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arXiv:1109.5529 · Journal of Statistical Mechanics Theory and Experiment

The inverse Ising problem for one-dimensional chains with arbitrary finite-range couplings

Giacomo Gori, Andrea Trombettoni

Published Oct 20, 2011• 12 citations
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Abstract

We study Ising chains with arbitrary multispin finite-range couplings, providing an explicit solution of the associated inverse Ising problem, i.e. the problem of inferring the values of the coupling constants from the correlation functions. As an application, we reconstruct the couplings of chain Ising Hamiltonians having exponential or power-law two-spin plus three- or four-spin couplings. The generalization of the method to ladders and to Ising systems where a mean-field interaction is added to general finite-range couplings is also discussed.

DOI
10.1088/1742-5468/2011/10/p10021

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We study Ising chains with arbitrary multispin finite-range couplings, providing an explicit solution of the associated inverse Ising problem, i.e. the problem of inferring the values of the coupling constants from the correlation functions. As an application, we reconstruct the couplings of chain Ising Hamiltonians having exponential or power-law two-spin plus three- or four-spin couplings. The generalization of the method to ladders and to Ising systems where a mean-field interaction is added to general finite-range couplings is also discussed.
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.
The kinetic Glauber–Ising spin chain is one of the very few exactly solvable models of non-equilibrium statistical mechanics. Nevertheless, existing solutions do not yield tractable expressions for two-time correlation and response functions of observables involving products of more than one or two spins. We use a new approach to solve explicitly the full hierarchy of differential equations for the correlation and response functions. From this general solution follow closed expressions for arbitrary multispin two-time correlation and response functions, for the case where the system is quenched from equilibrium at T i > 0 to some arbitrary T ⩾ 0. By way of application, we give the results for two- and four-spin two-time correlation and response functions. From the standard mapping, these also imply new exact results for two-time particle correlation and response functions in one-dimensional diffusion limited annihilation.
The one-dimensional Ising model with arbitrary nearest neighbor interactions and a magnetic field is formulated in terms of the spin correlation function relations derived in the preceding article I. The local magnetization and the two-spin correlation functions are shown to satisfy a set of coupled algebraic summation equations. The solution of this set is given in terms of a power series in hyberbolic tangents of the magnetic field strength, for chains of arbitrary length. For cases in which the interaction constants are random variables, explicit expressions are derived for the configurational averaged magnetization and the two-spin correlation functions to the lowest order in the magnetic field.
The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean- field approximations and I derive new analytical expressions for the Bethe approximation, which allow to solve the inverse Ising problem without running the Susceptibility Propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in presence of an external field.
Abstract The previously developed n-vicinity method [3] allows us to calculate accurately critical values of inverse temperatures for Ising models with short-range interaction. We generalize the method to the case of long-range interactions in spin systems and obtain theoretical formulas for the inverse temperatures in terms of the spin interaction constants. The comparison of our theoretical estimates with computer simulations for the two- and three-dimensional Ising models shows that the larger the dimension of the problem the better their agreement.
For pt. I see abstr. A11864 of 1974. A study is made of the ground state orderings of the one-dimensional Ising system, under the assumption that the interaction is of finite range r and has inversion symmetry. It is shown that the ground state energy is effected by the regular chain either of an irreducible block without or with one symmetry point of r or (r+1) sites, or of a symmetric irreducible block with two symmetry points. For the case of the one-dimensional Ising magnet under a uniform external field, it is effected by two, four, seven and twenty regular orderings, when the range r of the interaction is one, two, three and four, respectively.