aa r X i v : . [ nu c l - t h ] J a n Fermi polaron in low-density spin-polarized neutron matter
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 study the properties of a spin-down neutron impurity immersed in a low-density free Fermi gasof spin-up neutrons. In particular, we analyze its energy, effective mass and residue, and we comparethe results obtained with those of state-of-the-art quantum Monte Carlo calculations of the attractiveFermi polaron realized in ultracold atomic gases experiments. The calculations are performed withinthe Brueckner–Hartree–Fock approach using the chiral two-body nucleon-nucleon potential of Entemand Machleidt at N LO with a 500 MeV cut-off, and including only contributions from the S partialwave which is the dominating one in the low-density region considered. Contributions from three-nucleon forces are expected to be irrelevant at these densities and, therefore, are neglected in thecalculation. Our results for the energy ( E ↓ = − . E F ), effective mass ( m ∗↓ = 1 . m ) and residue( Z ↓ = 0 . ) of the impurity are in very good agreement with quantum Monte Carlo calculations,showing that a spin-down neutron impurity in a low-density free Fermi gas of spin-up neutronsexhibits properties very close to those of an attractive Fermi polaron in the unitary limit. The idea of the polaron, a quasiparticle arising fromthe dressing of an impurity strongly coupled to an en-vironment or a bath, was introduced in the pioneeringworks of Landau [1] and Pekar [2] to describe the prop-erties of conduction electrons in a dielectric crystal, andit was further elaborated by Fröhlich [3] and Feynman[4] who considered the ionic crystal or polar semicon-ductor as a phonon bath. Since this initial introduc-tion of the electron-phonon polaron concept in solid-statephysics, the polaron idea has been extended and gener-alized to other areas of physics with applications in field-effect transistors [5], high-temperature superconductors[6, 7], ultracold atomic gases [8, 9], nuclear physics [10–12], or even general relativity [13] and quantum-state re-duction [14]. In particular, experiments with population-imbalanced ultracold atomic gases, where the minorityatomic species plays the role of the impurity and themajority one that of the environment, have allowed theexperimental realization of polarons providing an excep-tional framework for studying the properties of quan-tum impurities. The possibility of varying the impurity-medium interaction from a weak to a strong couplingregime by means of Feshbach resonances [15] has per-mited to investigate, in a controlled way, how the impu-rity becomes “dressed” by the excitations of the medium.The properties of the polaron depend strongly on thequantum statistics of the majority atomic species. Ex-periments in which the minority atomic species is im-mersed in a bath of atoms of fermionic or bosonic naturehave been carried out revealing the existence of attrac-tive and repulsive Fermi [16–26] and Bose [27–38] po-larons. Whereas Fermi polarons are the paradigmatic re-alization of Landau’s fundamental idea of quasiparticle,the description of an impurity atom in a Bose-Einsteincondensate can be cast in the form of the Fröhlich’s po-laron Hamiltonian [3], where the role of the phonons isplayed by Bogoliubov excitations. The theoretical de-scription of Fermi and Bose polarons has achieved a sig- nificant progress thanks to the development and appli-cation of several numerical many-body techniques thatinclude among others, the renormalization group theory[39–41], exact diagonalization [42], field-theoretical di-agrammatic approaches [43], continuous-time quantumMonte Carlo [44], diagrammatic Monte Carlo methods[45–47] and variational approaches [48–53].Nowadays most of the interest is focused on the Bosepolaron, although the Fermi polaron is still an excitingarea of research. In this work, we study the propertiesof a spin-down ( ↓ ) neutron impurity immersed in a low-density free Fermi gas of spin-up ( ↑ ) neutrons and showthat it behaves as an attractive Fermi polaron in theunitary limit, i.e., the limit of infinite (negative) S -wavescattering length of a gas of fermions at vanishing smalldensity. Despite the fact that the neutron-neutron scat-tering length of the S partial wave is extremely large( a = − . fm), low-density neutron matter actuallynever reaches the unitary limit, although it shows proper-ties close to it [54–56]. A few years ago, Forbes et al. [11]extended the idea of the polaron to a system of stronglyinteracting neutrons. These authors studied the energyof the