Abstract
The ratios R 2 k = g 2 k / g k − 1 4 of renormalized coupling constants g 2 k entering the small-field equation of state approach universal values R * 2 k at criticality. They are calculated for the three-dimensional λ ϕ 4 field theory within the pseudo- ϵ expansion approach. Pseudo- ϵ expansions for R * 6 , R * 8 , R * 10 are derived in the five-loop approximation, numerical estimates are obtained with a help of the Pade–Borel–Leroy resummation technique. Its use gives R * 6 = 1.6488, the number which perfectly agrees with the most recent lattice result R * 6 = 1.649. For the octic coupling the pseudo- ϵ expansion is less favorable numerically. Nevertheless the Pade–Borel–Leroy resummation leads to the estimate R * 8 = 0.890 close to the values R * 8 = 0.87, R * 8 = 0.857 extracted from the lattice and field-theoretical calculations. The pseudo- ϵ expansion for R * 10 turns out to have big and rapidly increasing coefficients. This makes correspondent estimates strongly dependent on the Borel–Leroy shift parameter b and prevents proper evaluation of R * 10
Highlights
The critical behavior of the systems undergoing continuous phase transitions is characterized by a set of universal parameters including, apart from critical exponents, renormalized effective coupling constants g2k and the ratios R2k = g2k/gk4−1
The Hamiltonian of the Euclidean field theory describing the critical behavior of the 3D Ising-like systems reads: H= 1 2
One can expect that use of Padé–Borel–Leroy resummation technique will lead to high-precision numerical results in this case
Summary
The critical behavior of the systems undergoing continuous phase transitions is characterized by a set of universal parameters including, apart from critical exponents, renormalized effective coupling constants g2k and the ratios R2k = g2k/gk−. The critical behavior of the systems undergoing continuous phase transitions is characterized by a set of universal parameters including, apart from critical exponents, renormalized effective coupling constants g2k and the ratios R2k = g2k/gk4−1 These ratios enter the scaling equation of state via the small-magnetization expansion of the free energy: F(z, m). Into alternative series looking more friendly from the numerical point of view To do this we address the pseudo- expansion approach which proved to be very efficient numerically when used to evaluate the critical exponents and other universal parameters of various 3D and 2D systems [27,28,29,30,31,32,33,34,35,36,37,38,39,40]
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