Inverse problem for Ising connection matrix with long-range interaction

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of it eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions.


Introduction
In papers [1][2][3], we calculated eigenvalues of Ising connection matrices defined on d -dimensional hypercube lattices ( 1, 2,3... d = ). To provide the translation invariance we imposed periodic boundary conditions. In our calculations, we accounted for interactions not only with the nearest spins but with distant spins too. In papers [1,2] we analyzed isotropic interactions. The general case of anisotropic interactions we discussed in [3]. We succeeded to obtain analytical expressions for the eigenvalues of the above-described Ising connection matrices. For the d -dimensional system, the eigenvalues are polynomials of the degree d in the eigenvalues for the one-dimensional system with longrange interaction (see [2,3]). The coefficients of these polynomials are the constants of interaction between spins.
In the present paper, we solve an inverse problem formulated as follows. Suppose we know the spectrum of an Ising connection matrix and we have to answer two questions. Firstly, is it possible to restore the interaction constants that define the connection matrix whose spectrum matches the given one? Secondly, when a sequence of random numbers may be the spectrum of some connection matrix? In Section 2, we obtain the answers to these questions for the onedimensional Ising system. Then, we use the obtained results to analyze the two-and three-dimensional systems in Sections 3 and 4, respectively. Discussion and conclusions are in Section 5. In this section, we also discuss the possibilities to use the eigenvalues of the Ising connection matrices when calculating the partition functions.
There is extensive literature on the inverse Ising problem; see, for example, a rather full review published in [7]. When solving the inverse Ising problems the authors examine how with the aid the statistical inference method they can estimate the parameters of the Ising system -interaction constants and external magnetic fields -when they know empirical characteristics of a large number of random spin configurations. We would like to emphasize that although as in the papers cited in [7] we also restoring the parameters of the Ising systems, the setting of the problem and the method of its solution differ significantly. In our approach we inverse the exact formulas that express the connection matrix eigenvalues in terms of its matrix elements. However, when using the statistical inference method the input data are the observables such as magnetizations, correlations, and so on. The solution tools are also different. They are the Boltzmann equilibrium distribution, the principle of the maximal likelihood, the Bayes theorem and so on.

One-dimensional Ising model 1)
A one-dimensional Ising system is a linear chain of L interacting spins. To provide a translation invariance, let us close the chain in a ring. Then the last spin is also the nearest neighbor of the first spin. This means that each spin has two (on the left and right) nearest neighbors, two next nearest neighbors (the distance to which is twice as large), two next-next nearest neighbors, and so on. To be specific, we suppose that L is odd: 2 1 L l = + . Consequently, each spin has l pairs of the neighbors. Since we have in mind to discuss multidimensional lattices, we use the term "coordination spheres" to describe these pairs: first coordination sphere, second coordination sphere … l-th are symmetric, their eigenvalues are real. By { } 1 ( ) L k α α λ = we denote the eigenvalues of these matrices. It can be shown that [2,3]  The first eigenvalue of each matrix ( ) k J is equal to 2, and other eigenvalues are twice degenerate: 1, 2,.., : Consequently, for each k (if we do not take into account the first eigenvalue), the sequence is mirror-symmetrical about its middle (see Fig. 1). In what follows, we repeatedly use this symmetry property. Vertical line in the middle shows explicitly mirror symmetry of graphs.
The eigenvector (1) f with equal coordinates corresponds to the first eigenvalue 1 ( ) 2 k λ = : We can choose the two eigenvectors ( ) Since the eigenvectors of all the matrices ( ) k J are the same, it is easy to write down the eigenvalues of the connection matrix (1) The expression (4) is a generalization of the formula obtained previously in [4].
Second, due to the zero-valued elements at the diagonals of all the matrices ( ) k J the sum of the eigenvalues of the matrix 0 A has to be equal to zero. This means that Consequently, only l numbers 2 0 ( ) (4) can be arbitrary; the other eigenvalues are expressed through these numbers with the aid of the equalities (5) and (6).  (5) and (6). What are the connections k w between the spins that provide this spectrum?
To determine the unknowns k w , we have to solve the system (4) with the known left-hand side: We can obtain the answer in an explicit form. Let us generate an L -dimensional vector Λ  whose coordinates are the eigenvalues of the experimental spectrum { } . We also generate L -dimensional vectors ( ) k Λ whose coordinates are the eigenvalues of the matrices ( ) k J : Then we can rewrite the system of equations (7) in the vector form: It is evident that the vectors ( ) k Λ and the eigenvectors ( 1) k + f are collinear: Consequently, we can calculate the weights k w as scalar products of the vectors Λ  and ( ) k Λ : By doing that, we solve the inverse problem in the one-dimensional case.

