Abstract
Long-range quantum systems, in which the interactions decay as 1/r^α, are of increasing interest due to the variety of experimental set-ups in which they naturally appear. Motivated by this, we study fundamental properties of long-range spin systems in thermal equilibrium, focusing on the weak regime of α>D. Our main result is a proof of analiticity of their partition functions at high temperatures, which allows us to construct a classical algorithm with sub-exponential runtime exp(O(log²(N/ε))) that approximates the log-partition function to small additive error ε. As by-products, we establish the equivalence of ensembles and the Gaussianity of the density of states, which we verify numerically in both the weak and strong long-range regimes. This also yields constraints on the appearance of various classes of phase transitions, including thermal, dynamical and excited-state ones. Our main technical contribution is the extension to the quantum long-range regime of the convergence criterion for cluster expansions of Koteck\'y and Preiss.