neutron polaron using quantum Monte Carlo cal-culations that included contributions beyond the effectiverange of the impurity-fermion interaction, and comparedtheir results with those obtained from effective field the-ory calculations that also included these contributions.Furthermore, they used the neutron polaron energy toconstrain the time-odd part of nuclear energy densityfunctionals (EDFs) in order to obtain better descriptionsof polarized nuclear systems. Similarly, Roggero et al. [12] used also the quantum Monte Carlo method withchiral nucleon-nucleon interactions to analyze the energyof a proton impurity in low-density neutron matter find-ing that, for a wide range of densities, the behavior ofthe proton impurity is similar to that of a polaron in afully polarized unitary Fermi gas. The authors of thiswork, as those of Ref. [11], employed also their results to F [fm -1 ]-4-3-2-10 I m pu r it y e n e r gy E [ M e V ] This work ( η = - 0.63)Ref. [16] ( η = - 0.58(5))Ref. [17] ( η = - 0.64(7))Ref. [46] ( η = -0.615) F [MeV](a) (b) FIG. 1: Energy of a ↓ neutron impurity with ~k ↓ = ~ as a function of the Fermi momentum (panel a) and of the Fermi energy(panel b) of the free gas of ↑ neutrons. The results (full circles) are compared with those obtained when using the values of theproportionality constant η derived experimentally in Refs. [16, 17] (solid and dashed lines) and in the quantum Monte Carlocalculation of Ref. [46] (dotted line). impose tight constraints on the time-odd components ofSkyrme-type nuclear EDFs. In the present work, in addi-tion to the energy, we also analyze the effective mass andthe quasiparticle residue (or Z factor) of the ↓ neutronimpurity, and we compare the results obtained with state-of-the-art quantum Monte Carlo calculations of the at-tractive Fermi polaron realized in ultracold atomic gasesexperiments. Our calculation starts with the construction of the h ~k ↓ ~k ↑ | G ( ω ) | ~k ↓ ~k ↑ i G -matrix elements describing the in-medium interaction of the ↓ neutron impurity with oneof the ↑ neutrons of the free Fermi gas. To such end,using the chiral nucleon-nucleon potential of Entem andMachleidt at N LO with a 500 MeV cut-off [57] as thebare interaction between the ↓ and ↑ neutrons, we solvethe coupled-channel Bethe–Goldstone integral equation, h ~k σ ~k σ | G ( ω ) | ~k σ ~k σ i = h ~k σ ~k σ | V | ~k σ ~k σ i + X σ i σ j h ~k σ ~k σ | V | ~k σ i ~k σ j ih ~k σ i ~k σ j | Q | ~k σ i ~k σ j ih ~k σ i ~k σ j | G ( ω ) | ~k σ ~k σ i ω − E σ i ( ~k σ i ) − E σ j ( ~k σ j ) + iη , (1)including only contributions from the S partial wavewhich is the dominating one in the low-density region[54]. Contributions from three-nucleon forces are ex-pected to be irrelevant at these densities and, therefore,are neglected in the calculation [58–60]. In Eq. (1), V isthe bare nucleon-nucleon interaction, Q is the Pauli op-erator allowing only intermediate states compatible withthe Pauli principle, and ω is the starting energy, definedas the sum of the initial non-relativistic energies of theinteracting ↓ and ↑ neutrons. Whereas the energy of a ↑ neutron of the free Fermi gas is just kinetic that of the ↓ neutron impurity is E ↓ ( ~k ↓ ) = ~ k ↓ / m + Re [ U ↓ ( ~k ↓ )] where U ↓ ( ~k ↓ ) = X | ~k ↑ |≤ k F ↑ h ~k ↓ ~k ↑ | G ( ω = E ↓ ( ~k ↓ )+ ~ k ↑ m ) | ~k ↓ ~k ↑ i (2) is the on-shell Brueckner–Hartree–Fock (BHF) ↓ neutronpotential denoting the mean field “felt” by the impuritydo to its interaction with the free Fermi gas. We notethat the continuous prescription is adopted when solvingthe Bethe–Goldstone equation. We note also that Eqs.(1) and (2) are self-consistently solved.The energy of a Fermi polaron, defined as the changein the energy of the non-interacting Fermi gas when animpurity of zero momentum is added to it, shows in theunitary limit a universal behavior regardless the natureof the impurity, the bath and the details of their mutualinteractions, being simply proportional to the Fermi en-ergy, E pol = ηE F [61]. State-of-the-art quantum MonteCarlo calculations of Prokof’ev and Svistunov [46] pre-dict a value of the proportionality constant η = − . ,in very good agreement with the values η = − . [16]and η = − . [17] extracted from experiments withspin-polarized Li atoms with resonant interactions. InFig. 1 we show the energy of a ↓ neutron impurity with ~k ↓ = ~ as a function of the Fermi momentum (panel a)and of the Fermi energy (panel b) of the free gas of ↑ neu-trons. Our results (full circles) are compared with thoseobtained when using the values of the proportionalityconstant η derived experimentally in Refs. [16, 17] (solidand dashed lines) and in the calculation of Prokof’ev andSvistunov in Ref. [46] (dotted line). Fitting linearly theenergy of the impurity with E F , we obtain η = − . , invery good agreement with the results of these works. 