Entrainment effects in neutron-proton mixtures within the nuclear-energy density functional theory. I. Low-temperature limit
aa r X i v : . [ nu c l - t h ] F e b Entrainment effects in neutron-proton mixtures within thenuclear-energy density functional theory. I. Low-temperaturelimit.
N. Chamel and V. Allard Institute of Astronomy and Astrophysics,Universit´e Libre de Bruxelles, CP 226,Boulevard du Triomphe, B-1050 Brussels, Belgium (Dated: February 5, 2021)
Abstract
Mutual entrainment effects in cold neutron-proton mixtures are studied in the framework ofthe self-consistent nuclear energy-density functional theory. Exact expressions for the mass cur-rents, valid for both homogeneous and inhomogeneous systems, are directly derived from the time-dependent Hartree-Fock equations with no further approximation. The equivalence with the Fermi-liquid expression is also demonstrated. Focusing on neutron-star cores, a convenient and simpleanalytical formulation of the entrainment matrix in terms of the isovector effective mass is found,thus allowing to relate entrainment phenomena in neutron stars to isovector giant dipole reso-nances in finite nuclei. Results obtained with different functionals are presented. These includethe Brussels-Montreal functionals, for which unified equations of state of neutron stars have beenrecently calculated. . INTRODUCTION Neutron stars are unique celestial bodies in that their core is expected to contain neu-tron and proton superfluids, the former permeating also the inner part of the crust [1–7](see, e.g., Refs. [8–11] for reviews). Predicted before the actual discovery of these compactstars [12], nuclear superfluidity has found strong support from observations of pulsar fre-quency glitches [13, 14], and more recently from the rapid cooling of the young neutron starin Cassiopeia A supernova remnant [15–17] (but see also Ref. [18]). Superfluidity in neutronstars may leave its imprints on other astrophysical phenomena (see, e.g., Refs. [19, 20]).Although superfluid neutrons and protons in a cold mature neutron star can flow withdifferent velocities, their dynamics are not completely independent from each other. Despitethe absence of viscous drag, the neutron superfluid in the crust does not flow freely due toscattering by inhomogeneities. The neutron superfluid is thus effectively entrained by thecrust (see, e.g. Ref. [21] for a recent review). Likewise, neutrons and protons in the core aremutually coupled by nondissipative entrainment effects of the kind originally discussed byAndreev and Bashkin in the context of superfluid He- He mixtures [22]: the mass current ρ q ρ q ρ q of one nucleon species ( q = n, p for neutron, proton respectively) is found to depend onthe superfluid velocities V q V q V q of both species, i.e. ρ q ρ q ρ q = X q ′ ρ qq ′ V q ′ V q ′ V q ′ . (1)These effects may have important consequences for the global dynamics of a neutron star.For instance, electron scattering off the magnetic field induced by the circulation of entrainedprotons around individual neutron superfluid vortices leads to a very strong frictional cou-pling between the neutron superfluid in the core and the electrically charged particles [23].The (symmetric) entrainment matrix ρ qq ′ in neutron-proton mixtures has been previouslycalculated in the framework of the Fermi liquid theory [24–30]. An alternative approachbased on relativistic mean-field models has been followed in Refs. [31–33].In this paper, entrainment effects are studied within the self-consistent nuclear energy-density functional theory. In Section II, we derive the microscopic expressions for the neutronand proton mass currents in the framework of the time-dependent Hartree-Fock (TDHF)method (see, e.g. Refs. [34, 35] for recent reviews). Applications to neutron-star coresare discussed in Section III, where the entrainment matrix is calculated. The equivalence2ith the Fermi-liquid expression obtained earlier is explicitly demonstrated. Numericalresults are presented for extended Skyrme functionals, for which unified equations of stateof neutron stars have been recently calculated [36, 37]. Other functionals are also consideredfor comparison. II. MICROSCOPIC EXPRESSIONS OF THE MASS CURRENTS
