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Physical review. B, Solid state

Renormalization-group calculation of the critical-point exponent η for a critical point of arbitrary order

G. F. Tuthill, Jeffrey F. Nicoll, Eugene Stanley

Published Jun 1, 1975• 44 citations
Publisher

Abstract

The critical-point exponent $\ensuremath{\eta}$ for a critical point of order $\mathcal{O}$ in dimensions less than ${d}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}$, is calculated to leading nonvanishing order in the parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}{d}_{\mathcal{O}}\ensuremath{-}d$. The result is given for $n$-component isotropically interacting magnetic systems. For Ising systems, $n=1$, the result is ${\ensuremath{\eta}}_{\mathcal{O}}=\frac{{\ensuremath{\epsilon}}_{\mathcal{O}}^{2}4{(\mathcal{O}\ensuremath{-}1)}^{2}}{{({2\mathcal{O}}{\mathcal{O}})}^{3}}$. As $\mathcal{O}$ increases, the coefficient of ${\ensuremath{\epsilon}}_{\mathcal{O}}^{2}$ rapidly becomes very small, varying as ${2}^{\ensuremath{-}6\mathcal{O}}$ for $\mathcal{O}$ large. In the limit of large $n$, ${\ensuremath{\eta}}_{\mathcal{O}}$ for odd order points approaches a constant and, for even order points, is proportional to $\frac{1}{n}$.

DOI
10.1103/physrevb.11.4579

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The critical-point exponent \ensuremathη\ensuremath{\eta}\ensuremathη for a critical point of order O\mathcal{O}O in dimensions less than dO\ensuremath≡2O(O\ensuremath−1){d}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}dO​\ensuremath≡(O\ensuremath−1)2O​, is calculated to leading nonvanishing order in the parameter \ensuremathϵO\ensuremath≡dO\ensuremath−d{\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}{d}_{\mathcal{O}}\ensuremath{-}d\ensuremathϵO​\ensuremath≡dO​\ensuremath−d. The result is given for nnn-component isotropically interacting magnetic systems. For Ising systems, n=1n=1n=1, the result is \ensuremathηO=\ensuremathϵO24(O\ensuremath−1)2(2OO)3{\ensuremath{\eta}}_{\mathcal{O}}=\frac{{\ensuremath{\epsilon}}_{\mathcal{O}}^{2}4{(\mathcal{O}\ensuremath{-}1)}^{2}}{{({2\mathcal{O}}{\mathcal{O}})}^{3}}\ensuremathηO​=(2OO)3\ensuremathϵO2​4(O\ensuremath−1)2​. As O\mathcal{O}O increases, the coefficient of \ensuremathϵO2{\ensuremath{\epsilon}}_{\mathcal{O}}^{2}\ensuremathϵO2​ rapidly becomes very small, varying as 2\ensuremath−6O{2}^{\ensuremath{-}6\mathcal{O}}2\ensuremath−6O for O\mathcal{O}O large. In the limit of large nnn, \ensuremathηO{\ensuremath{\eta}}_{\mathcal{O}}\ensuremathηO​ for odd order points approaches a constant and, for even order points, is proportional to 1n\frac{1}{n}n1​.
Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order O\mathcal{O}O, O\mathcal{O}O phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points (O=2\mathcal{O}=2O=2) and Gaussian tricritical points (O=3\mathcal{O}=3O=3) to critical points of arbitrary order O\mathcal{O}O. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for O=3,4\mathcal{O}=3,4O=3,4 are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for O=3\mathcal{O}=3O=3, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.
The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute mmm-vector model, the n=0n=0n=0 limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic mmm-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mcm<{m}_{c}m<mc​, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for m=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
Critical exponents are calculated for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}$ small, using renormalization-group techniques. To order $\ensuremath{\epsilon}$ the exponent $\ensuremath{\gamma}$ is $1+\frac{1}{6}\ensuremath{\epsilon}$ for an Ising-like model and $1+\frac{1}{5}\ensuremath{\epsilon}$ for an XY\mathrm{XY}XY model.
The tricirical exponents ${\ensuremath{\eta}}_{t}$ and ${\ensuremath{\varphi}}_{t}$ are calculated through order ${\ensuremath{\epsilon}}^{3}$ and ${\ensuremath{\epsilon}}^{2}$, respectively, in the isotropic nnn-component model, where $\ensuremath{\epsilon}=3\ensuremath{-}d$. The estimate for ${\ensuremath{\eta}}_{t}$, in two dimensions for the Ising case, is 0.027; the series for ${\ensuremath{\varphi}}_{t}$ is quite ill behaved, producing a negative estimate at this order. Beginning at this order in $\ensuremath{\epsilon}$, the spherical-model limit fails to exist. A scaling function for the spin-spin correlation function, appropriate for nonexceptional paths of approach to the tricritical point in the disordered phase, is calculated through ${\ensuremath{\epsilon}}^{2}$; its large momentum expansion is shown to involve the crossover exponent. These results are generalized to points where $\ensuremath{\theta}$ coexisting phases become critical.
The zero-field critical behavior of ferromagnets above Tc{T}_{c}Tc​, with both isotropic exchange coupling and dipolar interactions, is studied by renormalization-group techniques in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions with n(=d)n (=d)n(=d) component spins. The critical exponents are calculated to order ${\ensuremath{\epsilon}}^{2}$, and are found to be numerically very close to their nondipolar analogs. In particular the correlation length exponent is given by $2\ensuremath{\nu}=1+(\frac{9}{34})\ensuremath{\epsilon}+(\frac{7013}{58956}){\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$. The value of the specific-heat exponent ${\ensuremath{\alpha}}_{s} (=2\ensuremath{-}d\ensuremath{\nu})$ seems inconsistent with experimental data (at d=3d=3d=3, $\ensuremath{\epsilon}=1$). The critical scattering function is shown to have the form ${\ensuremath{\Gamma}}^{\ensuremath{\alpha}\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})=C{t}^{\ensuremath{-}\ensuremath{\gamma}}\stackrel{^}{D}({\ensuremath{\xi}}^{2}{q}^{2})({\ensuremath{\delta}}_{\ensuremath{\alpha}\ensuremath{\beta}}\ensuremath{-}\frac{{q}^{\ensuremath{\alpha}}{q}^{\ensuremath{\beta}}}{{q}^{2}})$ to all orders in $\ensuremath{\epsilon}$. The exponents related to the leading correction to scaling, to crossovers to anisotropic exchange behavior, and to cubic dipolar behavior, are also calculated, and are found to differ significantly from their nondipolar counterparts.