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.To further confirm this behavior, in the next, we analyzealso the effective mass and the quasiparticle residue ofa ↓ neutron impurity with zero momentum. Results areshown, respectively, in panels a and b of Fig. 2 as a func-tion of the Fermi momentum of the ↑ neutron free Fermigas.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 calculated BHFone, E ↓ ( ~k ↓ ) . The quasiparticle residue is defined as Z ↓ = − ∂U ↓ ( ~k ↓ = ~ , E ′↓ ) ∂E ′↓ ! − E ′↓ = U ↓ ( ~k ↓ = ~ (3)where U ↓ ( ~k ↓ , E ′↓ ) is the off-shell BHF ↓ neutron potential,obtained from Eq. (2) by integrating off-shell G -matrixelements. It gives a measurement of the importance ofthe correlations. The smaller its value is, the more im-portant are the correlations. We find, as it is seen in Fig.2, that m ∗↓ ( Z ↓ ) increases (decreases) initially, reachesa maximum (minimum) at k F ∼ . fm − and then itdecreases (increases) at higher densities. To understandbetter the dependence of m ∗↓ and Z ↓ with k F , we notethat, in the Fermi momentum region where m ∗↓ and Z ↓ present their respective maximum and minimum, the av-erage interparticle spacing n − / (with n = k F / π thedensity of the ↑ neutron free Fermi gas) is of the or-der of the S neutron-neutron scattering length, i.e., n / | a | ∼ . Therefore, we can venture to say that thisregion establishes the border between a less correlatedand a more correlated regime of the system. In fact,note that the values of Z ↓ are in general larger in the k F region from to . fm − that for k F & . fm − ,indicating that in this region correlations are less impor-tant, and that the ↓ neutron impurity propagates morefreely in the ↑ neutron gas. We also notice that for Fermimomenta above ∼ . fm − , the S neutron-neutronscattering length is larger that the average interparticlespacing with values of the dimensionless quantity n / | a | I m pu r it y e ff ec ti v e m a ss m * / m 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. 2: Effective mass (panel a) and quasiparticle residue(panel b) of a ↓ neutron impurity with ~k ↓ = ~ as a functionof the Fermi momentum of the free gas of ↑ neutrons. ranging from at k F = 0 . fm − to . at k F = 0 . fm − . Although in the unitary limit it is extrictly ful-filled the condition n / | a | ≫ , these numbers indicateonce more that low-density neutron matter is close to it,at least for Fermi momenta in the range from ∼ . fm − to ∼ . fm − . Averaging the effective mass and thequasiparticle residue over the Fermi momentum in therange between . fm − and . fm − we find the meanvalues m ∗↓ = 1 . m and Z ↓ = 0 . for the effective massand the quasiparticle residue, respectively. Our resultsfor both quantities compare remarkably well with thoseof the full-many body analysis of Combescot and Giraud[62] who found m ∗↓ = 1 . m , and those of the diagram-matic Monte Carlo method employed by Vlietinck et al., [63] who obtained a value of 0.759 for the quasiparticleresidue. These results confirm once more the Fermi po-laron behavior exhibited by the ↓ neutron impurity in alow-density free Fermi gas of ↓ neutrons.Summarizing, in this work we have analyzed the prop-erties (energy, effective mass and quasiparticle residue)of a ↓ neutron impurity in a low-density free Fermi gas of ↑ neutrons. Calculations have been done within the BHFapproach using chiral two-body nucleon-nucleon forces atN LO including only contributions from the S partialwave. Our results have clearly shown that a ↓ neutronimpurity in a low-density free Fermi gas of ↑ neutronspresents properties very close to those of an attractiveFermi polaron in the unitary limit.The author is very grateful to Artur Polls for themany interesting and stimulating conversations they hadon this topic who, unfortunately, could not see thiswork finished. The author is also very grateful to Mar-cello Baldo, Silvia Chiacchiera and Albert Feijoo fortheir comments and careful reading of the manuscript.This work has been supported by the COST ActionCA16214, “PHAROS: The multi-messenger physics andastrophysics of compact stars”. [1] L. D. Landau, Phys. Z. Sowjetunion , 664 (1933).[2] S. I. Pekar, Zh. Eksp. Teor. Fiz. , 335 (1946).[3] H. Fröhlich, Adv. Phys. , 325 (1954).[4] R. P. Feynman, Phys. Rev. , 660 (1955).[5] I. N. 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