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arXiv:cond-mat/9912071 · Physical review. B, Condensed matter

Five-loop renormalization-group expansions for the three-dimensionaln-vector cubic model and critical exponents for impure Ising systems

D. V. Pakhnin, Aleksandr Ivanovich Sokolov

Published Jun 1, 2000• 60 citations
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Abstract

The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pad\'e-Borel-Leroy resummation under $n=0.$ These exponents are found to be $\ensuremath{\gamma}=1.325\ifmmode\pm\else\textpm\fi{}0.003,$ $\ensuremath{\eta}=0.025\ifmmode\pm\else\textpm\fi{}0.01,$ $\ensuremath{\nu}=0.671\ifmmode\pm\else\textpm\fi{}0.005,$ $\ensuremath{\alpha}=\ensuremath{-}0.0125\ifmmode\pm\else\textpm\fi{}0.008,$ and $\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.006.$ For the correction-to-scaling exponent, the less accurate estimate $\ensuremath{\omega}=0.32\ifmmode\pm\else\textpm\fi{}0.06$ is obtained.

DOI
10.1103/physrevb.61.15130

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The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pad'e-Borel-Leroy resummation under n=0.n=0.n=0. These exponents are found to be \ensuremathγ=1.325\ifmmode±\else\textpm\fi0.003,\ensuremath{\gamma}=1.325\ifmmode\pm\else\textpm\fi{}0.003,\ensuremathγ=1.325\ifmmode±\else\textpm\fi0.003, \ensuremathη=0.025\ifmmode±\else\textpm\fi0.01,\ensuremath{\eta}=0.025\ifmmode\pm\else\textpm\fi{}0.01,\ensuremathη=0.025\ifmmode±\else\textpm\fi0.01, \ensuremathν=0.671\ifmmode±\else\textpm\fi0.005,\ensuremath{\nu}=0.671\ifmmode\pm\else\textpm\fi{}0.005,\ensuremathν=0.671\ifmmode±\else\textpm\fi0.005, \ensuremathα=\ensuremath−0.0125\ifmmode±\else\textpm\fi0.008,\ensuremath{\alpha}=\ensuremath{-}0.0125\ifmmode\pm\else\textpm\fi{}0.008,\ensuremathα=\ensuremath−0.0125\ifmmode±\else\textpm\fi0.008, and \ensuremathβ=0.344\ifmmode±\else\textpm\fi0.006.\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.006.\ensuremathβ=0.344\ifmmode±\else\textpm\fi0.006. For the correction-to-scaling exponent, the less accurate estimate \ensuremathω=0.32\ifmmode±\else\textpm\fi0.06\ensuremath{\omega}=0.32\ifmmode\pm\else\textpm\fi{}0.06\ensuremathω=0.32\ifmmode±\else\textpm\fi0.06 is obtained.
The renormalization-group functions governing the critical behavior of the three-dimensional weakly-disordered Ising model are calculated in the five-loop approximation. The random fixed point location and critical exponents for impure Ising systems are estimated by means of the Padé-Borel-Leroy resummation of the renormalization-group expansions derived. The asymptotic critical exponents are found to be γ=1.325 ± 0.003, η=0.025 ± 0.01, ν= 0.671 ± 0.005, α=−0.0125 ± 0.008, β=0.344 ± 0.006, while for the correction-to-scaling exponent, a less accurate estimate ω=0.32 ± 0.06 is obtained.
Critical exponents for the three-dimensional O(n)-symmetric model with n>3 are estimated on the basis of six-loop renormalization-group (RG) expansions. A simple Pad\'e-Borel technique is used for the resummation of the RG series and the Pad\'e approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n expansion. The same value n=28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.
The critical thermodynamics of the two-dimensional NNN-vector cubic and MNMNMN models is studied within the field-theoretical renormalization group (RG) approach. The $\ensuremath{\beta}$ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various NNN are investigated. For N=2N=2N=2 it is found that the continuous line of fixed points running from the XYXYXY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both $\ensuremath{\beta}$ functions closer to each another. For the cubic model with $N\ensuremath{\geqslant}3$, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2N>2N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M)O(M)O(M) models (MNMNMN models with N=0N=0N=0) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the MNMNMN model with $M,N\ensuremath{\geqslant}2$ all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of MMM and NNN.
The critical behavior of the two-dimensional N-vector cubic model is studied within the field-theoretical renormalization-group (RG) approach. The beta-functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pade-Borel-Leroy and conformal mapping techniques. It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta-functions closer to each another. For N > 2 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N > 2 is an artifact of the perturbative analysis. In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.
The critical behavior of the two-dimensional N -vector cubic model is studied within the field-theoretical renormalization-group (RG) approach.The β functions and critical exponents are calculated in the five-loop approximation, RG series obtained are resummed using Pad é-Borel-Leroy and conformal mapping techniques.It is found that for N = 2 the continuous line of fixed points is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both β functions closer to each other.For N 3 the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point.This confirms the idea that the existence of the cubic fixed point in two dimensions under N > 2 is an artifact of the perturbative analysis.In the case N = 0 the results obtained are compatible with the conclusion that the impure critical behavior is controlled by the Ising fixed point.
RG functions are calculated up to 4-loop order in 3 and 2 dimensions to evaluate, within Chisholm-Borel resummation technique, critical exponents for cubic and impure uniaxial crystals and the order parameter critical dimensionality nc . For 3D impure Ising crystals γ = 1.326, v = 0.670, α = -0.011, β = 0.342, η = 0.034, while critical exponents of 3D cubic crystals practically coincide with those of the Heisenberg model; nc = 2.9. For 2D systems the fermion representation enables us to find that their critical behavior is identical to that of pure Ising model, apart some logarithmic factors.