Strongly paired fermions: Cold atoms and neutron matter
SStrongly paired fermions: Cold atoms and neutron matter
Alexandros Gezerlis , and J. Carlson Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA (Dated: November 2, 2018)Experiments with cold Fermi atoms can be tuned to probe strongly interacting fluids that arevery similar to the low-density neutron matter found in the crusts of neutron stars. In contrast totraditional superfluids and superconductors, matter in this regime is very strongly paired, with gapsof the order of the Fermi energy. We compute the T = 0 equation of state and pairing gap for coldatoms and low-density neutron matter as a function of the Fermi momentum times the scatteringlength. Results of quantum Monte Carlo calculations show that the equations of state are verysimilar. The neutron matter pairing gap at low densities is found to be very large but, except atthe smallest densities, significantly suppressed relative to cold atoms because of the finite effectiverange in the neutron-neutron interaction. PACS numbers: 21.65.-f, 03.75.Ss, 05.30.Fk, 26.60.-c
Strongly paired fermions are important in many con-texts: cold Fermi atom experiments, low-density neutronmatter, and QCD at the very high baryon densities po-tentially found in the center of massive neutron stars. De-veloping a quantitative understanding of strongly pairedFermi systems is important since they offer a uniqueregime for quantum many-body physics, relevant in verydifferent physical settings including the structure andcooling of neutron stars. Constraining neutron matterproperties can also be important in understanding theexterior of neutron-rich nuclei by constraining parame-ters of nuclear density functionals.Cold-atom experiments can provide direct tests of theequation of state and the pairing gap in the stronglypaired regime, and hence provide a crucial benchmarkof many-body theories in these systems. We consider asystem of two fermion species and a simple Hamiltonianof the form H = − (cid:126) m (cid:88) i ∇ i + (cid:88) i 15, but only at very low densities is the effec-tive range much smaller than the interparticle spacing.To the extent that the effects of finite range in the in-teraction can be neglected, cold atoms and neutron mat-ter are ‘universal’ in the sense that the properties of thesystem depend only upon the product of the Fermi mo-mentum and the scattering length. Experiments havebeen performed that probe the sound velocity [1] andcollective excitations [2], superfluidity [3] and criticaltemperature [4], phase separation and phase diagram[5, 6, 7, 8, 9] and RF response [10].We have performed fixed-node quantum Monte Carlo(QMC) calculations for both cold atoms and neutronmatter. In each case, the trial wave function is takento be of the Jastrow-BCS form with fixed particle num-ber and periodic boundary conditions:Ψ T = (cid:89) i FIG. 1: Zero-temperature equation of state for cold atomsand neutron matter. Near zero density we show the analyticalexpansion of the ground-state energy of a normal fluid, andat high density we show the cold atom result at unitarity( k F a = ∞ , arrow). QMC calculations are shown as circlesand squares for neutron matter and cold atoms, respectively. The interaction for cold atoms is taken as v ( r ) = − v (cid:126) m µ cosh ( µr ) , with µ = 24 /r , or an effective rangeof r / 12, with 1 /ρ = (4 / πr . The interaction rangeis small enough not to significantly affect the energyor pairing gap from the BCS regime to unitarity. Forthe neutron-neutron interaction, we take the s-wavepart of the AV18 [13] interaction. This interaction fitsnucleon-nucleon scattering very well at both low- andhigh-energies. For our purposes the important thing isthat the scattering length and effective range are cor-rectly described. We use this interaction only betweenspin up-down pairs, which sets the interaction in the L = 1 , M = ± L = 1 , M = 0 pairs perturbatively.This correction is 10% for the ground-state energy at thelargest densities considered, and typically much smaller.The correction to the pairing gap is always smaller thanthe statistical error in the calculation.The T = 0 equations of state for cold atoms and neu-tron matter are compared in Fig. 1. The horizontal axisis k F a , with the equivalent Fermi momentum k F for neu-tron matter shown along the top. The vertical axis is theratio of the ground-state energy to the free Fermi gasenergy ( E F G ) at the same density; it must go to one atvery low densities and decrease as the density