High-temperature expansion of the inverse-square Ising chain in a magnetic field: exact coefficients and finite-size reconstruction

The one-dimensional Ising chain with inverse-square interactions is a paradigmatic long-range system: it exhibits the unusual Thouless transition, with a discontinuity in the spontaneous magnetization, yet its full field-dependent thermodynamics remains unknown in closed form. We develop the high-temperature expansion of its free energy while retaining the dependence on a uniform magnetic field at every order. We prove that, after a natural rescaling of the coupling, each coefficient is a rational polynomial in the squared hyperbolic tangent of the reduced field. Remarkably, at each order $n$ the field-dependent free-energy coefficient of a finite, periodically repeated ring is an exact polynomial of degree at most $n$ in $L^{-2}$. The infinite-chain coefficient is its constant term and can therefore be reconstructed from $n+1$ finite rings without numerical extrapolation. Using this reconstruction, we determine the full field-dependent expansion through order 26: 26 polynomials containing 377 rational coefficient entries. We further derive exact formulas, valid at every order, for the coefficients of the two highest powers of the field variable using matching determinants and a cycle-Laplacian reduction. These results reveal an unexpected algebraic structure in the microscopic thermodynamics of the inverse-square chain and provide high-order information about its magnetic response.