From weak to strong: Constrained extrapolation of perturbation series with applications to dilute Fermi systems

C Wellenhofer,D R Phillips,A Schwenk

doi:10.1103/physrevresearch.2.043372

Abstract

We develop a method that uses truncation-order-dependent re-expansions constrained by generic strong-coupling information to extrapolate perturbation series to the nonperturbative regime. The method is first benchmarked against a zero-dimensional model field theory and then applied to the dilute Fermi gas in one and three dimensions. Overall, our method significantly outperforms Pad\'e and Borel extrapolations in these examples. The results for the ground-state energy of the three-dimensional Fermi gas are robust with respect to changes of the form of the re-expansion and compare well with quantum Monte Carlo simulations throughout the BCS regime and beyond.

Highlights

A common situation in physics is that properties of a system can be computed analytically in a weak-coupling expansion but only numerically at discrete points in the nonperturbative regime
Our main results are for the 3D Fermi gas, where we find that the order-dependent-mapping extrapolation (ODME) leads to well-converged extrapolants that are consistent with quantum Monte Carlo (QMC) within uncertainties
We have developed the ODME method to provide powerful weak-to-strong-coupling extrapolants constrained by limited data on strong-coupling behavior

Summary

Introduction

A common situation in physics is that properties of a system can be computed analytically in a weak-coupling expansion but only numerically at discrete points in the nonperturbative regime. The constrained extrapolation problem is to construct approximants that combine these two sources of information. Consider an observable F (x) defined relative to the noninteracting system, e.g., the ground-state energy E /E0. Its perturbation series (denoted PT), truncated at order N in the coupling x, reads N. While Eq (1) provides precise information about the behavior of F (x) as x → 0, it generally fails to yield viable approximations away from weak coupling. The PT is often a divergent asymptotic series, with large-order coefficients obeying, e.g., ck k→∼∞ k! Experiment or computational methods can give access to the behavior of F (x) at a specific point x0.

Methods

Results

Conclusion