In the following, we will consider cold neutron-proton mixtures at temperatures T muchlower than the critical temperatures of nuclear superfluidity. We shall further suppose thatcurrents are small compared to the critical currents for the breakdown of nuclear superflu-idity. With these assumptions, the influence of nuclear pairing on the entrainment matrixcan be safely ignored (see e.g. Ref. [29]). A. Time-dependent Hartree-Fock equations
The total energy E of a nucleon-matter element of volume V is supposed to be a functionalof the following local densities and currents:(i) the nucleon number density at position rrr and time tn q ( rrr, t ) = X σ = ± n q ( rrr, σ ; rrr, σ ; t ) , (2)(ii) the kinetic density at position rrr and time tτ q ( rrr, t ) = X σ = ± Z d r ′ r ′ r ′ δ ( rrr − r ′ r ′ r ′ ) ∇∇∇ · ∇ ′ ∇ ′ ∇ ′ n q ( rrr, σ ; r ′ r ′ r ′ , σ ; t ) , (3)(iii) and the momentum density (in units of ¯ h ) at position rrr and time tj q j q j q ( rrr, t ) = − i2 X σ = ± Z d r ′ r ′ r ′ δ ( rrr − r ′ r ′ r ′ )( ∇∇∇ − ∇ ′ ∇ ′ ∇ ′ ) n q ( rrr, σ ; r ′ r ′ r ′ , σ ; t ) , (4)where n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) is the density matrix in coordinate space (denoting the spin states by σ, σ ′ ; allowed values are 1 , − i , j , etc.) as [39]i¯ h ∂n ijq ∂t = X k ( h ikq n kjq − n ikq h kjq ) , (5)where the (Hermitian) Hamiltonian matrix h ijq is defined by h ijq = ∂E∂n jiq = ( h jiq ) ∗ (6)(the symbol ∗ denoting complex conjugation).As shown in Appendix A, the TDHF equations can be equivalently expressed in coordi-nate space asi¯ h ∂n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) ∂t = h q ( rrr, t ) n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) − h q ( r ′ r ′ r ′ , t ) ∗ n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) . (7)in which the single-particle Hamiltonian h q is given by h q ( rrr, t ) = −∇∇∇ · ¯ h m ⊕ q ( rrr, t ) ∇∇∇ + U q ( rrr, t ) − i2 (cid:20) I q I q I q ( rrr, t ) · ∇∇∇ + ∇∇∇ · I q I q I q ( rrr, t ) (cid:21) (8)with the various fields defined by the functional derivatives of the energy¯ h m ⊕ q ( rrr, t ) = δEδτ q ( rrr, t ) , U q ( rrr, t ) = δEδn q ( rrr, t ) , I q I q I q ( rrr, t ) = δEδj q j q j q ( rrr, t ) . (9) B. Mass currents, velocities and momenta
Due to neutron-proton interactions, the nucleon mass current ρ q ρ q ρ q is not simply given bythe momentum density ¯ hj q j q j q . The mass current can be rigorously calculated from the TDHFmatrix equations (7), whose diagonal part can be rearranged in the form of continuityequations for nucleons of type q after summing over spins following the seminal work ofRef. [40] ∂ρ q ( rrr, t ) ∂t + ∇∇∇ · ρ q ρ q ρ q ( rrr, t ) = 0 . (10)Using the Hamiltonian (8) and the definitions (2)-(4), we thus find ρ q ρ q ρ q ( rrr, t ) = mm ⊕ q ( rrr, t ) ¯ hj q j q j q ( rrr, t ) + ρ q ( rrr, t ) I q I q I q ( rrr, t )¯ h , (11)where m denotes the nucleon mass, ignoring the small difference between the neutron andproton masses. 4he energy E of a nucleon matter element can be decomposed into a kinetic term E kin = Z d rrr ¯ h m τ ( rrr, t ) , (12)where τ = τ n + τ p , a Coulomb term E Coul and a nuclear term E nuc , i.e. E = E kin + E Coul + E nuc . (13)Assuming nuclear isospin symmetry, E nuc remains unaffected if neutron and proton densitiesand currents are interchanged. It is convenient to introduce an isospin index equal to 0 forisoscalar quantities and 1 for isovector quantities. The former (also written without anysubscript) are sums over neutrons and protons (e.g. n ≡ n = n n + n p ) while the latter aredifferences between neutrons and protons (e.g. n = n n − n p ). Due to Galilean invariance, thenuclear-energy terms contributing to the mass currents, denoted by E j nuc , can only depend onthe combinations X ( rrr, t ) = n ( rrr, t ) τ ( rrr, t ) − j ( rrr, t ) and X ( rrr, t ) = n ( rrr, t ) τ ( rrr, t ) − j ( rrr, t ) ,as shown, e.g., Ref. [41]. Therefore, the functional derivatives of E j nuc with respect to τ q ( rrr, t )and j q ( rrr, t ) can be written as δE j nuc δτ q ( rrr, t ) = ¯ h m ⊕ q ( rrr, t ) − ¯ h m = " δE j nuc δX ( rrr, t ) − δE j nuc δX ( rrr, t ) n + 2 n q δE j nuc δX ( rrr, t ) , (14) δE j nuc δj q j q j q ( rrr, t ) = I q I q I q ( rrr, t ) = − jjj " δE j nuc δX ( rrr, t ) − δE j nuc δX ( rrr, t ) − j q j q j q δE j nuc δX ( rrr, t ) . (15)Using Eqs. (14) and (15), the mass current (11) can be expressed in terms of the momentumdensities only as ρ q ρ q ρ q ( rrr, t ) = ¯ hj q j q j q ( rrr, t ) ( h " δE j nuc δX ( rrr, t ) − δE j nuc δX ( rrr, t ) ρ ( rrr, t ) ) − ¯ hjjj ( rrr, t ) 2¯ h " δE j nuc δX ( rrr, t ) − δE j nuc δX ( rrr, t ) ρ q ( rrr, t ) . (16)While neutron and proton mass currents are not separately aligned with their correspond-ing momenta, it can be easily seen that the total mass current coincides with the totalmomentum density ρρρ ( rrr, t ) = ρ n ρ n ρ n ( rrr, t ) + ρ p ρ p ρ p ( rrr, t ) = ¯ hjjj ( rrr, t ) . (17)5he mean mass current ρ q ρ q ρ q ( t ) in the volume V is obtained by integrating the correspondinglocal current ρ q ρ q ρ q ( rrr, t ). Decomposing the density matrix in a single-particle basis (A2) usingEqs. (2), (4), and (11), the mean mass current can thus be written as ρ q ρ q ρ q ( t ) = 1 V Z d rrr ρ q ρ q ρ q ( rrr, t ) = mV X i,j n ijq v ( q ) ji v ( q ) ji v ( q ) ji , (18)where v ( q ) ji v ( q ) ji v ( q ) ji = X σ Z d rrr ϕ ( q ) j ( rrr, σ ) ∗ v ( q ) v ( q ) v ( q ) ( rrr, t ) ϕ ( q ) i ( rrr, σ ) (19)are the matrix elements of the velocity operator v ( q ) v ( q ) v ( q ) ( rrr, t ) = − i¯ h (cid:20) m ⊕ q ( rrr, t ) ∇∇∇ + ∇∇∇ m ⊕ q ( rrr, t ) (cid:21) + 1¯ hI q I q I q ( rrr, t ) . (20)That v ( q ) v ( q ) v ( q ) ( rrr, t ) is a velocity operator is confirmed by the application of the Ehrenfest theorem(see, e.g. Ref. [42]) v ( q ) ji v ( q ) ji v ( q ) ji = X σ Z d rrr ϕ ( q ) j ( rrr, σ ) ∗ h (cid:20) rrrh q ( rrr, t ) − h q ( rrr, t ) rrr (cid:21) ϕ ( q ) i ( rrr, σ ) . (21)In the canonical basis for which the density matrix is diagonal, i.e. n ijq = e n ( q ) i δ ij where e n ( q ) i represents the occupation number of the single-particle state i ( δ ij being the Kroneckersymbol), the mean mass current takes a particularly simple form ρ q ρ q ρ q = mV X i e n ( q ) i v ( q ) i v ( q ) i v ( q ) i , (22)with v ( q ) i v ( q ) i v ( q ) i ≡ v ( q ) ii v ( q ) ii v ( q ) ii denoting the mean velocity of the state i .The equations derived so far for the mass currents are very general since we only madeuse of the TDHF equations (7) with no further approximation. In particular, Eqs. (11), (16)and (22) are applicable to both homogeneous and inhomogeneous systems such as the coreand the crust of a neutron star respectively. C. Relation to the Fermi liquid theory
In systems that have some translational symmetry (but not necessarily homogeneous),any single-particle state can be labelled by a wave vector kkk . Assuming further that the systemis stationary, the TDHF equation (5) shows that the Hamiltonian and density matrices6ommute, and therefore share the same eigenstates. In other words, the single-particleHamiltonian is diagonal in the canonical basis h q ( rrr ) ϕ ( q ) kkk ( rrr, σ ) = ε ( q ) kkk ϕ ( q ) kkk ( rrr, σ ) . (23)As shown in Appendix C, the mean velocity v ( q ) k v ( q ) k v ( q ) k of a state kkk can be expressed as v ( q ) k v ( q ) k v ( q ) k = 1¯ h ∇ k ∇ k ∇ k ε ( q ) kkk . (24)The mean current is thus given by the familiar expression ρ q ρ q ρ q = mV X kkk e n ( q ) kkk v ( q ) k v ( q ) k v ( q ) k . (25)This demonstrates the equivalence between the definition of the mass currents in the Fermiliquid theory, namely Eqs. (24) and (25), and the expression (18) derived from the TDHFequations (7). III. ENTRAINMENT EFFECTS IN NEUTRON-STAR CORES
We focus here on homogeneous nucleon matter with stationary currents. All fields aretherefore spatially uniform and independent of time.
