High-temperature partition functions and classical simulatability of long-range quantum systems

Abstract

Long-range quantum systems, in which the interactions decay as $1/r^{\hat{I}}\pm$, 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 $\hat{I}\pm >D$. 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 $\exp (\mathcal{O}({\log }^2(N/\hat{I}\mu )))$ that approximates the log-partition function to small additive error $\hat{I}\mu$. 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 Kotecký and Preiss.

18 pages, 9 figures, comments welcome