Stability of a cubic fixed point in three dimensions: Critical exponents for genericN

Abstract

The detailed analysis of the global structure of the renormalization-group ͑RG͒ flow diagram for a model with isotropic and cubic interactions is carried out in the framework of the massive field theory directly in three dimensions ͑3D͒ within an assumption of isotropic exchange. Perturbative expansions for RG functions are calculated for arbitrary N up to four-loop order and resummed by means of the generalized Padé-Borel-Leroy technique. Coordinates and stability matrix eigenvalues for the cubic fixed point are found under the optimal value of the transformation parameter. Critical dimensionality of the model is proved to be equal to N c ϭ2.89Ϯ0.02 that agrees well with the estimate obtained on the basis of the five-loop expansion ͓H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 ͑1995͔͒ resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for Nу3, and the critical exponents controlling phase transitions in three-dimensional magnets should belong to the cubic universality class. The critical behavior of the random Ising model being the nontrivial particular case of the cubic model when Nϭ0 is also investigated. For all physical quantities of interest the most accurate numerical estimates with their error bounds are obtained. The results achieved in the work are discussed along with the predictions given by other theoretical approaches and experimental data.

Pseudo-ɛexpansion of six-loop renormalization-group functions of an anisotropic cubic model

“…[8] and [24] were obtained from the condition of the result stability in successive approximation orders. However, the numerical value of the fixed point coordinates were substituted into the expansions for the γ-functions (4), (5).…”

“…However, the numerical value of the fixed point coordinates were substituted into the expansions for the γ-functions (4), (5). To this end the most reliable numerical values of the stable fixed point coordinates were substituted into the resummed γ-functions and then an optimal value of the fit parameter in modified Padé-Borel resummation [24] and two fit parameters in the conformal mapping procedure [8] were chosen. The deviations between five-and six-loop results obtained within the resummation procedure with optimal fit parameter(s) value gave the error interval.…”

Multicritical phenomena inO(n1)⊕O(n2)Calabrese1, Pelissetto2, Vicari3 2003Phys. Rev. B167141421View full textAdd to dashboardCite

“…In Table I the Monte Carlo results of Ref. [57] are also shown; they were obtained by simulating the standard N-vector model and computing the RG dimension of the cubicsymmetric term i s [53,41,62,63,64,66,67,68,69,57,70]. The most accurate results have been provided by analyses of high-order perturbative fieldtheory expansions, which predict N c < ∼ 2.9 in three dimensions.…”

Randomly dilute Ising model: A nonperturbative approach

“…(1), the S i are Ising variables and the ε i are quenched, uncorrelated, random variables taking the value 1 with probability p and 0 with probability 1 − p. According to the Harris criterion 3 , the critical behavior of a pure system is altered by the disorder if its specific-heat exponent α is positive, as it is the case for the three-dimensional Ising model for which α ≃ 0.109, β ≃ 0.326, γ ≃ 1.24, ν ≃ 0.630 and η ≃ 0.0335 (see 4 Numerically, early Monte Carlo simulations had confirmed the relevance of disorder for the Ising model (see 2 ). The situation was however unclear since the critical exponents seemed to vary with p. Recently, Ballesteros et al 5 , by a careful infinite volume extrapolation of the data based upon an analysis of the correction-to-scaling terms, have reached the conclusion of dilution-independent critical exponents : α = −0.051 (16), β = 0.3546(28), γ = 1.342(10), ν = 0.6837(53) and η = 0.0374(45) for 0.4 ≤ p ≤ 0.9.…”

“…Different renormalization schemes have been used : minimal subtraction (MS) scheme 6,8,9,10,11 or massive scheme 12,13,14,15,16,17 directly in three dimensions where the β functions have been computed up to five 11 and six loops 15 respectively. In each scheme, two different kinds of expansions for the corresponding β functions have been considered : i) ǫ-expansion (actually √ ǫ-expansion) 6,8,9,10,11 in the MS scheme and pseudo-√ ǫ-expansion 18 in the massive scheme ii) expansion in the coupling constants in three dimensions 12,13,14,15,16,17 . It has, however, appeared that the series obtained in case i) are non-Borel summable and thus inappropriate for the quantitative evaluation of critical exponents behavior 9,10,19 , a result anticipated by several authors 20,21 within an analysis of the zero-dimensional problem (see however 22 ).…”