A. Andreev-Bashkin matrix in the Fermi liquid theory
The entrainment matrix was previously calculated in the framework of the Fermi liquidtheory by considering small perturbations of the static ground-state configuration [24]. Inthe presence of currents, the neutron and proton Fermi surfaces are shifted by a vector Q n Q n Q n and Q p Q p Q p respectively, which are related to the “superfluid velocities” by V q V q V q = ¯ hQ q Q q Q q m . (26)To first order in Q q /k ( q )F , where k ( q )F = (3 π n q ) / denotes the Fermi wave number, theinduced mass current, ρ q ρ q ρ q ≈ δρ q ρ q ρ q = mV X kkk ( δ e n ( q ) kkk v ( q ) k v ( q ) k v ( q ) k + e n ( q ) kkk δv ( q ) k v ( q ) k v ( q ) k ) , (27)7an be written in the form of Eq. (1) with the entrainment matrix [24] ρ qq ′ = √ ρ q ρ q ′ m q m ⊕ q m ⊕ q ′ δ qq ′ + F qq ′ ! , (28)where m ⊕ q is the (Landau) effective mass and F qq ′ are dimensionless ℓ = 1 Landau parame-ters. B. Andreev-Bashkin matrix in the TDHF theory
As we will now show the entrainment matrix can be calculated exactly in the TDHFtheory. Introducing the “superfluid velocity” V q V q V q = ¯ hρ q j q j q j q , (29)and using Eq. (16), the entrainment matrix is found to be given by ρ nn = ρ n " h (cid:18) δE j nuc δX − δE j nuc δX (cid:19) ρ p ρ pp = ρ p " h (cid:18) δE j nuc δX − δE j nuc δX (cid:19) ρ n ρ np = ρ pn = − ρ n ρ p h (cid:18) δE j nuc δX − δE j nuc δX (cid:19) . (30)Let us stress that the functional derivatives of E j nuc may generally depend on the nucleondensities and currents unless E j nuc is a linear combination of X and X or the functionalderivatives of E j nuc with respect to X and X cancel exactly. Unlike the Fermi-liquid ex-pression (28), the mass currents obtained from the TDHF expression (30) may thus dependnonlinearly on the superfluid velocities.The Fermi-liquid expression (28) is recovered by evaluating the functional derivativesof E j nuc with respect to X and X in the static configuration, i.e. by setting j q j q j q = 000 afterderivation. To verify that Eq. (30) reduces to (28), we need to calculate the Landau effectivemass and the ℓ = 1 Landau parameters in the TDHF theory. It follows immediately fromEq. (C9) that the Landau effective mass defined as (the subscript ’0’ indicating that thederivative is evaluated in the absence of currents with kkk lying on the corresponding Fermisurface) 1 m ⊕ q = 1¯ h k ( q )F dε ( q ) kkk dk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (31)8oincides with the effective mass appearing in the TDHF theory. We have thus used thesame symbol. The Landau parameters are obtained from the spin-averaged quasiparticleinteraction defined by f qq ′ ( kkk, kkk ′ ) = δ Eδ e n ( q ) kkk δ e n ( q ′ ) k ′ k ′ k ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = δε ( q ) kkk δ e n ( q ′ ) k ′ k ′ k ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (32)The quasiparticle interaction is further expanded into Legendre polynomials f qq ′ ( kkk, kkk ′ ) = X ℓ f qq ′ ℓ P ℓ (cos θ ) (33)where θ is the angle between the wave vectors kkk and kkk ′ lying on the corresponding Fermisurface. The dimensionless Landau parameters F qq ′ appearing in Eq. (28) are defined by F qq ′ = p N q N q ′ f qq ′ ℓ , (34)in which N q is the density of quasiparticle states at the Fermi surface, N q = m ⊕ q k ( q )F ¯ h π . (35)In the TDHF theory for homogeneous matter (see Appendix C), the quasiparticle energiesare given by Eq. (C9). From the general definition (32), it follows that only the term kkk · I q I q I q contributes to the ℓ = 1 Landau parameters. Using Eq. (15) and remarking from Eq. (4)that the momentum density (in the canonical basis) reduces to j q j q j q = X kkk kkk e n ( q ) kkk , (36)the term kkk · I q I q I q can be explicitly written as kkk · I q I q I q = − X k ′ k ′ k ′ kkk · k ′ k ′ k ′ e n ( q ) k ′ k ′ k ′ " δE j nuc δX + δE j nuc δX − X