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Ferroelectrics

Renormalization group study of the m-vector model between two and four dimensions

Yurij V. Holovatch

Published Feb 1, 1997• 6 citations
Publisher

Abstract

In the frames of field theoretical fixed dimension renormalization group scheme critical exponents of m vector model are calculated in the case when space dimension continuously changes from the lower critical dimension to the upper one.
DOI
10.1080/00150199708216170

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An exact renormalization equation is derived by making an infinitesimal change in the cutoff in momentum space. From this equation the expansion for critical exponents around dimensionality 4 and the limit $n=\ensuremath{\infty}$ of the nnn-vector model are calculated. We obtain agreement with the results of Wilson and Fisher, and with the spherical model.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.
We investigate the critical properties of d-dimensional magnetic systems with quenched extended defects, correlated in ${\ensuremath{\varepsilon}}_{d}$ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\varepsilon}}_{d}$ dimensions; both in the case of fixed dimension d=3d=3d=3 and when the space dimension continuously changes from the lower critical dimension to the upper one. Our renormalization-group calculations are performed in the minimal subtraction scheme. We analyze the two-loop renormalization-group functions for different fixed values of the parameters $d,{\ensuremath{\varepsilon}}_{d}.$ To this end, we apply the Chisholm-Borel resummation technique and report the numerical values of the critical exponents for the universality class of this system.
This is the third article in a sequence in which we reexamine the calculation of the critical exponents of the N-vector model from field theory. In the first article we have applied summation methods to the ε-expansion for which new terms have been calculated (up to order ε5). In the second article we have incorporated in the summation method, in the case of Ising-like systems (N = 1), the knowledge of the exact values of exponents in d = 2 dimensions. Since exact values are now also known in the case of polymers (N = 0), we apply here the same method to this case. Moreover, since for the exponent η for N = 1 the results coming from the ε-expansion and perturbation series at fixed dimension are slightly different, we have recalculated the exponents in the latter case using the variant of the summation method developed for the ε-expansion. The results do not change significantly showing both the reliability and the limits of the method.