aa r X i v : . [ nu c l - t h ] J a n Low-density neutron matter and the unitary limit
Isaac Vidaña Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Dipartimento di Fisica “Ettore Majorana”,Università di Catania, Via Santa Sofia 64, I-95123 Catania, Italy
We review the properties of neutron matter in the low-density regime. In particular, we reviseits ground state energy and the superfluid neutron pairing gap, and analyze their evolution fromthe weak to the strong coupling regime. The calculations of the energy and the pairing gap areperformed, respectively, within the Brueckner–Hartree–Fock approach of nuclear matter and theBCS theory using the chiral nucleon-nucleon interaction of Entem and Machleidt at N LO and theArgonne V18 phenomenological potential. Our results are compared with those of quantum MonteCarlo calculations for neutron matter and cold atoms. The Tan contact parameter in neutron matteris also calculated finding a reasonable agreement with experimental data with ultra-cold atoms onlyat very low densities. We find that low-density neutron matter exhibits a behavior close to thatof a Fermi gas at the unitary limit, although, this limit is actually never reached. We also reviewthe properties (energy, effective mass and quasiparticle residue) of a spin-down neutron impurityimmersed in a low-density free Fermi gas of spin-up neutrons, already studied by the author in arecent work, and include new results which confirm that these properties are very close to those ofan attractive Fermi polaron in the unitary limit.
INTRODUCTION
Pure neutron matter [1] is an ideal infinite nuclearsystem whose properties are of remarkable interest fora comprehensive understanding of neutron stars andneutron-rich nuclei. Particularly interesting are the prop-erties of neutron matter at low densities, since they arecrucial to understand the physics of the inner crust ofneutron stars [2], where the number density varies from ∼ − to ∼ . fm − and matter consists of a mix-ture of very neutron-rich nuclei (arranged in a Coulomblattice), electrons and a superfluid neutron gas. Low-density neutron matter, however, is a system less trivialthan one could expect at a first sight. The reason is thatat low densities the neutron-neutron interaction is dom-inated by the S partial wave which is very attractiveand, although it is not able to bind two neutrons, leadsto a well known virtual state which makes the neutron-neutron scattering length in this channel extremely large, a s = − . fm [3]. Therefore, even at very low den-sities, where the average distance between two neutrons( ∝ k − F with k F the Fermi momentum) is much largerthan the effective range of the S neutron-neutron in-teraction, r e = 2 . fm [4], neutron matter is still astrongly correlated system.Low-density neutron matter is similar to a unitaryFermi gas, an idealized system of spin-1/2 fermions with azero-range interaction having an infinite (negative) scat-tering length in which all its properties are simply pro-portional to the corresponding ones of a non-interactingFermi gas. The so-called unitary limit was introduced in1999 by George Bertsch [5], when he proposed a model oflow-density neutron matter with a zero-range interactionand a tuned to infinity scattering length. In this limit,or close to it, all the length-scales of a system drop outand the Fermi momentum becomes the only relevant one. Dilute fermionic systems with r e ≪ k − F ≪ | a s | , like neu-tron matter at low densities, exhibit universal propertiesclose to those of a unitary Fermi gas, regardless the na-ture of the particles that constitute the system and theirmutual interactions. Universality is expected to show inground state properties [6], collective excitations [7–13]and thermodynamical properties [14–18]. In particular,the ground state energy of any fermionic system close tothe unitary limit is expected to be E = ξ E F G where ξ isthe so-called Bertsch parameter and E F G = 3 ~ k F / m is the energy of the corresponding non-interacting Fermigas. Different theoretical calculations predict values of ξ in the range . − . [6, 19–24]. The best estima-tions of the value of the Bertsch parameter come fromquantum Monte Carlo (QMC) calculations which predict ξ = 0 . [25], 0.42(1) [26] and 0.40(1) [65]. Variational[28], finite volume Green’s function Monte Carlo [29] andBrueckner–Hartree–Fock (BHF) [30] calculations of theequation of state (EoS) of low-density neutron mattergive ξ ≈ . . Using unitary nucleon potentials, con-structed