We discuss high-order calculations in perturbative effective field theory for fermions at low energy scales. The Fermi-momentum or
kFas 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 effective field theory approach simplifies the perturbative computation of the ground-state energy of the diluted gas of repulsively interacting fermions allowing in the case of the unpolarized system to easily rederive the classic results up to the
(kFa0)2 order (where
kF is the system's Fermi momentum and
a0 the
s-wave scattering length) and (with more labor) to extend it up to the order
(kFa0)4. The analogous expansion of the ground-state energy of the polarized gas of spin
1/2 fermions is known only up to the
kFa0 order (where
kF stands for ${k}_{\mathrm{F}\ensuremath{\uparrow}}$ or ${k}_{\mathrm{F}\ensuremath{\downarrow}}$); the order
(kFa0)2 contribution has been computed (analytically) only using a specific (hard-core type) interaction potential. Here we show that the effective field theory method also allows to easily obtain the order
(kFa0)2 correction to the ground state of the polarized gas in a way applicable to all repulsive interactions.
The aim of this work is to study a dilute Fermi gas, whether it has a spin 1/2 or it has a high degeneracy. The starting point is to calculate the energy of the system. However, as we are dealing with a many body system, a perturbation series method will be applied. R. F. Bishop in reference [1] wrote the terms of the expansion up to fourth order in the gas parameter, which is defined as the Fermi momentum times the S-wave scattering length, but only for the case of a non-polarized gas. What we do in this present work is the generalization of that expansion up to third order including any possible polarization. The universal expansion, that is up to second order, have been computed analytically. For the third order terms was not possible, but thanks to numerical calculation we have obtained the energy for different polarizations. Therefore, this work will contain a combination of both analytical and numerical calculus. The final expression obtained for the energy depends on the polarization, the gas parameter and the degeneracy and stands for low density gases. Afterwards, thanks to knowing the relation between the energy and the polarization, the study of the ferromagnetic phase transition is presented. We see how the transition point changes due to the degeneracy and what kind of phase transition we obtain. For spin 1/2 a continuous transition is observed, while a discontinuous transition happens for higher degeneracy. To complement the phase transition study, different physical properties have been computed, such as, the magnetic susceptibility, and the Tan?s constant. Our study is extended to highly degenerate Fermi gases which are now realized in cold Fermi labs.
The dependence upon momentum
k and upon energy
E of the self-energy of a dilute Fermi gas is studied up to terms of order
(kFc)2, where
kF denotes the Fermi momentum and
c is the positive scattering length. Algebraic expressions are derived for the imaginary part
W(k;E) of the self-energy in the whole (
k,
E) plane. They are compared with a conjecture recently made by Orland and Schaeffer in their analysis of single-particle states in nuclei. The contributions of core polarization and of ground state correlations to the real part
V(k;E) of the self-energy are calculated with the help of subtracted dispersion relations which connect them with $W(k;{E}^{\ensuremath{'}})$. Algebraic expressions are derived for the momentum distribution in the correlated ground state. It is shown that the effective mass of a quasiparticle with momentum
k is equal to the bare particle mass at
k=0 and reaches a local maximum for
k close to
kF. This maximum is ascribed to the dependence of
V(k;E) upon
E, which is described in terms of an
E mass. We compute the contributions of core polarization and of ground state correlations to this
E mass. The dependence of
V(k;E) upon
k reflects the nonlocality of the self-energy. It is characterized by a
k mass that we also calculate. These results shed light on some nuclear matter properties and on the meaningfulness and limitation of nuclear matter calculations that have recently been performed in the framework of the Brueckner-Hartree-Fock approximation.NUCLEAR STRUCTURE Effective mass near the Fermi surface in dilute Fermi gas model.
The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.