Higher-order correlation functions of nonuniform Ising chain

Partition function of a uniform ring (hi = h, Ki = K) was originally calculated by Ising [1]. General expressions for distribution functions (DFs) 〈Si1 · · ·Sin〉 of an arbitrary order were suggested in [2]. Those expressions contain some unknown coefficients to be found from a cumbersome system of equations. They have been calculated only for n = 2, 3, 4, while for n = 3, 4 some of the obtained results were incorrect. However, for a nonuniform 1D model, the exact solutions exist only for a few special examples or in some limiting cases, such as zero temperature, certain disorder types, etc (see [3] and references therein). For instance, in [4] the averaged pair DFs were calculated for a binary substitution alloy at hi = 0. The aim of the present paper is to formulate the diagrammatic rules for calculating arbitrary order DFs and correlation functions (CFs) 〈Si1 · · ·Sin〉 c (cumulant average). We shall proceed from an exact expression for generating F -function (logarithm of partition function) of the Ising chain

In this paper we consider a nonuniform Ising chain (free ends) described by the Hamiltonian Partition function of a uniform ring (h i = h, K i = K) was originally calculated by Ising [1].General expressions for distribution functions (DFs) S i 1 • • • S in of an arbitrary order were suggested in [2].Those expressions contain some unknown coefficients to be found from a cumbersome system of equations.They have been calculated only for n = 2, 3, 4, while for n = 3, 4 some of the obtained results were incorrect.However, for a nonuniform 1D model, the exact solutions exist only for a few special examples or in some limiting cases, such as zero temperature, certain disorder types, etc (see [3] and references therein).For instance, in [4] the averaged pair DFs were calculated for a binary substitution alloy at h i = 0.The aim of the present paper is to formulate the diagrammatic rules for calculating arbitrary order DFs and correlation functions (CFs) S i 1 • • • S in c (cumulant average).We shall proceed from an exact expression for generating F -function (logarithm of partition function) of the Ising chain Here, the one-particle F i and two-particle F i,j F -functions are calculated with the one-particle H i and two-particle H i,i+1 Hamiltonians and read The fields α ϕ i , (α = +, −) depend on the Hamiltonian parameters for the righthand (+) or the left-hand (−) (in respect to the site i) branches of the chain and thus describe the effect of the right (left) branch of the chain on the ith spin.Obviously, − ϕ 1 = + ϕ N = 0.The rest of 2N − 2 unknown fields α ϕ i can be found from 2N − 2 independent relations (because of the imposed 2N relations Sp S i ρ i (S i ) = Sp S i ,Sr ρ ir (S i , S r )) between the one-particle ρ i and the two-particle ρ i,r density matrices Here, ρ i and ρ i,r are constructed based on H i and H i,r , respectively.They can be written in a different form coinciding with the expressions obtained in [3] Hereafter we use the following notations The following form of the system (6) will be used Relations (11) can be also obtained proceeding from the stationarity condition for F ({h}) with respect to α ϕ 1 ( ∂ ∂αϕ i F ({h}) = 0).In the case of an infinite chain and a uniform field (h i = h, − ϕ i = + ϕ i+1 = ϕ) the thermodynamic quantities in (7), ( 8) and (11) are independent of the site index i.Solving the equation (8) with respect to − h and substituting the obtained expression into (7) and (11), we get the known formulas Hereafter we consider only the right-hand (in respect to the site i) branch of a chain.Differentiating (11) with respect to h i+n , (n 0) and taking into account the fact that δ δh i+n − hi = δ i,i+n , we obtain an equation for δ δh i+n + hi = δ δh i+n hi .It can be solved by iterating procedure and permits to find the expression for the pair CF where The expression for ternary CF is derived by differentiating ( 14) i,i+n = −2F (2) i,i+k As follows from relations ( 14) and ( 16), the higher CFs invoke only the single-particle F (1) i and the pair F i,j CFs.It is convenient to present these relations in a diagram form (below we use the notations (2) First, we find the higher DFs from the recurrent relation Thus, we obtain expressions for S 1 S 2 S 3 and S 1 S 2 S 3 S 4 In a uniform case, these diagrams are rewritten as follows: In the particular case of the nearest neighbors, the DFs (18) and ( 19) coincide with those obtained in [5].The same formulae can be also derived by using the transfer matrix method.In contrast to the present results, all the terms in the expressions for S 1 S 2 S 3 and S 1 S 2 S 3 S 4 obtained in [2] occur only with the plus signs and are incorrect, as we have already mentioned earlier.
Starting from (17) and using the mathematic induction method, we can formulate general rules for calculating the DFs of an arbitrary order.The DFs S 1 S 2 • • • S n are the sums of all diagrams with n vertices, obeying the following conditions.
• A diagram should consist of free vertices i = S i and bonds i−j = S i S j c .• Each bond i−j gives the coefficient (−1) j−i−1 where (j − i − 1) is the number of vertices between the sites i and j.
Similar rules can be also derived by using the transfer matrix method.The higher CFs are obtained from (9), ( 14) and (16).For instance, As one can see, the CF, unlike the DEF, contains only the diagrams connected along the line 1, 2, • • • , n.A thorough diagrammatic analysis of the CFs permitted to formulate the following general rules.The CF S 1 S 2 • • • S n c is a sum of all diagrams with n vertices, obeying the following conditions.
• A diagram should consist of free vertices i = S i and bonds i−j = S i S j c .
• There is always a bond 1−n.The bonds i (i + 1) for the neighbouring vertices are absent.
• A prefactor at each diagram is (−2) (n−l−1) where l is the number of bonds in diagram.
• There is an additional coefficient, arising at the differentiation and defined by a special rule.
For instance, the expression for S 1 • • • S 8 c (for simplicity at h i = 0) reads

C
Here, the notation P (n) at the diagrams stands for a sum of all nonequivalent diagrams with the same symmetry of lines.The boxes contain digital notations of the diagrams.The index i in the upper row corresponds to one of the vertices i = S i or bonds i−j = S i S j c which are differentiated with respect to h k given by the index k in the lower row.Thus, the digital diagram reflects the differentiation procedure.For example, the expressions for S 1 S 2 S 3 c and S 1 S 2 S 3 S 4 c in digital notations are as follows: At a certain step of differentiation, the derivatives of different diagrams can be identical.Consider, for example, the origin of the last diagram inS 1 • • • S 8 c .The two different diagrams from S 1 • • • S 5c at the differentiation with respect to h 6 additional coefficient 2 emerged due to the differentiation.At the differentiation of the vertices S 3 and S 4 in the present diagram with respect to h 7 and h 8 , correspondingly, we obtain the last diagram in S 1 • • • S 8 c .