Two-dimensional Ising model 1)
In this case, the spins are in the nods of a square lattice of the size L L × . As previously, we set 2 1 L l = + and assume periodic boundary conditions. Then each spin has l pairs of neighbors along both the horizontal and the vertical axes. In addition, there are neighbors that are not on the same horizontal or vertical axes as the given spin.
The set of spins equally interacting with the given spin belongs to the same coordination sphere. In the case of an isotropic interaction, the coordination spheres consist of spins equally distant from the given spin. Then we can enumerate the coordination spheres in the ascending order of distances to the given spin. In the anisotropic case the interaction constants but not the distances define the spins belonging to the given coordination sphere.
When analyzing multidimensional Ising systems, we first have to distribute spins between the coordination spheres. This step is simple in the one-dimensional case: the pair of spins that are equidistant from the given spin belongs to the same coordination sphere. In the case of two-dimensional lattice, to describe the interaction between the spins spaced by m steps along the vertical axis and by k steps along the horizontal axis we introduce the interaction constant ( , ) w m k ; the values of m and k change independently from 0 to l . If the interaction is anisotropic ( , ) The difference between the coordination spheres in the isotropic and anisotropic cases influences symmetry properties of the spectrum.
Let us make a few necessary comments. Since there is no a self-action in the system, we always have (0, . This matrix completes the set of . All the eigenvalues of the matrix (0) J are equal to one. With the aid of these eigenvalues, we In subsection 2), we solve the inverse problem in the case of anisotropic interaction. The isotropic interaction is a subject of the last subsection of this Section.
2) In paper [3] we showed that a ( ) The vectors ( , ) m k Λ are mutually orthogonal. Let us define an 2 L -dimensional vector Μ whose coordinates are This equation allows us to express 11 µ , through the other ( 2) l l + independent numbers αβ µ from the sequence (16).
Consequently, the number of the independent values αβ µ equals exactly to the number of the orthogonal vectors ( , ) n m Λ taking part in the expansion (13).

3)
Finally, let us discuss briefly a two-dimensional Ising system with an isotropic interaction. Evidently, we again can use the equations (13) In paper [3], we showed that the ( ) The vectors αβγ F constitute a full set of the eigenvectors of any connection matrix of the three-dimensional Ising system and they do not depend on the type of the interaction constants ( ) The equation (22) The symmetry reasons allow us to restore all the other numbers αβγ µ . Consequently

Discussion and conclusions
Connection matrices define the most important characteristics of Ising systems -such as the energies of the states and their distribution, the free energy, and all the macroscopic properties defined by the free energy. All these functions are crucially dependent on the connection matrix whose main characteristics are its eigenvalues and eigenvectors. In papers [1][2][3], we obtained the expressions for the eigenvalues of the Ising connection matrix with an arbitrary long-range interaction in terms of its matrix elements. In the present paper, we solve the inverse problem: we suppose that we know the matrix spectrum and we have to determine the interaction constants providing this spectrum.
We would like to note that the statement of the problem itself is not obvious.
On the other hand, at the beginning of each Section we recall that all connection matrices of any d-dimensional Ising model are circulants and, consequently, all these matrices have the same set of the eigenvectors [6]. In other words, their eigenvectors are known by default. However, our analysis shows that when calculating the matrix elements the internal symmetry of the problem allows us not to use Eq. (26) but much more simpler and convenient formulas (see Eqs. (9), (14), and (23)). In addition, using the symmetry reasons, we obtain the number and positions of independent values in a given sequence that allows it to be the spectrum of some connection matrix. The problem is solved for the d-dimensional hypercube lattices and an arbitrary long-range interaction.
The following question arises: may the connection matrix eigenvalues be useful when calculating the partition function The first thing that comes to mind is to improve the transfer-matrix method for the partition function calculation using the expansion (26) of the matrix A in terms of the eigenvectors and the eigenvalues we obtained. Apparently, this is a hopeless idea. The basis of the transfer-matrix method is a transition from N spin variables to a larger their number 1 N + . However, an increase of the dimension of the problem leads to a complex restructuring of the eigenvalues and the eigenvectors of the matrix A in each term of Eq. (27).
On the other hand, the connection matrix eigenvalues themselves may be useful when calculating the partition functions. Indeed, let us expand each exponential in (27) into a formal Taylor series. Now let us rearrange the summands, combining in one sum the terms with the same power β . We showed that for matrices with zero diagonals the equation is true [8]. Our arguments show that the eigenvalues of the Ising connection matrix may be useful when calculating the partition function.