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arXiv:2011.02105 · physica status solidi (b)

Constrained Extrapolation Problem and Order‐Dependent Mappings

Corbinian Wellenhofer, Daniel R. Phillips, A. Schwenk

Published Mar 23, 2021• 3 citations

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.

DOI: 10.1002/pssb.202000554

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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.

We develop a method that uses truncation-order-dependent reexpansions constrained by generic strongcoupling 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 Padé 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 in the form of the reexpansion and compare well with quantum Monte Carlo simulations throughout the BCS regime and beyond.

We develop a method that uses truncation-order-dependent reexpansions constrained by generic strongcoupling 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 Padé 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 in the form of the reexpansion and compare well with quantum Monte Carlo simulations throughout the BCS regime and beyond.

We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or

{k}_{\mathrm{F}}{a}_{s}

expansion for the ground-state energy of the dilute Fermi gas is calculated to fourth order, both in cutoff regularization and in dimensional regularization. For the case of spin one-half fermions we find from a Bayesian analysis that the expansion is well converged at this order for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}0.5$. Furthermore, we show that Pad\'e-Borel resummations can improve the convergence for $|{k}_{\mathrm{F}}{a}_{s}|\ensuremath{\lesssim}1$. Our results provide important constraints for nonperturbative calculations of ultracold atoms and dilute neutron matter.

The superfluid transition temperature

{T}_{c}

of a unitary Fermi gas on a three-dimensional isotropic lattice with an attractive on-site interaction is investigated as a function of density

n

, from half filling down to $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ per unit cell, using a pairing fluctuation theory. We show that except at very low densities (

{n}^{1/3}<0.2

), where

{T}_{c}/{E}_{F}

is linear in

{n}^{1/3}

,

{T}_{c}/{E}_{F}

exhibits a significant higher-order nonlinear dependence on

{n}^{1/3}

. This calls for extra caution against possible significant error in the zero-density-limit value of

{T}_{c}/{E}_{F}

from typical quantum Monte Carlo (QMC) simulations, obtained by linearly extrapolating data points at intermediate and high densities. Our result,

{T}_{c}/{E}_{F}=0.256

, at the

n=0

limit, is close to, and should be compared with, the maximum

{T}_{c}/{E}_{F}

near the Bose-Einstein condensation regime obtained from QMC calculations.

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej (Phys Rev A 71:053605, 2005). In this paper, we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.

Dilute Fermi systems with large s-wave scattering length a_s exhibit universal properties if the interparticle spacing r_o greatly exceeds the range of the underlying two-body interaction potential. In this regime, r_o is the only relevant length scale and observables such as the energy per particle depend only on r_o (or, equivalently, the energy E_{FG} of the free Fermi gas). This paper investigates Bose and Fermi systems with non-vanishing angular momentum l using the lowest order constrained variational method. We focus on the regime where the generalized scattering length becomes large and determine the relevant length scales. For Bose gases with large generalized scattering lengths, we obtain simple expressions for the energy per particle in terms of a l-dependent length scale ξ_l, which depends on the range of the underlying two-body potential and the average interparticle spacing. We discuss possible implications for dilute two-component Fermi systems with finite l. Furthermore, we determine the equation of state of liquid and gaseous bosonic helium.