Correlation functions of quenched and annealed Ising systems

Spin correlation functions (up to the 3-site one) of disordered Ising model with the nearest neighbour interaction are calculated and investigated within a two-site cluster approximation for both quenched and annealed cases. The approach yields the exact results for the one-dimensional system. The long-range interaction is taken into account in the mean field approximation.


Introduction
During recent years the statistical theory of disordered spin models is an object of great interest due to their various properties and applications. In this paper we will consider an alloy of Ising systems (AIS) that can describe the processes of (magnetic or ferroelectric) ordering in alloys of magnets, magnets with nonmagnetic admixtures, solid solutions of ferroelectrics etc. This model is considered usually in two limiting cases, namely the case of annealed system (equilibrium type of disorder) and the case of quenched system (nonequilibrium disorder). There is no common opinion on the question, what type of disorder is realized in partially deuterrated ferroelectrics, e.g. Cs(D x H 1−x ) 2 P O 4 , K(D x H 1−x ) 2 P O 4 . This situation requires both annealed and quenched cases to be theoretically studied and compared with experimental data.
The exact results for both limits are known at the one-dimensional case [ 1]. Effect of dilution (noninteracting admixture) on the two-and threedimensional Ising model (quenched case) have been studied by Monte-Carlo method [ 2,3]. AIS can also be used as a test of approximative methods of statistical mechanics. It is known that the mean field approximation (MFA) yields quite satisfactory results for the Ising model. However it is not able to reproduce some essential properties of AIS [ 4] (such as percolation in diluted system and some differences between the properties of annealed and quenched systems).
In spite of great attention to AIS and great advance in its investigation (see review in [ 5,6]) the problem of calculation of AIS's correlation functions (CFs) for the case D ≥ 2 (within some acceptable approximation) is not solved up to now. In this paper we suggest the solution of this problem for both quenched and annealed systems generalizing the approach of [ 7] on disordered model. The short-range interactions are taken into account within the two-site cluster approximation (TCA). In the appendix the longrange interactions are also taken into consideration within MFA. where describes a quasispin subsystem on the sites of a simple lattice (i, j = 1 · · · N ) with the pair exchange interaction K ij in the site dependent field κ i . The set of spin variables {S i } represents a state of the spin subsystem The other part of the Hamiltonian includes "nonexchange" pair interaction V ij and field µ i . Each site contains a spin of a certain sort and the sort configuration is described by the set of variables {X iα } (α = 1 · · · Ω, Ω is a number of sorts): It should be noted that parameters of the Hamiltonian contain inverse temperature β. We suppose the pair interactions to be short-range V iα,jβ = βV αβ π ij ; βK iα,jβ = K αβ π ij ; π ij = 1, if i ∈ π j 0, in opposite case , (2.5) where π i denotes the set of the nearest neighbours of the site i (the first coordination sphere). In an appendix we will also take into account the long-range interactions in MFA.
The model is considered at the cases of annealed system and of quenched one. At the first case an equilibrium sort configuration is realized and the system is described by the density matrix as well as by the generating function where G is the grand thermodynamic potential of the system.
The generating function allows to calculate the correlation functions (CFs) of the system where S iα = S i X iα , the superscript c means cumulant averaging and The chemical potentials µ iα have to be found from the equations where c α is a concentration of the sort α spins. At the case of quenched type of disorder the sort configuration is fixed and independent of temperature, therefore thermodynamic averaging implies only a trace over spin degrees of freedom In order to obtain observable quantities one must perform also an averaging over sort configurations where the distribution ρ({X}) is determined by the conditions of system's freezing. Our approximation is sensitive to the following moments of this distribution: (2.14) Free energy of a quenched system is defined as follows (2.15) Function F x x = F x x ({κ}) that is the cumulant generating function in this case: (2.16)