increasesand the interactions become important. The curve atlower densities shows the analytical result [14] for normalmatter: E/E F G = 1 + π ak F + π (11 − ak F ) .This curve should be valid at very low densities. Whileit ignores the contributions of superfluidity, these are ex- ponentially small in (1/ k F a ).The neutron matter and cold atom equations of stateare very similar even for densities where the effectiverange is comparable to the interparticle spacing. Hencecold atom experiments can tell us something rather di-rectly about the neutron matter equation of state. Near k F a = − 10 the energy per particle is not too far fromQMC calculations [12, 15] and measurements [16] of theratio ξ of the unitary gas energy to E F G ; previous calcu-lations give ξ = 0 . r e = 0 and also AFMC calculations sug-gest that ξ = 0 . k F a = − 10 for neutron matter arecompatible with previous calculations of the neutronmatter equation of state at somewhat higher densities( k F ≥ − ) [18, 19, 20]. Results shown are for 66 par-ticles in periodic boundary conditions; calculations havealso been performed near N = 20, 44, and 90. Based onthese results, finite-size effects for N = 66 and beyondare expected to be quite small, of the order of a couplepercent. Calculations of the cold atom equation of stateare very similar to those reported previously in [15] and[21]; the energies reported here are slightly lower (up to ≈ 10% in some cases) because of larger system sizes andbetter optimizations.Realistic microscopic calculations that incorporatestrong pairing thus provide important constraints on theneutron matter equation of state in the subnuclear satu-ration density regime. Skyrme models or more generallydensity functionals are used, for example, to determinethe structure of neutron star crusts [22] and the neu-tron skin thickness of nuclei [23]. A realistic treatmentof these problems should incorporate the physics of therapid transition of neutron matter from nearly free par-ticles to a strongly paired system at very low densities.The pairing gap is the other fundamental zero-temperature property of superfluid systems. Calcula-tions of the s-wave pairing gap in neutron matter havevaried enormously over the past 20 years [24, 25]. Thedifficulties in accurately calculating corrections to theBCS pairing gaps in the strongly paired regime aresignificant, and hence calculations of the pairing gap[24, 25, 26, 27, 28, 29, 30] can differ by large factors(from 4 to 10) in the low-density regime. Cold atom ex-periments can provide a critical test of theories of thepairing gap in this regime. We first compare our cal-culations of the pairing gap in cold atoms and neutronmatter, and then compare with previous results.We calculate the pairing gap from the odd-even energystaggering ∆ = E ( N +1) − ( E ( N )+ E ( N +2)) / 2, where N is an even number of particles. Finite-size effects for thepairing gap are considerably larger than for the groundstate energy. In order to estimate the convergence of thegap to the continuum value with increasing N we have N D / E F BCS (continuum)BCS BCS projectedQMC FIG. 2: Neutron matter pairing gap at k F a = − 10 versusparticle number in periodic boundary conditions, BCS andQMC calculations. solved the BCS equations:∆( k ) = − (cid:88) k (cid:48) (cid:104) k | V | k (cid:48) (cid:105) ∆( k (cid:48) )2 (cid:112) (cid:15) ( k (cid:48) ) + ∆( k (cid:48) ) (cid:104) N (cid:105) = (cid:88) k (cid:34) − (cid:15) ( k ) (cid:112) (cid:15) ( k ) + ∆( k ) (cid:35) (3)in periodic boundary conditions for different (cid:104) N (cid:105) .The line in Fig. 2 is the continuum BCS result for k F a = − 10, and the open symbols are the solutions of theBCS equations for different (cid:104) N (cid:105) . The continuum resultsare nearly identical for the AV18 interaction and the sim-ple cosh potential adjusted to yield the same scatteringlength and effective range. For the finite systems BCS re-sults are shown for the cosh potential. Unlike the case ofcold atoms near unitarity, where − k F a >> r e ≈ N = 66, and oscillations fromthat point on are fairly small, comparable in size to thestatistical error in the QMC calculations. We also showas solid points the gaps obtained from particle-projectedBCS wave functions in variational Monte Carlo calcula-tions and the odd-even staggering formula. The projec-tion to definite particle number is a small effect.The lower points in Fig. 2 are QMC results for k F a = − 10. At very small values of N the gap is quitelarge, as is also seen in the