Calculations of the dynamical critical exponent using the asymptotic series summation method

Abstract

We consider how the Padé-Borel, Padé-Borel-Leroy, and conformal mapping summation methods for asymptotic series can be used to calculate the dynamical critical exponent for homogeneous and disordered Ising-like systems.

Borel resummation of the ɛ-expansion of the dynamical exponent z in model a of the ϕ 4(O(n)) theory

“…Comparing these values with those calculated for Padé approximations (15) (Table 11), we conclude that such a resummation method does not agree with the LOA approximation of the dynamical exponent [7], [8]. In [9], another regular expansion was used for the dynamical exponent (in the parameter g and in fixed integer dimensions of the space). In contrast to the ε-expansion, where the LOA approximation fixing the parameters b and a in formulas (7) and (8) is known, these parameters are free in the expansion in g. In [9], different methods were proposed for choosing the values of the parameter b in the PBL method.…”

“…In [9], another regular expansion was used for the dynamical exponent (in the parameter g and in fixed integer dimensions of the space). In contrast to the ε-expansion, where the LOA approximation fixing the parameters b and a in formulas (7) and (8) is known, these parameters are free in the expansion in g. In [9], different methods were proposed for choosing the values of the parameter b in the PBL method. It turns out that there several of them.…”

“…In [9], expansions of the dynamical exponent z in the same model A were similarly resummed using the renormalization group method directly in the two-and three-dimensional spaces. A specific feature of the present paper (in addition to a different regular expansion being used) is that the exact LOA approximation is known (in [9], the parameters a and b were considered arbitrary and were used to optimize the computations).…”

“…A specific feature of the present paper (in addition to a different regular expansion being used) is that the exact LOA approximation is known (in [9], the parameters a and b were considered arbitrary and were used to optimize the computations). We can therefore compare the resummation results (in the ε-scheme) obtained by taking the exact LOA approximation into account with the results "optimized" in the parameters.…”

On ergodic relaxation time in the three-dimensional Ising model

“…In particular, the value of the so-called dynamical (or critical slowing down) exponent z for the 3D Ising model was obtained in a range 1.95-2.17 (see [7][8][9]) using different numerical techniques. Actually the discussion on the z value for systems belonging to the 3D Ising universality class continues up to now (most recent data on z is limited to 2.017-2.10, see e. g. [10,11]). …”