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arXiv:cond-mat/0510055

A new perturbation theory for the superfluid Fermi gas in the molecular Bose-Einstein condensed state

Shina Tan

Published Oct 3, 2005• 0 citations
arXivHTMLPDF

Abstract

We demonstrate how solutions to quantum few-fermion scattering problems can be the point-of-departure of a new treatment of a generalized many-body wave function. Our focus is on a particular ansatz for the ground state wave function of a superfluid Fermi gas introduced earlier (cond-mat/0506293). Our method is perturbative in the sense that the probability amplitudes for few-fermion scattering processes are treated as small quantities; it is also NONPERTURBATIVE in the sense that whenever such scattering events occur, nonperturbative quantum few-fermion scattering physics dominates. This approach can be viewed as a new diagrammatic methodology, based on a wave function as distinct from a perturbation series in the interparticle interactions. Some generic properties of the wave function are studied, and its parameters in the Bose-Einstein condensed limit are computed beyond mean-field. These results enable us to predict many observable properties of this Fermi gas with well-controlled accuracy, such as the superfluid pairing function, the four-fermion and six-fermion correlation functions, the momentum distribution, and the two-body reduced density matrix, etc.

DOI
10.48550/arxiv.cond-mat/0510055

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We demonstrate how solutions to quantum few-fermion scattering problems can be the point-of-departure of a new treatment of a generalized many-body wave function. Our focus is on a particular ansatz for the ground state wave function of a superfluid Fermi gas introduced earlier (cond-mat/0506293). Our method is perturbative in the sense that the probability amplitudes for few-fermion scattering processes are treated as small quantities; it is also NONPERTURBATIVE in the sense that whenever such scattering events occur, nonperturbative quantum few-fermion scattering physics dominates. This approach can be viewed as a new diagrammatic methodology, based on a wave function as distinct from a perturbation series in the interparticle interactions. Some generic properties of the wave function are studied, and its parameters in the Bose-Einstein condensed limit are computed beyond mean-field. These results enable us to predict many observable properties of this Fermi gas with well-controlled accuracy, such as the superfluid pairing function, the four-fermion and six-fermion correlation functions, the momentum distribution, and the two-body reduced density matrix, etc.
The unitary Fermi gas is a many-body system of two-component fermions with zero-range interactions tuned to infinite scattering length. Despite much activity and interest in unitary Fermi gas and its universal properties, there have been great difficulties in performing accurate calculations of the superfluid condensate fraction and pairing wave function. In this paper, we present auxiliary-field lattice Monte Carlo simulations using a lattice interaction which accelerates the approach to the continuum limit, thereby allowing for robust calculations of these difficult observables. As a benchmark test, we compute the ground-state energy of 33 spin-up and 33 spin-down particles. As a fraction of the free Fermi gas energy EFG{E}_{\text{FG}}EFG​, we find E0/EFG=0.369(2),0.372(2),{E}_{0}/{E}_{\text{FG}}=0.369(2),0.372(2),E0​/EFG​=0.369(2),0.372(2), using two different definitions of the finite-system energy ratio, in agreement with the latest theoretical and experimental results. We then determine the condensate fraction by measuring off-diagonal long-range order in the two-body density matrix. We find that the fraction of condensed pairs is $\ensuremath{\alpha}=0.43(2)$. We also extract the pairing wave function and find the pair correlation length to be ${\ensuremath{\zeta}}_{p}{k}_{F}=1.8(3)\ensuremath{\hbar}$, where kF{k}_{F}kF​ is the Fermi momentum. Provided that the simulations can be performed without severe sign oscillations, the methods we present here can be applied to superfluid neutron matter as well as more exotic PPP-wave and DDD-wave superfluids.
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the sss-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
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We discuss the differences between one- and two-channel descriptions of fermionic gases with arbitrarily tunable attractive interactions; these two cases correspond to whether molecular bosonic degrees of freedom are omitted or included. We adopt the standard ground state wave function for the fermionic component associated with the BCS to BEC crossover problem: for weak attraction the system is in the BCS state while it crosses over continuously to a Bose-Einstein-condensed (BEC) state as the interaction strength is increased. Our analysis focuses on the BEC and near-BEC limit where the differences between the one- and two-channel descriptions are most notable, and where analytical calculations are most tractable. Among the differences we elucidate are the equations of state at general TTT below Tc{T}_{c}Tc​ and related particle density profiles. We find a narrowing of the density profile in the two-channel problem relative to the one-channel analog. Importantly, we infer that the ratio between bosonic and fermionic scattering lengths depends on the magnetic detuning and is generally smaller than its one-channel counterpart. Future experiments will be required to determine to what extent this ratio varies with magnetic fields, as predicted here.
We derive a resonance field-theory for ultra-cold Bose or Fermi gases that is valid when scattering resonances are present. Experiments with a ^85Rb condensate have shown new interesting physics close to resonance(N.R. Claussen, E.A. Donley, S.T. Thompson, C.E. Wieman, e-print cond-mat/0201400.). It is also expected that utilizing a Feshbach resonance can greatly enhance the critical temperature for superfluidity in a Fermi gas(M. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. 87), 120406 (2001).. Therefore it is important to treat the rich scattering physics properly inside a many-body theory. We show that the scattering length only is insufficient to describe a system close to resonance. Our many-body equations have build-in the two-body scattering equations, and we include the full scattering solution over the relevant energy range by means of a renormalized resonance scattering theory(S.J.J.M.F. Kokkelmans, J.N. Milstein, M.L. Chiofalo, R. Walser, and M.J. Holland, e-print cond-mat/0112283.).
We consider a superfluid state in a two-component gas of fermionic atoms with equal densities and unequal masses in the BCS limit. We develop a perturbation theory along the lines proposed by Gorkov and Melik-Barkhudarov and find that for a large difference in the masses of heavy (M)(M)(M) and light (m)(m)(m) atoms one has to take into account both the second-order and third-order contributions. The result for the critical temperature and order parameter is then quite different from the prediction of the simple BCS approach. Moreover, the small parameter of the theory turns out to be $({p}_{F}\ensuremath{\mid}a\ensuremath{\mid}∕\ensuremath{\hbar})\sqrt{M∕m}⪡1$, where pF{p}_{F}pF​ is the Fermi momentum and aaa the scattering length. Thus, for a large mass ratio M∕mM∕mM∕m the conventional perturbation theory requires significantly smaller Fermi momenta (densities) or scattering lengths than in the case of $M\ensuremath{\sim}m$, where the small parameter is ${p}_{F}\ensuremath{\mid}a\ensuremath{\mid}∕\ensuremath{\hbar}⪡1$. We show that three-body scattering resonances appearing at a large mass ratio due to the presence of three-body bound Efimov states do not influence the result, which in this sense becomes universal.