Dilute Fermi gas at fourth order in effective field theory

C. Wellenhofer, 2, ∗ C. Drischler, 4, † and A. Schwenk 2, 5, ‡ Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany Department of Physics, University of California, Berkeley, CA 94720 Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

The dilute Fermi gas has been a central problem for many-body calculations for decades [1][2][3][4][5][6][7][8]. Renewed interest in this problem has been triggered by striking progress with ultracold atomic gases. On the theoretical side, a systematic approach towards the dynamics of fermions (or bosons) at low energies has emerged in the form of effective field theory (EFT) [9][10][11][12][13][14]. Motivated by this, we revisit the expansion in the Fermi momentum k F of the ground-state energy density E(k F ) of a dilute gas of interacting fermions. Using perturbative EFT methods, we calculate E(k F ) up to fourth order in the expansion, including for the first time the complete fourth-order term. As we show, our results provide highorder benchmarks regarding the convergence behavior of the expansion, and allow to predict E(k F ) including systematic uncertainty estimates.
Short-ranged EFT represents a general framework for the dynamics of fermions (or bosons) at low momenta Q < Λ b , where Λ b denotes the breakdown scale. The EFT Lagrangian is given by the most general operators consistent with Galilean invariance, parity, and timereversal invariance. Assuming spin-independent interactions, the (unrenormalized) Lagrangian reads (see, e.g., Refs. [9][10][11][12][13][14]) where ψ are nonrelativistic fermion fields, ∇ is the Galilean invariant derivative, h.c. the Hermitian conjugate, and M the fermion mass. For natural systems the interaction terms of L EFT are ordered in powers of fields and derivatives (and suppressed with respect to the kinetic term), so in this case low-energy observables can be calculated systematically by ordering the various contributions in perturbation theory with respect to powers of Q/Λ b .
The ultraviolet (UV) divergences that appear beyond tree level in perturbation theory can be regularized by introducing a momentum cutoff Λ. By Galilean invariance, the cutoff must be attached to relative momenta p ( ) and Jacobi momenta q ( ) , respectively. The two-and threebody potentials emerging from L EFT are then given by Perturbative renormalization is carried out by introducing counterterms such that the divergent contributions are cancelled. In the two-body sector, this leads to where the cutoff-dependent parts are counterterms. In the natural case the low-energy constants C 0 , C 2 , and C 2 scale according to For the renormalized two-body potential the residual cutoff dependence due to terms O(1/Λ) in perturbation theory vanishes in the limit Λ → ∞. Matching the twobody low-energy constants to the effective-range expansion (ERE) then leads to (see, e.g., Ref. [10]) where a s and a p is the S-and P -wave scattering length, respectively, and r s is the S-wave effective range. In the two-body sector there are only power divergences, but in systems with more than two particles also logarithmic divergences can occur, starting at order (Q/Λ b ) 4 . The counterterm for the leading logarithmic divergences is provided by the leading term of the threebody potential V EFT . Neglecting O(1/Λ) terms, cutoff independence in the N -body sector with N 3 at order The coefficient of the ln Λ term in Eq.
, which can be obtained from the UV analysis of the two logarithmically divergent three-body scattering diagrams at order (Q/Λ b ) 4 , see Refs. [4,10,15]. Integrating Eq. (9) leads to The parameter D 0 (Λ 0 ) has to be fixed by matching to few-body data.
in the natural case. The scale Λ 0 is however completely arbitrary: from Eq. (9), D 0 (Λ 0 ) scales according EFT in many-body perturbation theory (MBPT) leads to the Fermi-momentum expansion for the ground-state energy density E(k F ) of the dilute Fermi gas, i.e., where n = g k 3 F /(6π 2 ) is the fermion number density and g is the spin multiplicity. The dependence of a given MBPT diagram on g is obtained by inserting a factor δ σ1,σ 1 δ σ2,σ 2 − δ σ1,σ 2 δ σ2,σ 1 for each vertex and summing over the spins σ The numerical values in the expression for C 3 (k F ) are the ones obtained by Kaiser [16,17] using semianalytic methods. We have reproduced these results. Our result for the fourth-order term is given by with Here, the effective-range contribution stems from the two second-order diagrams with one C 0 and one C 2 vertex (plus the corresponding tree-level counterterm), which can be evaluated using the semianalytic formula of Kaiser [18]. The remaining part of C 4 (k F ) corresponds to diagrams with four C 0 vertices and the tree-level contribution from V EFT . Setting Λ 0 = 1/|a s | one obtains from the nonanalytic part of γ 4 (k F ) the known form of the logarithmic term at fourth order [2,4,6,7,10]. Note again that Λ 0 is an arbitrary auxiliary scale: from Eq. (9), γ 4 (k F ) is independent of Λ 0 . Therefore, the logarithmic term at fourth order should not be treated as a separate contribution in the k F expansion.