k ′ k ′ k ′ kkk · k ′ k ′ k ′ e n ( q ′ ) k ′ k ′ k ′ " δE j nuc δX − δE j nuc δX . (37)The ℓ = 1 Landau parameters can be readily obtained by taking the derivatives of the aboveexpression with respect to e n ( q ) k ′ k ′ k ′ and e n ( q ′ ) k ′ k ′ k ′ : f qq = − " δE j nuc δX (cid:12)(cid:12)(cid:12)(cid:12) + δE j nuc δX (cid:12)(cid:12)(cid:12)(cid:12) ( k ( q )F ) , (38) f qq ′ = − " δE j nuc δX (cid:12)(cid:12)(cid:12)(cid:12) − δE j nuc δX (cid:12)(cid:12)(cid:12)(cid:12) k ( q )F k ( q ′ )F . (39)Inserting the corresponding dimensionless parameters in Eq. (28) leads to an expressionsimilar to Eq. (30) except that the derivatives are now evaluated for vanishing currents.9 . Entrainment and isovector effective mass Due to Galilean invariance, as embedded in Eq. (17), it can be easily seen from Eq. (30)that the entrainment matrix elements are not all independent but are related by the followingidentities ρ nn + ρ np = ρ n , ρ pp + ρ pn = ρ p . (40)This means that entrainment effects can be completely characterized by only one indepen-dent parameter, such as the dimensionless determinant of the entrainment matrixΥ = ρ nn ρ pp − ρ np ρ n ρ p . (41)This parameter directly appears in the perturbed hydrodynamical equations and is thereforeimportant for the study of oscillation modes (see, e.g., Refs. [43–45]). Introducing the asym-metry parameter η = ( n n − n p ) /n , the entrainment matrix elements can thus be equivalentlyexpressed as ρ nn = 12 ρ (1 + η ) − ρ (cid:0) − η (cid:1) (1 − Υ) , (42) ρ pp = 12 ρ (1 − η ) − ρ (cid:0) − η (cid:1) (1 − Υ) , (43) ρ np = 14 ρ (cid:0) − η (cid:1) (1 − Υ) = ρ pn . (44)The deviation of Υ from unity is a measure of the importance of entrainment effects. Thisparameter appears to have a simple physical meaning: it coincides with the inverse of theisovector effective mass defined by mm ⊕ v = (cid:18) mm ⊕ n − n n n p mm ⊕ p (cid:19) (cid:18) − n n n p (cid:19) − . (45)Introducing the isoscalar effective mass mm ⊕ s = 12 (cid:18) mm ⊕ n + mm ⊕ p (cid:19) , (46)the nucleon effective masses can be equivalently written as mm ⊕ q = 2 n q n mm ⊕ s + (cid:18) − n q n (cid:19) mm ⊕ v . (47)10he identity Υ = m/m ⊕ v can be directly demonstrated from Eq. (14) and the definition (45).This identity also holds in the Fermi-liquid theory if the Landau parameters are expressibleas f qq = f ( n, η )( k ( q )F ) (the function f being invariant under the interchange of neutronsand protons). In the TDHF theory, the parameter Υ is explicitly given byΥ = mm ⊕ v = 1 + 2¯ h (cid:18) δE j nuc δX − δE j nuc δX (cid:19) ρ . (48)This result is quite general and is applicable to any nuclear-energy density functional thatdepends on the nucleon densities n q ( rrr, t ), kinetic densities τ q ( rrr, t ) and momentum densities j q j q j q ( rrr, t ). The fact that the determinant Υ of the entrainment matrix is related to the isovec-tor effective mass is not unexpected since both quantities characterize similar phenomena,namely relative motions between neutrons and protons.In principle, the isovector effective mass can be extracted from measurements of isovec-tor giant dipole resonances in finite nuclei (the isovector effective mass being closely relatedto the enhancement factor κ of the energy-weighted sum rule m ). However, the valuesinferred from such analyses are model-dependent (see, e.g. Refs. [46, 47]). Alternatively,the isovector effective mass can be indirectly estimated from functionals fitted to variousnuclear data, as in Ref. [48]. In particular, the fit to essentially all nuclear masses seems tofavor values between m ⊕ v /m ∼ . m ⊕ v /m ∼ . m ⊕ v /m ∼ . − .
9. These values are consistent with those found in microscopic calculations(see, e.g. Ref. [50] for a recent review). Applications to neutron stars require the knowl-edge of the isovector effective mass at densities ranging from about ∼ .