ad hoc to have an infinite S neutron-neutronscattering length, the authors of Refs. [31, 32] studiedthe ground state energy of low-density neutron matter,obtaining values of ξ remarkably close to the QMC pre-dictions over a wide range of low densities, and showing,as expected, that low-density neutron matter behaves asa unitary Fermi gas as long as a s → −∞ .Unitary Fermi gases have been experimentally realizedwith ultra-cold trapped alkali atoms (with Li and Kbeing the most commonly used ones), where the effec-tive range of the interaction is r e ∼ − k − F , and thescattering length a s can be tuned from negative to pos-itive values with the help of magnetic fields, becominginfinity at the so-called Feshbach resonance [33]. Theseexperiments provide constraints on the properties of uni-tary Fermi gases and, therefore, indirectly also on thoseof low-density neutron matter. Experimental measure-ments of the Berstch parameter with ultra-cold atomicgases gives the values . [14], . [34], . [35], . +0 . − . [36], and . to . [37]. Thepossibility of varying in these experiments the interac-tion between the atomic species from a weak to a strongcoupling regime in a controlled way, has additionally al-lowed the study of the whole crossover from BCS pair-ing with weakly attractive ( a s < ) Cooper pairs tothe Bose–Einstein condensation (BEC) of bound dimers( a s > ) [38–40]. As it was mentioned before, althoughthe neutron-neutron interaction is very attractive in the S channel, it is unable to lead to the formation of abound dineutron state and, hence, a BEC phase does notexist in neutron matter. Nonetheless, by varying the den-sity, dilute neutron matter can go from the strong cou-pling regime close to the unitary limit to the weakly cou-pled BCS one. The importance in low-density neutronmatter of BCS-BEC crossover-like physics was pointedout by Matsuo in Ref. [41], where he studied the behav-ior of the strong spatial dineutron correlation, findingthat the density region region n ≈ (10 − − . n (where n ≈ . fm − is the nuclear saturation density) cor-responds to the domain of the BCS-BEC crossover. Itis known from a general argument (see e.g., Refs. [42–44]), which applies to any dilute fermionic system, thatthe pair correlations of fermions interacting with a largescattering length differ from what is considered in theconventional BCS theory. Corrections due to pair corre-lations in the normal phase of neutron matter have beenconsidered by several authors [45–50] using the Nozières–Schmitt–Rink approach [44], which is the simplest onethat interpolates correclty between the BCS and BCElimits. BCS-BEC crossover effects and the existence,above the critical temperature T c for the transition tothe superfluid state, of a pseudo-gap in neutron matterhave been also recently studied within the in-medium T -matrix formalism by Durel and Urban in Ref. [51]. Atthe BCS-BEC transition point, i.e. at the unitary limit,the superfluid pairing gap ∆ is expected to be propor-tional to the free Fermi energy, E F = ~ k F / m . Ultra-cold atoms experiments with imbalance Fermi gases of Li found ∆ = (0 . ± . E F [52–54], in contrast withconventional superfluids or superconductors where thepairing gap is very weak, of the order of ∼ . ofthe Fermi energy. QMC calculations of the neutron S pairing gap by Gezerlis and Carlson [55] found a max-imum value of ∆ of ∼ . E F at the Fermi momentum k F ∼ . fm − ( n ∼ × − fm − ). This maximumvalue of the gap corresponds to a strong coupling situa-tion ( ( k F a s ) − ∼ − . ) close to that found in a unitaryFermi gas.Experiments with population-imbalanced ultra-coldatomic gases, have allowed also to study the properties ofpolarized unitary gases and quantum impurities leading,particularly, to the experimental realization of attractive and repulsive Fermi and Bose polarons, quasiparticlesarising from the dressing of an impurity strongly cou-pled to a bath of particles of fermionic or bosonic nature.In the unitary limit, the energy of a Fermi polaron showsalso a universal behavior, being E pol = ηE F [56] with η ≈ − . [57–59]. A few years ago, Forbes et al. [60]extended the idea of the polaron to a system of stronglyinteracting neutrons and studied the energy of the neu-tron polaron with the QMC method. Similarly, Roggero et al. [61] used also this method to analyze the energy ofa proton impurity in low-density neutron matter findingthat, for a wide range of densities, the behavior of theproton impurity is similar to that of a