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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical behavior of energy-energy, strain-strain, higher-harmonics,and similar correlation functions

Roland R. Netz, Amnon Aharony A

Published Mar 1, 1997• 7 citations
Publisher

Abstract

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}-|${\mathrm{Ax}}_{0}$[(T-${\mathrm{T}}_{\mathrm{c}}$)/${\mathrm{T}}_{\mathrm{c}}$${]}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$, with the same n-dependent exponent \ensuremath{\nu} as for the correlation length characterizing fluctuations of the primary order parameter, which is denoted by \ensuremath{\xi}. The amplitude ratio X-|Ax\ensuremath{\equiv}(\ensuremath{\xi}-|${\mathrm{Ax}}_{0}$/${\ensuremath{\xi}}_{0}$${)}^{2}$ is universal, and we obtain ${\mathrm{X}}_{\mathrm{T}}$=${\ensuremath{\gamma}}_{\mathrm{T}}$/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 ${\mathrm{X}}_{\mathrm{E}}$=\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 ${\mathrm{X}}_{\mathrm{T}}$ with n=2 and takes the value ${\mathrm{X}}_{2}$=\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-${\mathrm{A}}_{2}$ 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.

DOI
10.1103/physreve.55.2267

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Abstract One of the fundamental problems of statistical mechanics and its quantum field theory formulation is the characterization of the order parameters and the computation of their correlation functions. Besides the intrinsic interest of this problem, the correlation functions are the key quantities in the determination of the universal ratios of the renormalization group and therefore they can have direct experimental confirmation (see Chapter 8). In general, the computation of correlation functions is a difficult task, usually achieved with partial success through perturbative methods.

