ref. 5, of the renormalized coupling in the Ising universality class.

Figure 7

The same as in Fig. 6, but for the bcc lattice.

Figure 8

The renormalized coupling $g({\xi }^2,g_0)$ as a function of $G1=g_0∕(g_0+1)$ for several values of the correlation length square ${\xi }^2$ on the sc lattice. The values of ${\xi }^2$ are indicated beside the corresponding curves. The dashed line represents the central value of our estimate $g=23.52\left(5\right)$, obtained in Ref. 5, of the renormalized coupling in the Ising universality class.

Figure 9

The same as in Fig. 8, but for the bcc lattice.

Figure 10

Contour plot of the renormalized coupling $g=g({\xi }^2,g_0)$ in the $G_4$, $X_4$ plane for the sc lattice. Here $G_4=g_0∕(g_0+4)$ and $X_4={\xi }^2∕({\xi }^2+4)$. The values of $g({\xi }^2,g_0)$ are shown beside the corresponding level curves.

Figure 11

The same as in Fig. 10, but for the bcc lattice.