polaron in a fullypolarized unitary Fermi gas. Using the BHF approach,very recently, in Ref. [62] the author of the present workhave analyzed the energy, effective mass and quasiparticleresidue of a spin-down neutron impurity in a low-densityfree Fermi gas of spin-up neutrons, showing that theseproperties are in remarkable agreement with those of theattractive Fermi polaron in the unitary limit realized inultra-acold atomic gases experiments.In this work we review the properties of neutron mat-ter in the low-density regime. Particularly, we revise itsground state energy and the superfluid neutron pairinggap, and analyze their evolution from the weak to thestrong coupling regime. We use the well known BHFapproach and the BCS theory to calculate, respectively,the ground state energy and the pairing gap, employ-ing as bare nucleon-nucleon (NN) interactions the chiralone of Entem and Machleidt at N LO with a 500 MeVcut-off [63] (hereafter referred to simply as chiral inter-action) and the Argonne V18 (AV18) [64] phenomeno-logical potential. Our results are compared with thoseof quantum Monte Carlo calculations for neutron matterand cold atoms. Finally, we also review the properties ofa spin-down neutron impurity immersed in a low-densityfree Fermi gas of spin-up neutrons including new results,based on AV18 potential, and compared then with thoseobtained by the author in Ref. [62] for the chiral interac-tion.The manuscript is organized in the following way. Theground state energy of low-density neutron matter andthe superfluid neutron pairing gap are presented, respec-tively, in Secs. 2 and 3, whereas, the properties of a spin-down neutron impurity in a low-density free Fermi gasof spin-up neutrons are shown in Sec. 4. Finally, a briefsummary and the main conclusions of this work are givenin Sec. 5.
GROUND STATE ENERGY
We start this section by showing in Fig. 1 the groundstate energy (in units of E F G ) of low-density neutronmatter and cold atoms as a function of the dimensionlessparameter − k F a s . Full circles and squares display theresults for neutron matter of our BHF calculation per-formed using the chiral interaction (full circles) and theAV18 potential (full squares). Only contributions fromthe S partial wave are included in the calculation. Fulltriangles and diamonds correspond, respectively, to theQMC results for neutron matter (full triangles) and coldatoms (full diamonds) obtained by Gezerlis and Carlsonin Ref. [65]. The QMC results of neutron matter shownhere include, as our BHF calculation, contributions onlyfrom the S partial wave and were obtained also withthe AV18 potential. The impact of P-wave interactionswas explored by these authors in Ref. [55] using the Ar-gonne V4 potential [66]. They found (see figure 3 of Ref.[55]) that at − k F a s = 10 the energy of neutron matterincreases by ∼ while this increase was only of ∼ at − k F a s = 5 , confirming that at very low densities thedominant role is played by the S-wave, as it was said inthe introduction. For the cold atom case, Gezerlis andCarlson considered an hyperbolic cosinus interaction po-tential of the form v ( r ) = − v ~ m µ cosh ( µr ) , (1)where the strength v was adjusted to obtained valuesof − k F a s from to , and µ was taken such that theeffective range of the potential was much smaller thanthe interatomic distance.Coming back to the figure, the arrow indicates the av-erage value, ξ ≈ . , of the experimental measurements[14, 34–37] of the Bertsch parameter with ultra-coldatomic gases at the unitary limit ( i.e., for − k F a s → ∞ ),whereas the continuous line shows the well known ex-treme low-density limit ( − k F a s ≪ ) of Lee and Yang[67] EE F G = 1 + 109 π k F a s + 421 π (11 − ln k F a s ) . (2)In the figure, for comparison, we also show the recentdensity functional proposed by Lacroix in Ref. [68], EE F G = 1 + 109 π k F a s − π k F a s / [1 − ξ ( k F r e )] (3)with ξ ( k F r e ) = 1 − (1 − ξ ) − ξ + k F r e η e (4)where the two parameters ξ and η e of this functional arefixed to reproduce both the universal properties of a uni-tary Fermi gas, and the Lee–Yang limit at extremely lowdensities. In particular, we show the Lacroix’s functionalfor ξ = 0 . assuming that r e = 0 , therefore, being ther. h.s. of Eq. (4) simply reduced to ξ , which makes thevalue of η e irrelevant in this case.We note first that, for neutron matter, our BHF calcu-lation gives results very similar for the two NN employed. F a s E / E F G BHF chiralBHF AV18QMC QMC cold atomsEq. (3) unitary limitLee-Yang limit } neutron matter FIG. 1: Ground state energy (in units of E F G ) of low-densityneutron matter as a function of the dimensionless parameter − k F a s . Results are shown for our BHF calculation (full cir-cles and squares) and the QMC one (full triangles) of Ref.