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Physical review. B, Condensed matter

Cubic N -vector model and randomly dilute Ising model in general dimensions

Kathie E. Newman, Eberhard K. Riedel

Published Jan 1, 1982• 153 citations
Publisher

Abstract

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, $N$. 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 $N$ and $d$ 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 $N$-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 $d$, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At $d=3$, 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 $N$-component cubic model, the spin dimensionality ${N}_{c}$, at which the isotropic and cubic fixed points change stability, is determined as a function of $d$. The results support ${N}_{c}>3$ for three dimensions.

DOI
10.1103/physrevb.25.264

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Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, 2.8\ensuremath≤d\ensuremath≤42.8\ensuremath{\le}d\ensuremath{\le}42.8\ensuremath≤d\ensuremath≤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≤N<\ensuremath∞\ensuremath{-}2\ensuremath{\le}N<\ensuremath{\infty}\ensuremath−2\ensuremath≤N<\ensuremath∞), the face-and corner-ordered cubic (1\ensuremath≤N<\ensuremath∞1\ensuremath{\le}N<\ensuremath{\infty}1\ensuremath≤N<\ensuremath∞), and, via the replica method for N\ensuremath→0N\ensuremath{\rightarrow}0N\ensuremath→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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ 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 \ensuremathϵ12{\ensuremath{\epsilon}}^{\frac{1}{2}}\ensuremathϵ21​ fixed point. At d=3d=3d=3, the short truncation yields \ensuremathα\ensuremath≈0.11\ensuremath{\alpha}\ensuremath{\approx}0.11\ensuremathα\ensuremath≈0.11 for the pure Ising and \ensuremathα\ensuremath≈\ensuremath−0.09\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09\ensuremathα\ensuremath≈\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.
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, $\ensuremath{\nu}$ and $\ensuremath{\eta}$, and several of the correction-to-scaling exponents, ${\ensuremath{\Delta}}_{m}$, to high precision. Results for N=0,1,2,and3N=0, 1, 2, \mathrm{and} 3N=0,1,2,and3 are tabulated. For the Ising case (N=1N=1N=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 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.
The detailed analysis of the global structure of the renormalization-group (RG) 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 (3D) 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 Pad\'e-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 ${N}_{c}=2.89\ifmmode\pm\else\textpm\fi{}0.02$ that agrees well with the estimate obtained on the basis of the five-loop $\ensuremath{\varepsilon}$ expansion [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)] resummed by the above method. As a consequence, the cubic fixed point should be stable in 3D for N> 3,N>~3,N> 3, 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 N=0N=0N=0 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.
We consider the Ginzburg-Landau MN\mathrm{MN}MN model that describes MNMNMN-vector cubic models with O(M)O(M)O(M)-symmetric couplings. We compute the renormalization-group functions to six-loop order in d=3.d=3.d=3. We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model (M=1).(M=1).(M=1). We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For M> 2M>~2M> 2 we show that the O(M)O(M)O(M) fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.
We study the combined effect of cubic anisotropy and quenched uncorrelated impurities on multicomponent spin models. For this purpose, we consider the field-theoretical approach based on the Ginzburg-Landau-Wilson ${\ensuremath{\varphi}}^{4}$ Hamiltonian with cubic-symmetric quartic interactions and quenched randomness coupled to the local energy density. We compute the renormalization-group functions to six loops in the fixed-dimension (d=3)(d=3)(d=3) perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).