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Physical review. B, Condensed matter

Critical exponents by the scaling-field method: The isotropic N -vector model in three dimensions

Kathie E. Newman, Eberhard K. Riedel

Published Dec 1, 1984• 103 citations
Publisher

Abstract

A numerical technique, termed the scaling-field method, is developed for solving by successive approximation Wilson's exact renormalization-group equation for critical phenomena in three-dimensional spin systems. The approach uses the scaling-field representation of the Wilson equation derived by Riedel, Golner, and Newman. A procedure is proposed for generating in a nonperturbative and unbiased fashion sequences of successively larger truncations to the infinite hierarchy of scaling-field equations. A "principle of balance" is introduced and used to provide a self-consistency criterion. The approach is then applied to the isotropic $N$-vector model. Truncations to order 13 (10, when $N=1$) scaling-field equations yield the leading critical exponents, $\ensuremath{\nu}$ and $\ensuremath{\eta}$, and several of the correction-to-scaling exponents, ${\ensuremath{\Delta}}_{m}$, to high precision. Results for $N=0, 1, 2, \mathrm{and} 3$ are tabulated. For the Ising case ($N=1$), the estimates $\ensuremath{\nu}=0.626\ifmmode\pm\else\textpm\fi{}0.009$, $\ensuremath{\eta}=0.040\ifmmode\pm\else\textpm\fi{}0.007$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\equiv}{\ensuremath{\Delta}}_{400}=0.54\ifmmode\pm\else\textpm\fi{}0.05$ are in good agreement with recent high-temperature-series results, though exhibiting larger confidence limits at the present level of approximation. For the first time, estimates are obtained for the second and third correction-to-scaling exponents. For example, for the Ising model the second "even" and first "odd" correction-to-scaling exponents are ${\ensuremath{\Delta}}_{422}=1.67\ifmmode\pm\else\textpm\fi{}0.11$ and ${\ensuremath{\Delta}}_{500}=1.5\ifmmode\pm\else\textpm\fi{}0.3$, respectively. Extensions necessary to improve the accuracy of the calculation are discussed, while applications of the approach to anisotropic $N$-vector models are described elsewhere. Finally, the scaling-field method is compared with other techniques for the high-precision calculation of critical phenomena in three dimensions, i.e., high-temperature-series, Monte Carlo renormalization-group, and field-theoretic perturbation expansions.