08 fm − (crust-coretransition) up to several times saturation density. The variations of the isovector effectivemass with density as predicted by functionals LNS [51] and Sk χm ∗ [52] are shown in Fig. 1.These two functionals were directly fitted to microscopic calculations based on the extendedBrueckner-Hartree-Fock approach for the former and on chiral effective field theory for thelatter. These results are compared to those obtained using the Brussels-Montreal function-als [49, 53]. These functionals have been employed to calculate a series of equations of stateof dense matter in all regions of a neutron star in a unified and thermodynamically consistentway [36, 37]. These functionals, which were derived from extended Skyrme effective interac-tions containing terms that are both momentum and density dependent (see Appendix B),were precision fitted to all measured masses of nuclei with Z, N ≥ ∼ . − . h ( C τ − C τ ) ρ , (49)where C τ and C τ are constant parameters for standard Skyrme functionals, and are functionsof the density n for the extended Skyrme functionals discussed above [49, 55, 59–61]. Explicitformulas for these coefficients are given in Appendix B.The entrainment matrix calculated from standard Skyrme effective interactions is found tocoincide with that obtained earlier using the Fermi-liquid expression (28) with correspondingLandau parameters F qq ′ and effective masses m ⊕ q [26]. This stems from the fact that themass currents ρ q ρ q ρ q depend linearly on the superfluid velocities V q V q V q (the entrainment matrix isindependent of V q V q V q ). However, this may not be necessarily the case for more complicatednuclear-energy density functionals. In particular, the exact expression (30) will differ fromthe Fermi-liquid approximation whenever the nuclear energy functional contains terms thatare not simply proportional to the fields X and X . Examples of such functionals havebeen proposed in Ref. [62]. 12 n [fm -3 ] BSk19BSk21/BSk24BSk25BSk20/BSk26
FIG. 1. (Color online) Variation of the isovector effective mass m ⊕ v /m with density n in nucleonmatter for the extended Skyrme functionals BSk19, BSk20, BSk21, BSk24, BSk25, and BSk26[49, 53]. The upper and lower black solid lines are results from the LNS [51] and Sk χm ∗ [52]functionals, which were fitted to calculations based on extended Brueckner-Hartree-Fock approachand chiral effective field theory respectively. IV. CONCLUSIONS
We have derived exact expressions for the local nucleon mass currents ρ q ρ q ρ q ( rrr, t ) at anyposition rrr and time t in a cold neutron-proton mixture directly from the TDHF equationswithout any further approximation. We have also shown how to relate the spatially averagedmass currents to the group velocities of single-particle quantum states, demonstrating in thisway the equivalence between TDHF theory and previous analyses based on the Fermi liquidapproximation. Our expressions are very general and are applicable to both homogeneousand inhomogeneous nuclear systems.Focusing on the core of a neutron star, we have shown that the neutron-proton entrain-ment matrix can be conveniently expressed in terms of its dimensionless determinant Υ,13 n [fm -3 ] SLy4/UNEDFeMSL07eMSL08eMSL09
