Universal Large-order asymptotic behavior of the Strong-coupling and High-Temperature series expansions

Abstract

For theories that exhibit second order phase transition, we conjecture that the large-order asymptotic behavior of the strong-coupling ( High-Temperature) series expansion takes the form $\sigma^{n} n^{b}$ where $b$ is a universal parameter. The associated critical exponent is then given by $b+1$. The series itself can be approximated by the hypergeometric approximants $_{p}F_{p-1}$ which can mimic the same large-order behavior of the given series. Near the tip of the branch cut, the hypergeometric function $_{p}F_{p-1}$ has a power-law behavior from which the critical exponent and critical coupling can be extracted. The conjecture has been tested in this work for the perturbation series of the ground state energy of the Yang-Lee model as a strong-coupling form of the $\mathcal{PT}$-symmetric $i\phi^3$ theory and the High-Temperature expansion within the Ising model. From the known $b$ parameter for the Yang-Lee model, we obtained the exact critical exponents which reflects the universality of $b$. Very accurate prediction for $b$ has been obtained from the many orders available for the High-Temperature series expansion of the Ising model which in turn predicts accurate critical exponent. Apart from critical exponents, the hypergeometric approximants for the Yang-Lee model show almost exact predictions for the ground state energy from low orders of perturbation series as input.