The structure factor associated with general biquadratic correlation functions is calculated for an n-component order parameter using \ensuremath{\epsilon}-expansion techniques in d=4-\ensuremath{\epsilon} dimensions. The results apply to energy-energy and strain-strain correlations as well as to correlations of higher harmonics in density wave systems. We find the correlations of these secondary order parameters to be characterized by a correlation length \ensuremath{\xi}-|Ax=\ensuremath{\xi}-|
Ax0[(T-
Tc)/
Tc${]}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$, with the same n-dependent exponent \ensuremath{\nu} as for the correlation length characterizing fluctuations of the primary order parameter, which is denoted by \ensuremath{\xi}. The amplitude ratio X-|Ax\ensuremath{\equiv}(\ensuremath{\xi}-|
Ax0/${\ensuremath{\xi}}_{0}$
)2 is universal, and we obtain
XT=${\ensuremath{\gamma}}_{\mathrm{T}}$/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for quadratic order parameters transforming like a traceless spin tensor in n-component space (with ${\ensuremath{\gamma}}_{\mathrm{T}}$ characterizing the divergence of the corresponding susceptibility) and
XE=\ensuremath{\alpha}/6\ensuremath{\gamma}+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) for energy-energy correlations, where \ensuremath{\alpha} and \ensuremath{\gamma} denote the usual specific heat and susceptibility critical exponents, respectively. The universal amplitude ratio for the second harmonic in density wave systems is given by
XT with n=2 and takes the value
X2=\ensuremath{\epsilon}/20-${\mathrm{\ensuremath{\epsilon}}}^{2}$/100+O(${\mathrm{\ensuremath{\epsilon}}}^{3}$), thus being very small. This naturally explains previously puzzling experimental results for the critical behavior of the second harmonic structure factor at the nematic--smectic-
A2 transition of a thermotropic liquid crystal. Applications to sound attenuation in liquids or solids close to critical transitions and to colloidal interactions in near-critical binary mixtures are briefly discussed.
The energy-density-energy-density correlation function is studied in the three-dimensional Ising model in zero external field, for various spins on the fcc lattice. An extension of the linked-cluster technique is used to obtain high-temperature series expansions for the correlation function and its associated functions, to at least eighth order in the high-temperature interaction variable. Analysis of the series provides values for their critical indices and amplitudes. These values are used to verify the predictions of several theories of critical behavior. The critical index of the coherence length of the energy density correlation function is found to be equal to that of the spin-spin correlation function for all spins within the accuracy obtained here. For spins higher than ~ this critical index appears to obey strong scaling. An analysis of universality indicates that only one scale factor is needed to transform the energy density correlation function from one spin to another. This prediction is checked here and found to be good within our present accuracy. The current-algebra hypothesis predicts definite relations between the leading singularities of various correlation functions at the critical point. This prediction is checked directly and consistent values of the leading coefficients are obtained. The shape of the energy density correlation function at large distances from the origin, both at and above the critical point, is determined. The results are sufficiently accurate to rule out the Ornstein-Zernike form in zero field for T > T , but not sufficient to eliminate the possibility c of anisotropy at T = T .
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).
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
Using a renormalized linked-cluster-expansion method, we have extended to order ${\ensuremath{\beta}}^{23}$ the high-temperature series for the susceptibility $\ensuremath{\chi}$ and the second-moment correlation length $\ensuremath{\xi}$ of the spin-1/2 Ising models on the sc and the bcc lattices. A study of these expansions yields updated direct estimates of universal parameters, such as exponents and amplitude ratios, which characterize the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}.$ Our best estimates for the inverse critical temperatures are ${\ensuremath{\beta}}_{c}^{\mathrm{sc}}=0.221654(1)$ and ${\ensuremath{\beta}}_{c}^{\mathrm{bcc}}=0.1573725(6).$ For the susceptibility exponent we get $\ensuremath{\gamma}=1.2375(6)$ and for the correlation length exponent $\ensuremath{\nu}=0.6302(4).$ The ratio of the critical amplitudes of $\ensuremath{\chi}$ above and below the critical temperature is estimated to be ${C}_{+}{/C}_{\ensuremath{-}}=4.762(8).$ The analogous ratio for $\ensuremath{\xi}$ is estimated to be ${f}_{+}{/f}_{\ensuremath{-}}=1.963(8).$ For the correction-to-scaling amplitude ratio we obtain ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$