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

TL;DR

This paper proposes a universal asymptotic form for large-order behavior of series expansions near phase transitions, using hypergeometric functions to accurately predict critical exponents and energies across models.

Contribution

It introduces a universal conjecture linking large-order asymptotics to hypergeometric approximants for phase transition series.

Findings

Confirmed universality of the parameter b across models

Accurately predicted critical exponents for the Ising and Yang-Lee models

Achieved precise ground state energy predictions from low-order series

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.