Effective potential of the three-dimensional Ising model: the pseudo-$\epsilon$ expansion study

Abstract

The ratios $R_{2k}$ of renormalized coupling constants $g_{2k}$ that enter the effective potential and small-field equation of state acquire the universal values at criticality. They are calculated for the three-dimensional scalar $\lambda {\varphi }^4$ field theory (3D Ising model) within the pseudo-$\epsilon$ expansion approach. Pseudo-$\epsilon$ expansions for the critical values of $g_6$, $g_8$, $g_{10}$, $R_6=g_6/g_4^2$, $R_8=g_8/g_4^3$ and $R_{10}=g_{10}/g_4^4$ originating from the five-loop renormalization group (RG) series are derived. Pseudo-$\epsilon$ expansions for the sextic coupling have rapidly diminishing coefficients, so addressing Padé approximants yields proper numerical results. Use of Padé--Borel--Leroy and conformal mapping resummation techniques further improves the accuracy leading to the values $R_6^{\ast }=1.6488$ and $R_6^{\ast }=1.6490$ which are in a brilliant agreement with the result of advanced lattice calculations. For the octic coupling the numerical structure of the pseudo-$\epsilon$ expansions is less favorable. Nevertheless, the conform-Borel resummation gives $R_8^{\ast }=0.868$, the number being close to the lattice estimate $R_8^{\ast }=0.871$ and compatible with the result of 3D RG analysis $R_8^{\ast }=0.857$. Pseudo-$\epsilon$ expansions for $R_{10}^{\ast }$ and $g_{10}^{\ast }$ are also found to have much smaller coefficients than those of the original RG series. They remain, however, fast growing and big enough to prevent obtaining fair numerical estimates.

arXiv comment12 pages, 4 figures, 5 tables. arXiv admin note: text overlap with arXiv:1601.00148

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