Constrained extrapolation problem and order-dependent mappings

Abstract

We consider 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, 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, we show that 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 extrapolation method (MaxEnt). However, we show 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, we propose order-dependent-mapping extrapolation (ODME) as a simple, practical, and general method for the CEP. We find 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.a Paper prepared for the Festschrift in honor of the 60th birthday of Prof. D.A. Drabold.

Effective field theory for dilute Fermi systems at fourth order

“…For investigations of extrapolations of the Fermi-momentum expansion that incorporate strong-coupling constraints, see Refs. [71,72]. Note that in this work, and in Fig.…”

“…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 explicit strong-coupling constraints [71,72]. 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.…”

“…5, we use the same labels (CB and PCB) for (unconstrained) conformal Borel approximants as used in Ref. [72] for the corresponding strong-coupling constrained Borel approximants. The results obtained from the Padé[1, 1] and [2,2] approximants (which were already studied in Ref.…”