[65]. The energy of cold atoms (full diamonds), obtainedalso by the authors of Ref. [65], is shown for comparison.The continuous line displays the Lee–Yang limit (see Eq. (2))for − k F a s ≪ [67]. The arrow indicates the average value, ξ ≈ . , of the experimental measurements [14, 34–37] of theBertsch parameter with ultra-cold atomic gases at the unitarylimit. The dotted line shows the density functional proposedby Lacroix (see Eq. (3)) in Ref. [68]. Furthermore, we notice also that our results are in quitegood agreement with the QMC ones over the whole rangeof values of the dimensionless parameter − k F a s consid-ered. This indicates that not only the details of the NNinteraction are irrelevant for neutron matter a very lowdensities, but also those of the approach employed tosolve the many-body problem seem to be quite unimpor-tant (see, e.g. , figure 4 of Ref. [55], where it is shown thatresults for neutron matter obtained with different meth-ods agree within in the range < − k F a s < ). Itcan be also seen in the figure that our BHF calculationas well as the QMC one extrapolate properly to the Lee–Yang limit at extremely low densities, and that both arereasonably well reproduced by the Lacroix’s functional ofEq. (3) for − k F a s ≤ .In the unitary limit QMC results of the ratio E/E
F G ,in the case of cold atoms, approach the value . ingood agreement with experimental measurements of theBerstch parameter [14, 34–37]. As we already said, neu-tron matter never reaches strictly the unitary limit, al-though is close to it. In particular, Baldo and Maieron[30], on the basis of the Brueckner–Bethe–Goldstonemany-body theory, showed that in the range of densi-ties corresponding to the Fermi momenta . < k F < . -2.5 -2 -1.5 -1 -0.5 0(k F a s ) -1 C / k F BHF chiral BHF AV18momentum distributionphotoemission spectroscopyradio frequency spectroscoystatic structure factor Ref. [73] -k F a s E / E F G } Ref. [72]
FIG. 2: Tan contact parameter as a function of ( k F a s ) − inunits of the Fermi momentum. Results for neutron matterfrom our BHF calculation with the two NN interactions con-sidered (full circles and squares) are compared with data fromexperiments with ultra-cold fermionic gases of K (open cir-cles, triangles and squares) [72] and Li (open diamonds) [73].The insert shows the fit of our BHF results for the groundstate energy of neutron matter with the Lacroix’s functionalof Eq. (3) (dashed lines). fm − the energy of neutron matter turns to be very closeto one half of E F G . A similar result was found also inthe variational and finite volume Green’s function MonteCarlo calculations of Refs. [28] and [29]. As it is seen inthe figure, our BHF results show an almost constant valueof the ratio
E/E
F G in the range . < − k F a s < . ( . < k F < . fm − ). Making a linear fit of ourresults in this range we obtain, respectively, the values E/E
F G = 0 . and . for the chiral interaction andthe AV18 potential, in agreement with the results theseworks. The linear dependence of the neutron matter en-ergy with E F G , found in our BHF calculation in thisFermi momentum range, can be understand from an ar-gument pointed out by Carlson et al. in Ref. [29] thatwe briefly review here.The interaction energy is proportional to the density( ∝ k F ) times the volume integral of the G -matrix which isrelated to the bare interaction V through the well knownBrueckner equation, written schematically as Gφ = V ψ , (5)where φ and ψ are the unperturbed and perturbed two-neutron wave functions. At low densities all the relevantrelative momenta are small and, therefore, one has φ = 1 and, in vacuum, beyond the effective range of the inter-action, ψ = 1 − a s /r e . In addition, since for neutron matter one has − a s /r e > one can approximate ψ sim-ply by − a s /r e , and, therefore, G ≈ − a s V /r e . It is easyto see then that in the range . < − k F a s < . , the G -matrix is proportional to ( k F r e ) − . Consequently, itsvolume integral becomes, in this range, proportional to k − F and the interaction energy proportional to k F , as itis the case of the energy of the non-interacting Fermi gas.To finish this section we show in Fig. 2 our results forthe Tan contact parameter [69–71] of