Phase transition to the state with nonzero average helicity in dense neutron matter
aa r X i v : . [ nu c l - t h ] D ec Phase transition to the state with nonzero average helicity in dense neutron matter
A. A. Isayev ∗ Kharkov Institute of Physics and Technology, Academicheskaya Street 1, Kharkov, 61108, UkraineKharkov National University, Svobody Sq., 4, Kharkov, 61077, Ukraine
J. Yang † Department of Physics and the Institute for the Early Universe,Ewha Womans University, Seoul 120-750, Korea
The possibility of the appearance of the states with a nonzero average helicity in neutron matteris studied in the model with the Skyrme effective interaction. By providing the analysis of the self-consistent equations at zero temperature, it is shown that neutron matter with the Skyrme BSk18effective force undergoes at high densities a phase transition to the state in which the degeneracywith respect to helicity of neutrons is spontaneously removed.
PACS numbers: 21.65.Cd, 21.10.Hw, 26.60.-c, 21.30.Fe
The issue of spontaneous appearance of spin polarizedstates in nuclear matter is a topic of a great current in-terest due to relevance in astrophysics. In particular, thescenarios of supernova explosion and cooling of neutronstars are essentially different, depending on whether nu-clear matter is spin polarized or not. On the one hand,the models with the Skyrme effective nucleon-nucleon(NN) interaction predict the occurrence of spontaneousspin instability in nuclear matter at densities in the rangefrom ̺ to 4 ̺ for different parametrizations of the NNpotential [1]-[9] ( ̺ ≃ .
16 fm − is the nuclear saturationdensity). On the other hand, for the models with therealistic NN interaction, no sign of spontaneous spin in-stability has been found so far at any isospin asymmetryup to densities well above ̺ [10]-[16]. In order to recon-cile two different approaches, based on the effective andrealistic NN interactions, recently a new parametriza-tion of the Skyrme interaction, BSk18, has been pro-posed [17], aimed to avoid the spin instability of nuclearmatter at densities beyond the nuclear saturation den-sity. This is achieved by adding new density-dependentterms to the standard Skyrme force. The advantage ofthe BSk18 parametrization is that it also preserves thehigh-quality fits to the mass data obtained with the con-ventional Skyrme force as well as it satisfactorily repro-duces the results of microscopic neutron matter calcula-tions (equation of state [18], S pairing gap [19]). Hence,this Skyrme parametrization has a good potentiality inthe studies of various neutron star properties [20].In terms of Landau Fermi liquid parameters, the fer-romagnetic instability of neutron matter is preventedby the requirement G > −
1, where G is the zerothcoefficient in the expansion of the dimensionless spin-spin interaction amplitude on the Legendre polynomi-als, G = P l G l P l (cos θ ) [21]. Although this conditioncan hold true for all relevant densities, nevertheless, the ∗ [email protected] † [email protected] first coefficient G , with increasing density, can becomelarge and negative, so that the condition G < − h σp i 6 = 0, is formed. Such a possibility was firststudied with respect to an electron liquid in metals inRef. [22] and, later, in the framework of a microscopicmodel, in Ref. [23]. Our primary goal here is to developthe proper formalism for the description of the stateswith a nonzero average helicity in neutron matter and toprovide the corresponding analysis of the self-consistentequations for the BSk18 Skyrme force.The nonsuperfluid states of neutron matter are de-scribed by the normal distribution function of neutrons f κ κ = Tr ̺a + κ a κ , where κ ≡ ( p , σ ), p is momentum, σ is the projection of spin on the third axis, and ̺ is thedensity matrix of the system [8, 9]. The self-consistentmatrix equation for determining the distribution function f follows from the minimum condition of the thermody-namic potential [24, 25] and is f = { exp( Y ε + Y ) + 1 } − ≡ { exp( Y ξ ) + 1 } − . (1)Here the single particle energy ε and quantity