Constrained Extrapolation Problem and Order‐Dependent Mappings

Abstract

The problem of extrapolating the perturbation series for the dilute Fermi gas in three dimensions to the unitary limit of infinite scattering length and into the BEC region is considered, using the available strong‐coupling information to constrain the extrapolation problem. In this constrained extrapolation problem (CEP), the goal is to find classes of approximants that give well‐converged results already for low perturbative truncation orders. First, it is shown that the standard Padé and Borel methods are too restrictive to give satisfactory results for this CEP. A generalization of Borel extrapolation is given by the so‐called maximum‐entropy (MaxEnt) extrapolation method. However, it is shown that MaxEnt requires extensive elaborations to be applicable to the dilute Fermi gas and is, thus, not practical for the CEP in this case. Instead, order‐dependent‐mapping extrapolation (ODME) as a simple, practical, and general method for the CEP is proposed. It is found that the ODME approximants for the ground‐state energy of the dilute Fermi gas are robust with respect to changes of the mapping choice and agree with results from quantum Monte Carlo simulations within uncertainties.

Effective field theory for dilute Fermi systems at fourth order

“…5. Accurate results throughout the entire BCS regime with negative k F a s (and into the BEC region) can however be obtained via resummations that incorporate constraints on the behavior for k F a s → −∞ from QMC computations [78,79]. Given the technical challenges that arise beyond fourth order, it is unlikely that the k F a s expansion will be evaluated to even higher precision in the near future.…”