Quenched case
First we carry out a cluster expansion of the generating function F x x . For this purpose we write the Hamiltonian H x ({S}) in the form where We introduce here the parameters rφi , which play a role of the effective field acting on the spin S i from the nearest neighbour at the site r. Summation (ij) in (3.1) spans pairs of the nearest neighbour sites. We can write function F x in the form Here we use the notations The first term K 1 of a cluster expansion [ 6,8] of ln Q has the form ln Q = ln exp (3.6) where z = j π ij is a number of the nearest neighbour sites and So the generating function F x x restricted by the first term of the cluster expansion is obtained in the form Parameters rφiα are found due to the minimization of the free energy It gives the following self-consistency equations for the average value of spin m where j ∈ π i and (3.12) Figure 1: Order parameters of pure and dilute (w 11 = c 2 1 ) systems on plane square lattice (z = 4).
Relations (3.10), (3.11) can be written in the form: iα,r x . (3.14) Here the notations are introduced. Below we give the explicit expressions for the quantities F i x , F ij x and their derivatives which will be used later: The expression (3.11) contains N zΩ equations for the same number of variables rφiα . At the case of uniform field (κ iα −→ κ α ) parameters rφiα lose their site dependence and (3.11) reduces to Ω equations for the same number of fields and we obtain a well-known result of Bethe approximation and cluster variation method for average value of spin We can obtain CFs of any order with differentiation of (3.10), (3.11) and it is an advantage of presented approach. It follows from (3.13) that Hereafter the summation over sort indices (Greek letters) is not written explicitly so the sum over γ symbol is omitted in (3.20). We have just used the notation ). It is convenient to write the relation (3.20) in the matrix formm where the matrix notation is introduced: In order to calculate the quantities rφ iα,jβ it is necessary to use the relation (3.14). Differentiating it with respect to κ jβ we obtain: Taking into account (3.26), the equation (3.24) can be rewritten in the following form: Replacing the indices i ↔ r in (3.27) we get the second equation which forms together with (3.27) the set of equations for the quantities rj . Excluding iφ (1) rj from this set and expressing theκ Later on we sum the relation (3.29) over r ∈ π i and take into account (3.22), (3.21). It results in the following Ornstein-Zernike-type equation for the pair CF m In the case of uniform field the matricesF are independent of site indices, the CFm (2) ij depends on the difference of the sites' 1, 2 coordinates and we can solve the equation (3.31) carrying out the Fourier transformation V e is a volume of the elementary cell, D being the dimension of the lattice (space). The solution has the form For the hypercubic lattices one has where a is the lattice spacing. The expression (3.34) in the limit c 1 = 1 yields the TCA result for the ideal (one-sort) system [ 7]. Taking q = 0 in (3.34) one gets the earlier result [ 4,6]. The known exact formula for the one-dimensional system in zero external field [ 1] also follows from (3.34).
We will obtain the expression for the 3-site CF m 1α,2β,3γ differentiating equation (3.31) which can be written in the form (3.37) (here free site indices are denoted with numbers and summation over repeated sort indices is implied). The differentiation with respect to κ 3γ yieldṡ U 1α,1δ,3γ m (2) 1δ,2β + U 1α,1δ m 1α,2β,3γ = δm (2) 1α,2β (3.39) In (3.39) the threepoles are introduced which allows us to write (3.38) in more convenient form: (3.40) It should be noted that equality of site indices in threepoles and matrices implies also the symmetry with respect to corresponding sort indices, i.e.
Differentiating expressions (3.30) and (3.32) we find (3.42) When obtaining the formula (3.42) the identities of the following type are used: They can be obtained for any matrixÂ by means of differentiation of the identityÂÂ −1 = 1. Now we shall obtain the expressions for the threepoles which have appeared in (3.42) 1δ,3γ . In the diagram form it can be represented as The other threepoles we find in the same way: 1α,rβγ x = S 1α S rβ S rγ c ρ 1r x .  ij . The same procedure is used to calculate the threepoleĠ. Using the obtained expressions in (3.40) we get the equation for the 3-site CF: In (3.51)-(3.54) "t.i." means twin to the previous (in parentheses) item, i.e. item where the matrix substitutionÊ ↔Ĝ has been carried out. The last (in braces) item of (3.51) is an example of t.i. to the previous (in parentheses) one. Assuming uniform field and performing the Fourier transformation we can obtain from (3.50) the following expression for the 3-site CF , (3.56) where the equality which follows from (3.31) has been taken into account.