FIG. 2. (Color online) Same as Fig. 1 for the extended Skyrme functionals eSML07, eSML08, andeSML09 [55]. For comparison, predictions from the standard Skyrme functionals SLy4 [56, 57] andUNEDF [58] are also shown. whose deviation from unity measures the importance of entrainment effects. This quantitydepends solely on the nucleon number density n and is found to coincide with the inverse ofthe isovector effective mass. This formulation thus allows to relate entrainment phenomenain neutron stars to isovector giant dipole resonances in finite nuclei. We have calculated theisovector effective mass for various semi-local nuclear energy-density functionals. These in-clude the precision-fitted Brussels-Montreal functionals, for which unified equations of stateof neutron stars have been already calculated [36, 37]. Comparing results to those obtainedfrom microscopic calculations, the functionals BSk24 and BSk25 appear to be particularlywell-suited for dynamical simulations of superfluid neutron stars.14 ppendix A: Coordinate-space formulation of TDHF Following the general definition of the density matrix, n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) = h Ψ( t ) | c q ( r ′ r ′ r ′ , σ ′ ) † c q ( rrr, σ ) | Ψ( t ) i , (A1)where | Ψ( t ) i is the many-nucleon wave function at time t , c q ( rrr, σ ) † and c q ( rrr, σ ) are thecreation and destruction operators for nucleons of charge type q at position rrr with spin σ ,the coordinate-space and discrete-basis representations are related by n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) = X i,j n ijq ( t ) ϕ ( q ) i ( rrr, σ ) ϕ ( q ) j ( r ′ r ′ r ′ , σ ′ ) ∗ (A2) n ijq ( t ) = X σ,σ ′ Z d rrr d r ′ r ′ r ′ n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) ϕ ( q ) i ( rrr, σ ) ∗ ϕ ( q ) j ( r ′ r ′ r ′ , σ ′ ) , (A3)denoting by ϕ ( q ) i ( rrr, σ ) the single-particle basis wavefunctions. Making use of the complete-ness relations X i ϕ ( q ) i ( rrr, σ ) ∗ ϕ ( q ) i ( r ′ r ′ r ′ , σ ′ ) = δ ( rrr − r ′ r ′ r ′ ) δ σσ ′ , (A4)the TDHF equations (5) can thus be alternatively written asi¯ h ∂n q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) ∂t = X σ ′′ Z d r ′′ r ′′ r ′′ (cid:20) h q ( rrr, σ ; r ′′ r ′′ r ′′ , σ ′′ ; t ) n q ( r ′′ r ′′ r ′′ , σ ′′ ; r ′ r ′ r ′ , σ ′ ; t ) − n q ( rrr, σ ; r ′′ r ′′ r ′′ , σ ′′ ; t ) h q ( r ′′ r ′′ r ′′ , σ ′′ ; r ′ r ′ r ′ , σ ′ ; t ) (cid:21) , (A5)with the Hamiltonian matrix defined by h q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) = X i,j h ijq ( t ) ϕ ( q ) i ( rrr, σ ) ϕ ( q ) j ( r ′ r ′ r ′ , σ ′ ) ∗ . (A6)In cases for which the energy E is a functional of local densities and currents, the Hamiltonianmatrix can be calculated as h ijq ( t ) = Z d rrr " δEδn q ( rrr, t ) ∂n q ( rrr, t ) ∂n jiq ( t ) + δEδτ q ( rrr, t ) ∂τ q ( rrr, t ) ∂n jiq ( t ) + δEδj q j q j q ( rrr, t ) ∂j q j q j q ( rrr, t ) ∂n jiq ( t ) . (A7)Using Eqs. (2), (3), (4) and (A2) in (A7), and integrating by parts, the Hamiltonian matrixcan be written in the form h ijq ( t ) = X σ,σ ′ Z d rrr d r ′ r ′ r ′ ϕ ( q ) i ( rrr, σ ) ∗ ϕ ( q ) j ( r ′ r ′ r ′ , σ ′ ) h q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) , (A8)15 q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) = h q ( rrr, t ) δ ( rrr − r ′ r ′ r ′ ) δ σσ ′ , (A9)with the Hamiltonian operator h q ( rrr, t ) defined by Eq. (8). From the Hermiticity property h ijq = ( h jiq ) ∗ , we have h q ( rrr, σ ; r ′ r ′ r ′ , σ ′ ; t ) = h q ( r ′ r ′ r ′ , t ) ∗ δ ( rrr − r ′ r ′ r ′ ) δ σσ ′ . (A10)Note that the order of the factors in Eqs. (A9) and (A10) matters: the Hamiltonian operatesonly on the Dirac distribution. Inserting Eqs. (A9) and (A10) in (A5) leads to Eq. (7). Appendix B: Nuclear energy-density functionals and Skyrme effective interactions