DOI
10.1103/physrevb.30.6615

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arXiv:cond-mat/0107401•Physical review. B, Condensed matter•Oct 16, 2001•1 citation
A numerical technique, termed the scaling-field method, is developed for solving by successive approximation Wilson's exact renormalization-group equation for critical phenomena in three-dimensional spin systems. The approach uses the scaling-field representation of the Wilson equation derived by Riedel, Golner, and Newman. A procedure is proposed for generating in a nonperturbative and unbiased fashion sequences of successively larger truncations to the infinite hierarchy of scaling-field equations. A "principle of balance" is introduced and used to provide a self-consistency criterion. The approach is then applied to the isotropic NNN-vector model. Truncations to order 13 (10, when N=1N=1N=1) scaling-field equations yield the leading critical exponents, \ensuremathν\ensuremath{\nu}\ensuremathν and \ensuremathη\ensuremath{\eta}\ensuremathη, and several of the correction-to-scaling exponents, \ensuremathΔm{\ensuremath{\Delta}}_{m}\ensuremathΔm​, to high precision. Results for N=0,1,2,and3N=0, 1, 2, \mathrm{and} 3N=0,1,2,and3 are tabulated. For the Ising case (N=1N=1N=1), the estimates \ensuremathν=0.626\ifmmode±\else\textpm\fi0.009\ensuremath{\nu}=0.626\ifmmode\pm\else\textpm\fi{}0.009\ensuremathν=0.626\ifmmode±\else\textpm\fi0.009, \ensuremathη=0.040\ifmmode±\else\textpm\fi0.007\ensuremath{\eta}=0.040\ifmmode\pm\else\textpm\fi{}0.007\ensuremathη=0.040\ifmmode±\else\textpm\fi0.007, and \ensuremathΔ1\ensuremath≡\ensuremathΔ400=0.54\ifmmode±\else\textpm\fi0.05{\ensuremath{\Delta}}_{1}\ensuremath{\equiv}{\ensuremath{\Delta}}_{400}=0.54\ifmmode\pm\else\textpm\fi{}0.05\ensuremathΔ1​\ensuremath≡\ensuremathΔ400​=0.54\ifmmode±\else\textpm\fi0.05 are in good agreement with recent high-temperature-series results, though exhibiting larger confidence limits at the present level of approximation. For the first time, estimates are obtained for the second and third correction-to-scaling exponents. For example, for the Ising model the second "even" and first "odd" correction-to-scaling exponents are \ensuremathΔ422=1.67\ifmmode±\else\textpm\fi0.11{\ensuremath{\Delta}}_{422}=1.67\ifmmode\pm\else\textpm\fi{}0.11\ensuremathΔ422​=1.67\ifmmode±\else\textpm\fi0.11 and \ensuremathΔ500=1.5\ifmmode±\else\textpm\fi0.3{\ensuremath{\Delta}}_{500}=1.5\ifmmode\pm\else\textpm\fi{}0.3\ensuremathΔ500​=1.5\ifmmode±\else\textpm\fi0.3, respectively. Extensions necessary to improve the accuracy of the calculation are discussed, while applications of the approach to anisotropic NNN-vector models are described elsewhere. Finally, the scaling-field method is compared with other techniques for the high-precision calculation of critical phenomena in three dimensions, i.e., high-temperature-series, Monte Carlo renormalization-group, and field-theoretic perturbation expansions.
A scaling‐field representation of Wilson’s exact renormalization‐group equation is derived and used to calculate critical exponents for the continuous‐spin Ising model in three dimensions. Here the results for two successive truncations of the infinite hierarchy of scaling‐field equations are presented. The truncation that retains only the two most relevant scaling fields yields the critical exponents ν?0.55 and η=0; the truncation retaining the four most relevant fields yields the improved values ν?0.60 and η?0.06. Approximations to the full hierarchy of scaling‐field equations introduce a weak dependence of the critical exponents on redundant parameters in the renormalization‐group Hamiltonian. That question is also discussed.
A scaling-field representation of Wilson's exact renormalization-group equation is derived and used for the approximate calculation of critical exponents for the continuous-spin Ising model in three dimensions. The truncation of the hierarchy of scaling-field equations that retains only the four most relevant scaling fields yields the critical exponents $\ensuremath{\nu}\ensuremath{\approx}0.60$, $\ensuremath{\eta}\ensuremath{\approx}0.06$, and ${\ensuremath{\Delta}}_{1}\ensuremath{\approx}\ensuremath{-}0.47$ (where ${\ensuremath{\Delta}}_{1}$ denotes Wortis's correction-to-scaling exponent).
Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, $2.8\ensuremath{\le}d\ensuremath{\le}4$, and number of spin components, NNN. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation. Fixed points studied include the isotropic and decoupled Ising ($\ensuremath{-}2\ensuremath{\le}N<\ensuremath{\infty}$), the face-and corner-ordered cubic ($1\ensuremath{\le}N<\ensuremath{\infty}$), and, via the replica method for $N\ensuremath{\rightarrow}0$, the quenched random Ising fixed point. Variations of NNN and ddd are used to link the results to exact results or results from other calculational methods, such as $\ensuremath{\epsilon}$ expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic NNN-vector model. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of ddd, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At d=3d=3d=3, the short truncation yields $\ensuremath{\alpha}\ensuremath{\approx}0.11$ for the pure Ising and $\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09$ for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the NNN-component cubic model, the spin dimensionality Nc{N}_{c}Nc​, at which the isotropic and cubic fixed points change stability, is determined as a function of ddd. The results support Nc>3{N}_{c}>3Nc​>3 for three dimensions.
In a recent article we have shown that, by applying sophisticated summation methods to Wilson-Fisher's ε-expansion, it is possible from the presently known terms of the series to obtain accurate values of critical exponents for the 0 ( n ) symmetric n-vector model : these values are consistent with the best estimates obtained from three-dimensional Renormalization Group calculations and, in the case of Ising-like systems, with the exactly known two-dimensional values of the Ising model. The controversial conjecture has been recently formulated that some fractal lattices could interpolate regular lattices in non-integer dimensions. Numerical calculations have been done for the Ising model. To allow for direct comparison with Renormalization Group values, we present here estimates for exponents in non-integer dimensions d(1
A renormalization-group scheme is developed for the three-dimensional O(2N)O(2N)O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a 1/N1/N1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at z=0,z=0,z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at z=0.z=0.z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of lny.\mathrm{ln}y.lny.