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

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 as well as discussed.

💡 Research Summary

The paper addresses the inverse Ising problem for one‑dimensional spin chains with arbitrary finite‑range multispin interactions. Starting from the most general translationally invariant Hamiltonian
(H(\sigma)= -\sum_{\mu\subset{1,\dots,R}} j_{\mu}, O_{\mu}(\sigma)),
where each (\mu) denotes a set of spin indices (allowing one‑, two‑, three‑, …‑spin couplings) and (R) is the maximal interaction distance, the authors consider a periodicity (\rho) (the size of the elementary cell). The observable quantities are the correlation functions (g_{\mu}= \langle O_{\mu}\rangle).

The core of the method is to express the entropy per cell as a function of the full set of correlations. For a block of (Q) consecutive spins the probability of a configuration (\tau_Q) can be written, after inverting the linear relations between (g_{\mu}) and the probabilities, as
(p(\tau_Q)=2^{-Q}\sum_{\mu\subset{1,\dots,Q}} g_{\mu}, O_{\mu}(\tau_Q)).
The Boltzmann entropy for that block is (s(Q)=-\sum_{\tau_Q} p(\tau_Q)\log p(\tau_Q)). Because the physical unit cell contains (\rho) spins, the entropy per cell is obtained by subtracting the contribution of the traced‑out (R-\rho) spins:

\