neutron matter asa function of ( k F a s ) − . The contact parameter of aninfinite spin saturated system is given by C = 1 n πma s ( ~ c ) dεda s , (6)where n = k F / π and ε = nE are, respectively, thedensity and the energy density of the system. For com-putational reasons, to calculate C we have first fitted ourBHF results for the ground state energy E of neutronmatter by using the Lacroix’s functional of Eq. (3) taking r e = 2 . fm. A good fit of our results for the energy isobtained using the parameters ξ = 0 . and η = 0 . in the case of the chiral interaction, and ξ = 0 . and η = 0 . for the AV18 potential. The results ofthe fit for are shown by the dashed lines in the insetof the figure. Our result for the contact parameter inneutron matter is compared with the experimental datafrom measurements with ultra-cold fermionic atoms ofthe momentum distribution (open circles), photoemis-sion spectroscopy (open triangles) and radio frequencyspectroscopy (open squares) of a gas of K [72], andthe static structure factor (open diamonds) of a gas of Li [73]. A reasonable good agreement between our re-sults for neutron matter and the experimental data fromultra-cold atoms is found only in the very low-densityregime ( ( k F a s ) − < − ), being the differences very largein the range − . < ( k F a s ) − < − . , correspondingto . < k F < . fm − . SUPERFLUID PAIRING GAP
We consider now the superfluid pairing gap of neutronmatter at low densities, which is an important quantityto understand the properties of neutron-rich nuclei [74]and neutron star cooling [75]. In particular, we have per-formed a mean-field BCS calculation of the S pairinggap using the chiral interaction and the AV18 potential.The results of this calculation are shown (in units of E F )as a function of the dimensionless parameter − k F a s inFig. 3, together with those of the QMC ones obtainedby Gezerlis and Carlson (full triangles) [65] using alsothe AV18 potential. The superfluid pairing gap for coldatoms (full diamonds) [65], calculated also by these twoauthors with the hyperbolic cosinus interaction of Eq.(1), is shown for comparison. The continuous lines at F a s ∆ / E F BCS chiralBCS AV18QMC QMC cold atomsBCS weak coupling limitGMB finite polarization correction } neutron matter unitary limit FIG. 3: Superfluid S pairing gap (in units of E F ) of low-density neutron matter as a function of the dimensionless pa-rameter − k F a s . BCS results (full circles and squares), ob-tained with the two NN interactions considered, are showntogether with the QMC ones (full triangles) of Ref. [65]. Thesuperfluid pairing gap for cold atoms (full diamonds), calcu-lated also by the authors of Ref. [65], is shown for comparison.The continuous lines at very low values of − k F a s display, re-spectively, the BCS result in the weak coupling limit and theGorkov and Melik-Barkhudarov [76] finite polarization correc-tion. The arrow indicates the value of ∆ /E F at the unitaritylimit extracted from ultra-cold Fermi atoms experiments withimbalance mixtures of Li [52–54]. very low values of the parameter − k F a s show, respec-tively, the well known analytic BCS result in the weakcoupling limit ∆ BCS ( k F ) = 8 e ~ k F m exp (cid:18) π k F a s (cid:19) , (7)where e is the Euler’s number, and the Gorkov andMelik–Barkhudarov (GMB) [76] finite polarization cor-rection, ∆ ( k F ) = 1(4 e ) / e ~ k F m exp (cid:18) π k F a s (cid:19) , (8)due to the inclusion of induced interactions that reducethe gap even at weak coupling. As it is seen, QMC re-sults for both neutron matter and cold atoms extrapolateproperly to the GMB result whereas our BCS calculation,as expected, does it towards the BCS weak coupling limit.We already said in the introduction that close to theunitary limit the superfluid pairing gap is expected tobe ∆ = δE F , where the proportionality constant δ wasfound ∼ . in ultra-cold Fermi atoms experiments withimbalance mixtures of Li [52–54]. This value is indi-cated in the figure with an arrow. At the unitary limit, the QMC result for cold atoms of Gezerlis and Carlsonis . E F , in good agreement with these experiments.For neutron matter, the QMC calculation predicts, aswe also mentioned in the introduction, a maximum valueof the pairing gap of ∼ . E F at − k F a s ∼ , whichis ∼ of the value of the BCS result found by thesame authors in Ref. [65]. Our BCS calculation predictsa maximum value of ∆ of ∼ . E F at − k F a s = 4 . and of ∼ . E F at − k F a s ∼ . when using the chi-ral interaction