The diagrams I(1-6) can be calculated using the semianalytic expressions derived by Kaiser [16], which can be obtained from the usual MBPT expressions [19] by applying various partial-fraction decompositions and the Poincaré-Bertrand transformation formula [21]. Table I. Results for the regular contributions to C4(kF). Diagrams with * ( * * ) have UV power (logarithmic) divergences, which are subtracted by the respective counterterm contributions. Diagrams with * * * have infrared singularities. The uncertainty estimates take into account both the statistical Monte-Carlo uncertainties and variations of the (numerical) cutoff. The g factors are listed without the generic g(g − 1) factor.
For the numerical evaluation of the IA diagrams it is more convenient to use single-particle momenta instead of relative momenta, because then the phase space is less complicated. We have carried out the numerical calculations using the Monte-Carlo framework introduced in Ref. [22] to evaluate high-order many-body diagrams.
The UV power divergences correspond to particleparticle ladders and are cancelled by counterterm contributions from the the first-, second-, and third-order diagrams obtained by removing the ladders. The diagrams with logarithmic divergences II(5,6), IIA1 and III1 are shown in Fig. 1. Using dimensionless momenta i ≡ k i /(αk F ) one can analytically extract (in II5 II6 III7 IIA1 III1 III8 Figure 1. Hugenholtz diagrams representing the fourth-order contributions with logarithmic divergences II(5,6), IIA1, and III1. Also shown are the other diagrams that are part of the sum III(1+7+8).
the limit Λ → ∞) from each diagram a contribution ∼ ln(Λ/(αk F )). The parameter α is arbitrary, and can be set to α = 1. Adding the logarithmic part of the tree-level contribution from V EFT , this leads to the logarithmic part of C 4 (k F ) given in Eq. (16).
Our results for the various contributions to the regular part of C 4 (k F ) are listed in Table 1. The numerical values for the diagrams without divergences are similar (but small differences are present) to the ones published by Baker in Table IV of Ref. [6]. 3 The contributions that involve logarithmic divergences, II5, II6, IIA1, and III(1+7+8), have the largest numerical uncertainties. For g = 2 more precise results can be given for II5+IIA1 and II6+III(1+7+8), because then no logarithmic divergences occur.
For spin one-half fermions, the logarithmic term at fourth order (and beyond, up to a certain order N log ) is Pauli blocked, so in that case the k F expansion is (for N < N log ) given by where δ = k F a s and E 0 = 3nk 2 F /(10M ). The coefficients X n are completely determined by the ERE parameters.
The results for N ∈ {2, 3, 4} are plotted in Fig. 2. For comparison, we also show results obtained from quantum Monte-Carlo (QMC) calculations [23][24][25]. Overall, the perturbative results are very close to the QMC results for |δ| 0.5 and start to deviate strongly for |δ| 1. In the LO case, the relative error with respect to the QMC point at δ = −0.5 is 4.5% at first, 0.8% at second, 0.4% at third, and 0.1% at fourth order. In the HS case X 4 is very small and the N = 3 and N = 4 curves are almost indistinguishable.
In Fig. 2 we also plot uncertainty bands obtained by setting X N +1 = ± max[X ν≤N ]. Going to higher orders in that scheme reduces the width of the bands in the perturbative region |δ| 1. For |δ| 0.5 the bands are very small for N = 4, which supports the conclusion that the expansion is well-converged at fourth order in this regime.
Finally, for the LO case we also plot the results obtained from the Padé [1,1] and [2,2] approximants. Only diagonal Padé approximants have a meaningful unitary limit. The Padé [2,2] results are very close to the QMC points for δ −1.2, while the Padé [1,1] ones are in better agreement with the QMC points close to the unitary limit δ → −∞. The range for the normal (i.e., non-superfluid) Bertsch parameter obtained from the Padé [1,1] and [2,2] approximants, ξ n ∈ [0.33, 0.54], is consistent with the value ξ n = 0.45 extracted from experiments with cold atomic gases [27]. Altogether, these results may indicate that Padé approximants converge in a larger region, compared to the Fermi-momentum expansion. To further investigate this one would need to construct the subsequent Padé[ν, ν] approximants, which require the expansion coefficients up to order 2ν 6. The main challenge (apart from the large increase of the number of diagrams) herein lies in the evaluation of diagrams with logarithmic divergences.
In summary, using EFT methods we have calculated the complete fourth-order term in the Fermi-momentum expansion for the ground-state energy of a dilute Fermi gas. A detailed study of the convergence behavior and comparison against QMC calculations for the case of spin one-half fermions showed that this (asymptotic) expansion is well-converged at this order for |k F a s | 0.5, and exhibits divergent behavior for |k F a s | 1. Our results provide important high-order benchmarks for many problems in many-body physics, ranging from cold atomic gases to dilute nuclear matter and neutron stars.
We thank R.F. Bishop, R.J. Furnstahl, A. Gezerlis, K. Hebeler, S. König, K. McElvain, and A. Tichai for useful discussions, and S. Gandolfi as well as S. Pilati for sending us their QMC results. This work is supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Projektnummer 279384907-SFB 1245, the US Department of Energy, the Office of Science, the Office of Nuclear Physics, and SciDAC under awards DE-SC00046548 and DE-AC02-05CH11231. Computational resources have been provided by the Lichtenberg high performance computer of the TU Darmstadt.