Nuclear-energy density functionals can be obtained from the HF method using extendedSkyrme effective interactions of the form v ( rrr i , rrr j ) = t (1 + x P σ ) δ ( rrr ij ) + 12 t (1 + x P σ ) 1¯ h (cid:2) p ij δ ( rrr ij ) + δ ( rrr ij ) p ij (cid:3) + t (1 + x P σ ) 1¯ h ppp ij · δ ( rrr ij ) ppp ij + 16 t (1 + x P σ ) n ( rrr ) α δ ( rrr ij )+ 12 t (1 + x P σ ) 1¯ h (cid:2) p ij n ( rrr ) β δ ( rrr ij ) + δ ( rrr ij ) n ( rrr ) β p ij (cid:3) + t (1 + x P σ ) 1¯ h ppp ij · n ( rrr ) γ δ ( rrr ij ) ppp ij + i¯ h W (ˆ σ i ˆ σ i ˆ σ i + ˆ σ j ˆ σ j ˆ σ j ) · ppp ij × δ ( rrr ij ) ppp ij , (B1)where rrr ij = rrr i − rrr j , rrr = ( rrr i + rrr j ) / ppp ij = − i¯ h ( ∇∇∇ i − ∇∇∇ j ) / σ i ˆ σ i ˆ σ i andˆ σ j ˆ σ j ˆ σ j are Pauli spin matrices, P σ is the two-body spin-exchange operator, and n ( rrr ) denotes theaverage nucleon number density. The terms proportional to t and t are absent in standardSkyrme functionals. Although the use of effective interactions imposes some restrictionson the form of the functional, it guarantees the cancellation of self-interaction errors [63](nonetheless, the functional may still be contaminated by many-body self-interactions errors,see, e.g. Ref. [64] and references therein). Parameters are usually determined by fittingvarious experimental and theoretical nuclear data.The nuclear energy is expressible as E nuc = R d rrr E Sky ( rrr ). The nuclear terms contributingto the mass currents take a very simple form E j Sky = C τ X + C τ X , (B2)16here the coefficients C τ and C τ are given by [61] C τ ( n ) = 316 t + 14 t (cid:18)
54 + x (cid:19) + 316 t n β + 14 t (cid:18)
54 + x (cid:19) n γ (B3) C τ ( n ) = − t (cid:18)
12 + x (cid:19) + 18 t (cid:18)
12 + x (cid:19) − t (cid:18)
12 + x (cid:19) n β + 18 t (cid:18)
12 + x (cid:19) n γ . (B4)The coefficients C τ and C τ coincide with the functional derivatives of the E j nuc with respectto X and X respectively, i.e. δE j nuc δX = C τ , δE j nuc δX = C τ . (B5). Appendix C: Group velocity in translationally invariant systems
In nuclear systems with some translational symmetry (this includes the crystalline crustand the homogeneous core of a neutron star), the single-particle wave functions are givenby Bloch waves [65] ϕ ( q ) kkk ( rrr, σ ) = 1 √ V exp(i kkk · rrr ) χ ( σ ) X GGG e ϕ ( q ) kkk ( GGG ) exp(i
GGG · rrr ) , (C1)where GGG are reciprocal lattice vectors and χ ( σ ) denotes the Pauli spinor. The HF equations(23) can thus be written as X G ′ G ′ G ′ e h ( q ) kkk ( GGG, G ′ G ′ G ′ ) e ϕ ( q ) kkk ( G ′ G ′ G ′ ) = ε ( q ) kkk e ϕ ( q ) kkk ( GGG ) , (C2) e h ( q ) kkk ( GGG, G ′ G ′ G ′ ) = 1 V Z d rrr e − i( kkk + GGG ) · rrr h q ( rrr ) e i( kkk + G ′ G ′ G ′ ) · rrr . (C3)Making use of the normalization of the wave functions X GGG | e ϕ ( q ) kkk ( GGG ) | = 1 , (C4)the single-particle energy is given by ε ( q ) kkk = X GGG,G ′ G ′ G ′ e ϕ ( q ) kkk ( GGG ) ∗ e h ( q ) kkk ( GGG, G ′ G ′ G ′ ) e ϕ ( q ) kkk ( G ′ G ′ G ′ ) . (C5)17ccording to the Hellmann-Feynman theorem [66], we have1¯ h ∇ k ∇ k ∇ k ε ( q ) kkk = 1¯ h X GGG,G ′ G ′ G ′ e ϕ ( q ) kkk ( GGG ) ∗ (cid:20) ∇ k ∇ k ∇ k e h ( q ) kkk ( GGG, G ′ G ′ G ′ ) (cid:21) e ϕ ( q ) kkk ( G ′ G ′ G ′ ) . (C6)Using Eq. (C3), it can be easily seen that Eq. (C6) coincides with the general definition(21), thus demonstrating 1¯ h ∇ k ∇ k ∇ k ε ( q ) kkk = v ( q ) k v ( q ) k v ( q ) k . (C7)In the limit of homogeneous nucleon matter as in the core of a neutron star, e ϕ ( q ) kkk ( GGG ) = 1for G = 0 and e ϕ ( q ) kkk ( GGG ) = 0 otherwise, i.e. the single-particle wave functions reduce to planewaves ϕ ( q ) kkk ( rrr, σ ) = 1 √ V exp(i kkk · rrr ) χ ( σ ) . (C8)In this case, the single-particle energy and the velocity can be readily calculated. Substitut-ing Eq. (C8) in Eq. (23) yields ε ( q ) kkk = ¯ h k m ⊕ q + U q + kkk · I q I q I q . (C9)Differentiating leads to v ( q ) k v ( q ) k v ( q ) k = 1¯ h ∇ k ∇ k ∇ k ε ( q ) kkk = ¯ hkkkm ⊕ q + I q I q I q ¯ h . (C10) ACKNOWLEDGMENTS
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