or the AV18 potential, respectively. It isinteresting to note that while for the ground state en-ergy both NN interactions give very similar results, thisis not the case for the pairing gap for which their pre-dictions are slightly different. The maximum values ofthe gap found in both QMC and BCS calculations corre-spond to a strong coupling situation where the behaviorof neutron matter, with a Fermi momentum of ∼ . fm − in the QMC case or ∼ . − . fm − in the BCSone, can be considered close to that of a unitary Fermigas. The reader should note, however, that although themaximum value of the gap obtained with our BCS calcu-lation seems to be in agreement with experimental datafrom ultra-cold atoms experiments, this is not the casebecause the BCS is just a mean-field calculation whichdoes not include the effects of medium polarization thatare very important even in the low-density regime andreduce the value of the gap. Therefore, in the case of ourBCS calculation, our previous statement regarding thevicinity of neutron matter to the unitary limit should beconsidered only qualitatively. NEUTRON POLARON
In this final section, we review the recent analysis of theenergy, effective mass and quasiparticle residue of a spin-down ( ↓ ) neutron impurity immersed in a low-density freeFermi gas of spin-up ( ↑ ) neutrons, made by the authorof the present work in Ref. [62] using the BHF approach,where he showed that the ( ↓ ) neutron impurity behavesbasically as an attractive Fermi polaron in a unitary gas.New results, based on AV18 potential are included in thisreview and are compared with those shown for the chiralinteraction in Ref. [62].The energy of the ↓ neutron impurity with zero mo-mentum is shown in Fig. 4 as a function of the Fermimomentum (panel a) and of the Fermi energy (panel b)of the free gas of ↑ neutrons for the two NN interac-tions considered here, the chiral one (full circles) and theAV18 potential (full squares). Note that both interac-tions predict essentially the same results. Note in addi-tion that the linear behavior shown by the energy of aFermi polaron in the unitary limit, E pol = ηE F [56], isclearly seen also in the case of the energy the ↓ neutronimpurity, where a value of the proportionality constant η = − . ( η = − . ) is found using the chiral interac- F [fm -1 ]-4-3-2-10 I m pu r it y e n e r gy E [ M e V ] Chiral ( η = - 0.63)AV18 ( η = - 0.64)Ref. [57] ( η = -0.615)Ref. [58] ( η = - 0.58(5))Ref. [59] ( η = - 0.64(7)) F [MeV](a) (b) FIG. 4: Energy of a ↓ neutron impurity with zero momentum as a function of the Fermi momentum (panel a) and of the Fermienergy (panel b) of the free gas of ↑ neutrons. The results obtained with the two NN interactions considered, chiral (full circles)and AV18 (full squares), are compared with those obtained when using the values of the proportionality constant η derived inthe QMC calculation of Ref. [57] (solid line) and experimentally in Refs. [58, 59] (dashed and dotted lines). tion (AV18 potential). These numbers are in very goodagreement with the results of state-of-the-art QMC calcu-lations η = − . [57] and the values η = − . [58]and η = − . [59] extracted from experiments withspin-polarized Li atoms with resonant interactions. Ourresult shows that a ↓ neutron impurity in a low-densityfree Fermi gas of ↑ neutrons presents a behavior similarto that of attractive Fermi polaron in the unitary limit,being irrelevant the details of the interaction between theimpurity and the free Fermi gas. To further confirm thisbehavior, in the next, we analyze also the effective massand the quasiparticle residue of a ↓ neutron impurity withzero momentum.The effective mass of a ↓ neutron impurity with zeromomentum, m ∗↓ , can be extracted by assuming that itsenergy is quadratic for low values of its momentum ~k ↓ ,and fitting this parabolic energy to the energy calculatedwithin the BHF approach. The quasiparticle residue isdefined as Z ↓ = − ∂U ↓ ( ~k ↓ = ~ , E ′↓ ) ∂E ′↓ ! − E ′↓ = U ↓ ( ~k ↓ = ~ (9)where U ↓ ( ~k ↓ , E ′↓ ) is the off-shell BHF ↓ neutron poten-tial. It gives a measurement of the importance of thecorrelations. The more important the correlations are,the smaller is its value. Results for both quantities areshown in panels a and b of Fig. 5 as a function of theFermi momentum of the ↑ neutron free Fermi gas for thetwo NN interactions considered. Note that also in thiscase both interactions predict almost the same results,indicating once more the irrelevance of the interactiondetails in