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Physical review. B, Condensed matter

Extension of the random-phase-approximation theory of ferromagnetism in a magnetic field applicable at all temperatures

A. Czachor, A. Holas

Published Mar 1, 1990• 9 citations
Publisher

Abstract

Generalization of the random-phase-approximation (RPA) theory of ferromagnetism for spins S=1/2, based on a refinement of the decoupling for a relevant three-spin Green's function, is presented. With the reasonable choice of two characteristic functions of the present theory---the virtual spectrum \ensuremath{\Delta}(\ensuremath{\omega}) and the decoupling factor \ensuremath{\kappa}(\ensuremath{\sigma}), where \ensuremath{\omega} is an excitation energy and \ensuremath{\sigma} is the relative magnetization---the theory provides a fair description of magnetic phenomena not only at temperature T\ensuremath{\rightarrow}0, in the vicinity of the Curie temperature ${\mathit{T}}_{\mathit{C}}$, or at high T, but at all temperatures and external magnetic fields H. A relation between some parameters of the functions \ensuremath{\Delta}(\ensuremath{\omega}) and \ensuremath{\kappa}(\ensuremath{\sigma}), and the critical indices of the static scaling theory, has been established. The form of the universal scaling function W, following from our theory, has been found. For the power-law virtual spectrum \ensuremath{\Delta}\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathit{s}}$ at \ensuremath{\omega}\ensuremath{\rightarrow}0, 0s1, we have found the nonanalytic increase of the magnetization versus the field at T${\mathit{T}}_{\mathit{C}}$:\ensuremath{\sigma}(T,H)-\ensuremath{\sigma}(T,0)\ensuremath{\propto}${\mathit{H}}^{\mathit{s}}$. It reduces to the classic result of Holstein and Primakoff for the three-dimensional isotropic Heisenberg ferromagnet having s=1/2. Assuming this power-law spectrum at all energies up to some cutoff energy, we have obtained a rather good agreement of the present predictions for \ensuremath{\sigma}(T,H) with experimental data for Ni, for all temperatures and fields available. Some model virtual spectra are found to lead to interesting models of ferromagnetism, with the magnetization and the susceptibility functions given in terms of explicit analytic formulas.

DOI
10.1103/physrevb.41.4674

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Generalization of the random-phase-approximation (RPA) theory of ferromagnetism for spins S=1/2, based on a refinement of the decoupling for a relevant three-spin Green's function, is presented. With the reasonable choice of two characteristic functions of the present theory---the virtual spectrum \ensuremath{\Delta}(\ensuremath{\omega}) and the decoupling factor \ensuremath{\kappa}(\ensuremath{\sigma}), where \ensuremath{\omega} is an excitation energy and \ensuremath{\sigma} is the relative magnetization---the theory provides a fair description of magnetic phenomena not only at temperature T\ensuremath{\rightarrow}0, in the vicinity of the Curie temperature TC{\mathit{T}}_{\mathit{C}}TC​, or at high T, but at all temperatures and external magnetic fields H. A relation between some parameters of the functions \ensuremath{\Delta}(\ensuremath{\omega}) and \ensuremath{\kappa}(\ensuremath{\sigma}), and the critical indices of the static scaling theory, has been established. The form of the universal scaling function W, following from our theory, has been found. For the power-law virtual spectrum \ensuremath{\Delta}\ensuremath{\propto}\ensuremathωs{\mathrm{\ensuremath{\omega}}}^{\mathit{s}}\ensuremathωs at \ensuremath{\omega}\ensuremath{\rightarrow}0, 0s1, we have found the nonanalytic increase of the magnetization versus the field at TTC{\mathit{T}}_{\mathit{C}}TC​:\ensuremath{\sigma}(T,H)-\ensuremath{\sigma}(T,0)\ensuremath{\propto}Hs{\mathit{H}}^{\mathit{s}}Hs. It reduces to the classic result of Holstein and Primakoff for the three-dimensional isotropic Heisenberg ferromagnet having s=1/2. Assuming this power-law spectrum at all energies up to some cutoff energy, we have obtained a rather good agreement of the present predictions for \ensuremath{\sigma}(T,H) with experimental data for Ni, for all temperatures and fields available. Some model virtual spectra are found to lead to interesting models of ferromagnetism, with the magnetization and the susceptibility functions given in terms of explicit analytic formulas.
Expressions for general correlation functions in Ising ferromagnets are derived in terms of the Lee-Yang theorem. By the use of these expressions, the asymptotic forms of correlation functions in the presence of magnetic field are analyzed, assuming those in the absence of the external field. In particular, the critical indices of spin-spin, energy-density-energy-density, energy-density-spin correlation functions are related with the critical indices of thermal susceptiblity χT0, specific heat, susceptibility χH0, and magnetization. These results agree in form with those predicted by the scaling law approach.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
We formulate the spectral-density method in classical statistical mechanics in strict analogy with the known quantum version, and we apply it to the classical Heisenberg ferromagnetic model in an external field. A new formula for the magnetization in the classical formulation of spin-vector Green functions is derived for arbitrary spatial dimensionality. Furthermore, the static properties of the one- and three-dimensional cases are considered in detail. We obtain accurate results over a wide range of temperatures with the external field for the one-dimensional model and without field for the three-dimensional model. In the first case very good agreement with the exact numerical transfer-matrix data is also found in the region of higher temperature where the interacting-boson approach fails. In particular the zero-field susceptibility results are found to be consistent with the exact results obtained by Fisher.
A generalized random-phase approximation (RPA) for decoupling the double-time Green's functions (DTGF) equations of motion of spin systems is developed. The decoupling parameters are chosen by fixing the moments of both the commutator and anticommutator decoupled spectral functions, thus establishing a well-defined relationship between the approximation scheme and the fundamental properties of DTGF. It is shown that the standard RPA must be supplemented by a condition on correlations to provide a well-defined, complete approximation scheme. The procedure is demonstrated by application to the fully anisotropic S=1/2 Heisenberg model in a field.
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.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.