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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{\mathrm{Ax}}_{0}Ax0​[(T-Tc{\mathrm{T}}_{\mathrm{c}}Tc​)/{\mathrm{T}}_{\mathrm{c}}$${]}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}, with the same n-dependent exponent \ensuremath{\nu} as for the correlation length characterizing fluctuations of the primary order parameter, which is denoted by \ensuremath{\xi}. The amplitude ratio X-|Ax\ensuremath{\equiv}(\ensuremath{\xi}-|Ax0{\mathrm{Ax}}_{0}Ax0​/{\ensuremath{\xi}}_{0}$${)}^{2} is universal, and we obtain XT{\mathrm{X}}_{\mathrm{T}}XT​=\ensuremathγT{\ensuremath{\gamma}}_{\mathrm{T}}\ensuremathγT​/6\ensuremath{\gamma}+O(\ensuremathϵ3{\mathrm{\ensuremath{\epsilon}}}^{3}\ensuremathϵ3) for quadratic order parameters transforming like a traceless spin tensor in n-component space (with \ensuremathγT{\ensuremath{\gamma}}_{\mathrm{T}}\ensuremathγT​ characterizing the divergence of the corresponding susceptibility) and XE{\mathrm{X}}_{\mathrm{E}}XE​=\ensuremath{\alpha}/6\ensuremath{\gamma}+O(\ensuremathϵ3{\mathrm{\ensuremath{\epsilon}}}^{3}\ensuremathϵ3) 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{\mathrm{X}}_{\mathrm{T}}XT​ with n=2 and takes the value X2{\mathrm{X}}_{2}X2​=\ensuremath{\epsilon}/20-\ensuremathϵ2{\mathrm{\ensuremath{\epsilon}}}^{2}\ensuremathϵ2/100+O(\ensuremathϵ3{\mathrm{\ensuremath{\epsilon}}}^{3}\ensuremathϵ3), 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{\mathrm{A}}_{2}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.
Cluster-Monte-Carlo simulations are used to calculate the structure factors associated with order-parameter and energy correlations for square and cubic Ising models, from which the corresponding correlation lengths ξ and ξE, respectively, are determined. The amplitude ratio XE≡ξE2 /ξ2 is asymptotically universal but strongly influenced by the crossover from mean-field behavior. The numerically determined XE therefore does not reach its universal asymptotic value but agrees well with ε-expansion predictions including corrections to scaling. The consequences for experiments on near-critical fluids and colloidal systems are briefly discussed.
The approach of Chapter 11, in which we have kept a large momentum or short-distance cut-off, is sufficient for the needs of computing the various universal indices and functions that characterize criticality. All calculations were made in the limit in which the ratio ξ/a of the correlation length ξ to the lattice spacing a is large. In the previous chapters, this was done by keeping a fixed and increasing ξ . The renormalized theory relies on the opposite strategy: it deals directly with the scaling theory in which one takes the limit of a vanishingly small a . I have included here a chapter on the renormalized theory for several reasons: • There is no ultraviolet (or lattice) cut-off any more and, getting rid of one dimensionful parameter considerably simplifies the integrations, especially at higher orders. • All the critical exponents and scaling functions can be computed at T c , i.e., in a massless theory, even when one considers temperature-dependent laws, such as those concerning the correlation length, the magnetic susceptibility or the spontaneous magnetization. Integrals in a massless theory without cut-off are by far simpler, and they are often given by mere dimensional analysis. • However, the main reason is to understand the meaning of renormalizability from the viewpoint both of critical phenomena and of high-energy physics. This will be done in the following section.
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 ${a}_{\ensuremath{\xi}}^{+}{/a}_{\ensuremath{\chi}}^{+}=0.87(6).$
High-temperature series expansions are used to examine the dependence of critical-point exponents upon lattice anisotropy (different interaction strengths in different directions in the lattice). The two-spin correlation function ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is calculated to tenth order in (1kBT\frac{1}{{k}_{B}T}kB​T1​) for the Ising Hamiltonian ${\mathcal{H}}_{l \mathrm{anis}}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{S}_{i}^{z}{S}_{j}^{z}$ for a wide range of anisotropy parameters $R\ensuremath{\equiv}\frac{{J}_{z}}{{J}_{\mathrm{xy}}}$ and for both the sc and fcc lattices; here the first summation is over all pairs of nearest-neighbor sites whose relative displacement vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{\mathrm{ij}}$ has no zzz component, while the second summation is over all other pairs of nearest-neighbor sites. Hence for R=0R=0R=0, both the sc and fcc lattices reduce to the two-dimensional square lattice, while in the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, the sc becomes a one-dimensional linear chain and the fcc becomes two noninteracting three-dimensional bcc lattices. The series for ${C}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ are then used to obtain series of corresponding lengths for the specific heat, susceptibility, and second moment. Analysis of these series yields results consistent with the universality hypothesis of critical-point exponents. Specifically, it is found that when lattice anisotrophy is introduced, the critical-point exponents studied (the susceptibility exponent $\ensuremath{\gamma}$ and the correlation length exponent $\ensuremath{\nu}$) do not appear to change from their values for an isotropic lattice. The problem of next-nearest-neighbor interactions is treated using similar methods in Paper II of this series (and briefly discussed in this paper).
Simple but rigorous derivations are presented for the inequalities2−η≤dγ′/(2β+γ′)≤dγ′/(2−α′),2−η≤d(δ−1)/(δ+1),γ≤(2−η)ν.(Proofs of the first two inequalities have been given by Buckingham and Gunton.1) Here d is the dimensionality and α′, β, γ, γ′ and δ are the exponents characterizing a ferromagnet near its critical point,2 while η and ν describe the decay of the corresponding spin-spin correlation functions2 Γ2(r)=〈S0zSrz〉, according to ∑ |r|<RΓ2(r)∼R2−η(T=Tc,H=0,R→∞),κ1∼(T−Tc)ν,(T→Tc,H=0),where the inverse range of correlation κ1 may be defined by κ1−2=[Σrr2Γ2(r)]/[ΣrΓ2(r)]. The corresponding exponent ηE for the energy-energy correlation function (essentially Γ4(r, a)−[Γ(a)]2, where Γ4(r, a)=〈S0zSazSrzSr+az〉) is proved to satisfy 2−ηE≤dα′,2−ηE≤dαc/(1+ζ+αc),where the specific heat CM at T=Tc diverges with the magnetization as M−αc while the energy derivative | ∂U/∂M |Tc varies as Mζ. (Mean Field or classical values are αc=0, ζ=1.) The proofs are based on the ``intuitively obvious'' properties: (a) positivity, namely, Γ1=〈S0z〉≥0,Γ(r)−Γ12≥0,Γ4(r, a)−[Γ2(a)]2≥0, and (b) monotonic increase of Γ1, Γ2 and Γ4 under increase of magnetic field and decrease of temperature. These properties are known3 to be rigorously valid for Ising models of arbitrary spin, lattice structure and ferromagnetic coupling (Jij≥0). Their validity for real magnets together with the experimental observation δ≤4.7 leads to the significant conclusion η≥0.05.
Abstract One of the fundamental problems of statistical mechanics and its quantum field theory formulation is the characterization of the order parameters and the computation of their correlation functions. Besides the intrinsic interest of this problem, the correlation functions are the key quantities in the determination of the universal ratios of the renormalization group and therefore they can have direct experimental confirmation (see Chapter 8). In general, the computation of correlation functions is a difficult task, usually achieved with partial success through perturbative methods.