BCS calculations. This isdue to the coarse description of the Fermi surface in suchsmall systems; the momentum grid spacing in occupiedstates is similar in magnitude to the Fermi momentum.When the pairing is very strong, as in cold atoms in theunitary regime, this coarse description is not too critical. k F [fm -1 ] D / E F Neutron MatterCold Atoms - k F a BCS-neutronsBCS-atoms QMC Unitarity FIG. 3: Superfluid pairing gap versus k F a for cold atoms( r e ≈ 0) and neutron matter ( | r e /a | ≈ . However for weaker coupling or the larger effective rangein neutron matter this becomes more important. Thegap in both BCS and QMC calculations reaches a mini-mum near 44 particles (near the midpoint between closedshells at 38 and 54), and then increases to values near thecontinuum limit. Pairing gap results for N = 66 − 92 areconsistent within the statistical errors.For all values of N the gap is considerably smaller thanthe BCS results. For comparison, at unitarity in coldatoms BCS calculations give a gap of 0 . E F while theQMC result is 0 . E F .[31] These calculations are ingood agreement with recent polarized cold atom exper-iments [9, 32]. For cold atoms the BCS equations willproduce the exact gap in the BEC limit where the pairsare strongly bound. No such limit is relevant for a finite-range interaction.In Fig. 3 we plot the pairing gap as a function of k F a for both cold atoms and neutron matter. BCS calcula-tions are shown as solid lines, and QMC results are shownas points with error bars. QMC pairing gaps are shownfrom calculations of N = 66 − 68 particles. For coldatoms away from unitarity the pairing gaps are smallerthan calculated previously [21], due to more complete op-timizations and because these larger simulations reducethe finite-size effects.For very weak coupling, − k F a << 1, the pairinggap is expected to be reduced from the BCS value bythe polarization corrections calculated by Gorkov [33]∆ / ∆ BCS = (1 / e ) / . Because of finite-size effects, itis difficult to calculate pairing gaps using QMC in theweak coupling regime. The QMC calculations at the low-est density, k F a = − 1, are roughly consistent with thisreduction from the BCS value. At slightly larger yet still k F [fm -1 ] D ( M e V ) BCSChen [26]Wambach [27]Schulze [28]Schwenk [29]Fabrocini [30]AFDMC [30]QMC - k F a FIG. 4: Superfluid pairing gap versus k F a for neutron mattercompared to previous results. small densities, where − k F a = O (1) but k F r e << k F a = − . 5, where k F r e ≈ . 35, support this expec-tation. Beyond that density the effective range becomesimportant and the QMC results are significantly reducedin relation to the cold atoms where r e ≈ et al . [30]. These calcu-lations are somewhat similar to those reported here. Thedisadvantage of the AFDMC approach is that it doesnot provide a variational bound to the energy, and hencethe wave functions are chosen from another approach.In the calculation of Ref. [30] the wave function wastaken from a correlated basis function approach that in-cluded a BCS initial state. The pairing in that variationalstate is unusually large, and in fact increases as a frac-tion of E F when the density is lowered. The advantageof the AFDMC approach is that it includes the full in-teraction rather than the simple s-wave interaction usedhere. AFDMC calculations with larger particle numbersare underway.[34]In summary, we have calculated the T = 0 equationsof state and pairing gaps for cold atoms and neutronmatter. These systems are quite similar in that both are very strongly paired, and both have pairing gaps of theorder of the Fermi energy. Experiments on the cold-atomequation of state would be very valuable in constrainingthe neutron matter equation of state. Pairing gaps inneutron matter are found to be suppressed compared tocold atoms and BCS theory, but much larger than inmost other approaches. Again, cold-atom experimentscould provide very valuable tests of many-body theories.It could be very important to explore finite-range effectsexperimentally using other atomic systems. Note added in proof. We recently became aware of newcalculations of the equation of state and pairing gap forcold atoms using auxiliary field Quantum Monte Carlotechniques [35]. Their results are similar to, but slightlydifferent than, those presented here.A.G. wishes to express his gratitude to V. R. Pandhari-pande for initial guidance. The authors would also like tothank K. E. Schmidt for valuable discussions. 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