the low-density regime. As it can be seen in the figure, initially m ∗↓ ( Z ↓ ) increases (decreases), thenit reaches a maximum (minimum) at k F ∼ . fm − andfinally it decreases (increases) at higher densities. Wenotice that for k F ∼ . fm − , where m ∗↓ and Z ↓ presenttheir respective maximum and minimum, the average in-terparticle spacing n − / (with n = k F / π the densityof the ↑ neutron free Fermi gas) is of the order of the S neutron-neutron scattering length, i.e., n / | a | ∼ . Wecan venture to say that k F ∼ . fm − establishes theborder between a less correlated and a more correlatedregime of the system. In fact, note that the values of Z ↓ are in general larger in the k F region from to . fm − than for k F & . fm − , indicating that in this regionof very low-densities correlations are less important, andthat the ↓ neutron impurity propagates more freely in the ↑ neutron gas. We also notice that for Fermi momentaabove ∼ . fm − , the S neutron-neutron scatteringlength is larger that the average interparticle spacingwith values of the dimensionless quantity n / | a | rang-ing from at k F = 0 . fm − to . at k F = 0 . fm − .Although in the unitary limit it is strictly fulfilled thecondition n / | a | ≫ , these numbers indicate once morethat low-density neutron matter is close to it, at least forFermi momenta in the range from ∼ . fm − to ∼ . fm − . Averaging the effective mass and the quasiparticleresidue over the Fermi momentum in the range between . fm − and . fm − we find, respectively, the meanvalues m ∗↓ = 1 . m and Z ↓ = 0 . using the chiral inter-action, and m ∗↓ = 1 . m and Z ↓ = 0 . , in the case of theAV18 potential. The results for both quantities compareremarkably well with those of the full-many body analysisof Combescot and Giraud [77] who found m ∗↓ = 1 . m ,and those of the diagrammatic Monte Carlo method em- I m pu r it y e ff ec ti v e m a ss m * / m ChiralAV18
F [fm -1 ]0.70.80.91 I m pu r it y qu a s i p a r ti c l e r e s i du e Z (a)(b) FIG. 5: Effective mass (panel a) and quasiparticle residue(panel b) of a ↓ neutron impurity with zero momentum as afunction of the Fermi momentum of the free gas of ↑ neutrons.Results are shown for the two NN interactions considered. ployed by Vlietinck et al., [78] who obtained a value of0.759 for the quasiparticle residue. These results confirmonce more the Fermi polaron behavior exhibited by the ↓ neutron impurity in a low-density free Fermi gas of ↓ neutrons. SUMMARY AND CONCLUSIONS
In this work we have reviewed the properties of neu-tron matter at low-densities. In particular, using the wellknown BHF approach of nuclear matter and the BCStheory we have calculated, respectively, the ground stateenergy and the superfluid neutron pairing gap. Resultshave been obtained for two NN interactions, the chiralone of Entem and Machleidt at N LO with a 500 MeV cut-off and the Argonne V18 phenomenological potential,and have been compared with those of quantum MonteCarlo calculations for neutron matter and cold atoms.We have found that the energy of neutron matter withFermi momenta in the range . < k F < . fm − isabout one half of the energy of a non-interacting Fermigas, in agreement with previous BHF, variational and fi-nite volume Green’s function Monte Carlo calculationsof low-density neutron matter. This result indicates thatin this range of low densities neutron matter is close tothe unitary limit although, actually, it never reaches it.We have determined also the Tan contact parameter inneutron matter finding that only at very low densitiesthere is a reasonable good agreement between our re-sults and experimental data form ultra-cold atoms. Wehave found that out BCS calculation predicts a maxi-mum value of the pairing gap of ∼ . E F for a Fermimomentum of ∼ . fm − . However, although this valueis close to that found at the unitary limit in experimentsultra-cold Fermi gases, this does not mean that there isa good agreement between our calculation and experi-mental data because the BCS calculation does not takeinto account medium polarization effects which very im-portant even at low densities and reduce the value ofthe gap. Finally, we have have reviewed the properties(energy, effective mass and quasiparticle reside) of a ↓ neutron impurity in a low-density free Fermi gas of ↑ neu-trons. Our results have shown that this impurity presentsproperties close to those of an attractive Fermi polaronin the unitary limit. Acknowledgements
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