Annealed case
In this case we shall introduce into Hamiltonian additional variation parametersψ. The quantity rψiα plays the role of effective field acting on the where Further, the technique of the previous section leads to the generating function beeing (within TCA) of the following form: where The conditions ∂F ∂ rφiα = 0; ∂F ∂ rψiα = 0 (4.7) lead to where i.e. again equality of the one-site and intracluster unary CFs is implied. We note that equations (4.9) like the ones for the chemical potential (2.10) are not independent due to the identity α X iα = 1 . Therefore the number of independent fields rψiα and chemical potentials µ iα decreases. We put In order to shorten following formulae let us intoduce a notation S = {S, X}, i.e.
With this new notations (4.8) and (4.9) can be written as 14) The same technique as for quenched case yields a result for a pair CF in the same form: In equation (4.15) the sort indices lay in the restricted interval α = 1 · · · 2Ω− 1 due to (4.11), (4.12); and the matrices (4.17) have the dimension (2Ω − 1) × (2Ω − 1). The exact result [ 1] for the one-dimensional system (z = 2) is a partial case of the formula (4.17).
The relation (4.14) is equivalent to and it yields F Therefore (4.15) can be written in more simple form (4.20) An expression for the 3-site CFs also remains the same form as in the quenched case , (4.21) with A, B, C, D given by (3.51)-(3.54) and intracluster CFs

Discussion
The great fluctuations of quantities under discussion make the results of "effective field" theories (like TCA and MFA) worse. Thus for the present model a great difference in interactions (e.g., K αβ = Kδ 1α (sort 1 is diluted by noninteracting impurties)) is expected to worse the quality of TCA results whereas great z, when interaction K iα,jβ couples many sites, improves these results. Taking into account that TCA gives exact results for the one-dimensional system (z = 2), the diluted system on plane square lattice seems to be the most difficult test for TCA, which can discover all its shortcomings. That is why below we concentrate our attention at this case. Now let us consider a numerical investigation of the obtained results. In the quenched case we assume for simplicity w αβ = c α c β (complete chaos: X iα X jβ x = X iα x X jβ x ). The figures 2, 3 show model's phase diagrams of two-sort system for different parameters of the Hamiltonian. Curie temperature T c and temperature of alloy spinodal segregation T s are found from the condition M (2) αβ ( q = 0, T ) → ∞. Apart from a complicated form of the TCA phase diagrams 2 with respect to those of MFA 3 one can also remark the rule, that great values of the quantitiesK = K 11 + K 22 − 2K 12 , J = J 11 + J 22 − 2J 12 enhance both segregation and spin alignment, whereas greatṼ 11 = V 11 + V 22 − 2V 12 ,Ĩ 11 = I 11 + I 22 − 2I 12 enhance segregation and therefore enhance spin alignment, ifK > 0,J > 0, and supress alignment, ifK < 0,J < 0.
Temperature dependence of order parameters m ). In TCA this takes place for the case w 11 = c 1 only (all interacting spins constitute one infinite cluster).
One can see in the figure 6 that within TCA pair CF of the quenched diluted system m (2) 11 ( q) is not zero at T = 0 and it yields infinite rise of the susceptibility at zero temperature χ( q) 11 ( q)/T ) whereas MFA gives χ( q) → 0 similarly to the case of the pure system. The TCA behavior of χ( q) and σ (1) can be attributed to interacting spins out of the infinite cluster of interacting spins (finite size cluster effect [ 9]). Indeed, one can find that susceptibility of the isolated spins, which are always present in diluted system, diverges at T = 0. Finite size cluster effects are observed also in other approximations (e.g., χ( 0) of the quenched diluted system in the effective field approximation of Kaneyoshi et.al. [ 9]). Figure 5 shows, that significant differencies between annealed and quenched CFs appear at low temperatures.