Y arethe matrices in the space of κ variables, with Y κ κ = Y δ κ κ , Y = 1 /T , and Y = − µ /T being the Lagrangemultipliers, µ being the chemical potential of neutrons,and T being the temperature.For the BSk18 interaction, the ferromagnetic instabil-ity is avoided by adding to the standard Skyrme forcethe additional density-dependent terms. Nevertheless,although spontaneous spin polarization is excluded at alldensities relevant for neutron stars, there is still the possi-bility, related to the spontaneous appearance of the statewith a nonzero average helicity λ ≡ h σ p i , p = p /p, (2)where h . . . i ≡ tr f.../ tr f , tr ... being the trace in the spaceof κ variables. For this state, the single particle energyof neutrons reads ε ( p ) = ε ( p ) σ − ∆( p ) σp , (3)where σ i are the Pauli matrices in the spin space and2∆( p ) is the energy splitting between the neutron spectrawith different helicities (positive and negative). It followsfrom Eqs. (1) and (3), that the distribution function ofneutrons has the structure f ( p ) = f ( p ) σ + f || ( p ) σp , (4)where f = 12 { n ( ω + ) + n ( ω − ) } , f || = 12 { n ( ω + ) − n ( ω − ) } . (5)Here n ( ω ) = { exp( Y ω ) + 1 } − and ω ± = ε − µ ∓ ∆ . (6)The quantity ω ± , being the exponent in the Fermi dis-tribution function n , entering Eqs. (5), plays the roleof the quasiparticle spectrum. There are two branchesof the quasiparticle spectrum, corresponding to neutronswith definite helicity, σp = ± f should satisfythe normalization condition2 V X p f ( p ) = ̺, (7)where ̺ is the total density of neutron matter. The av-erage helicity λ plays the role of the order parameter ofa phase transition to the state, in which the majority ofneutron spins are oriented along, or opposite to their mo-menta. By calculating the traces in Eq. (2), it is easy tofind that λ = P p f || ( p ) P p f ( p ) . (8)In order to get the self–consistent equations for thecomponents of the single particle energy, one has to setthe energy functional of the system, which reads [9, 24] E ( f ) = E ( f ) + E int ( f ) , (9) E ( f ) = 2 X p ε ( p ) f ( p ) , ε ( p ) = p m ,E int ( f ) = X p { ˜ ε ( p ) f ( p ) + ˜ ε i ( p ) f i ( p ) } , where˜ ε ( p ) = 12 V X q U n ( k ) f ( q ) , k = p − q , (10)˜ ε i ( p ) = 12 V X q U n ( k ) f i ( q ) , f i ( q ) = f || ( q ) q i . (11) Here ε ( p ) is the free single particle spectrum, m isthe bare mass of a neutron, U n ( k ) , U n ( k ) are the nor-mal Fermi liquid (FL) amplitudes, and ˜ ε , ˜ ε i are the FLcorrections to the free single particle spectrum. Usingequation δE = tr ε ( f ) δf [21], we get the self-consistentequations in the form ξ ( p ) = ε ( p ) + ˜ ε ( p ) − µ , (12) ξ i ( p ) ≡ − ∆( p ) p i = ˜ ε i ( p ) . (13)To obtain numerical results, we utilize the BSk18parametrization of the Skyrme interaction, developedin Ref. [17] and generalizing the conventional Skyrmeparametrizations. In the conventional case, the ampli-tude of NN interaction reads [26]ˆ v ( p , q ) = t (1 + x P σ ) + 16 t (1 + x P σ ) ̺ α (14)+ 12 ~ t (1 + x P σ )( p + q ) + t ~ (1 + x P σ ) pq , where P σ = (1 + σ σ ) / t i , x i and α are some phenomenological parameters spec-ifying a given parametrization of the Skyrme interaction.The Skyrme interaction suggested in Ref. [17] has theformˆ v ′ ( p , q ) = ˆ v ( p , q ) + ̺ β ~ t (1 + x P σ )( p + q ) (15)+ ̺ γ ~ t (1 + x P σ ) pq . In Eq. (15), two additional terms are the density-dependent generalizations of the t and t terms of theusual form.The normal FL amplitudes U , U can be expressed interms of the Skyrme force parameters. For conventionalSkyrme force parametrizations, their explicit expressionsare given in Refs. [24, 25]. As follows from Eqs. (14) and(15), in order to obtain the corresponding expressions forthe generalized Skyrme interaction (15), one should usethe substitutions t → t + t ̺ β , t x → t x + t x ̺ β , (16) t → t + t ̺ γ , t x → t x + t x ̺ γ . (17)Therefore, the FL amplitudes are related to the pa-rameters of the Skyrme interaction (15) by formulas U n ( k ) = 2 t (1 − x ) + t ̺ α (1 − x ) + 2 ~ [ t (1 − x )(18)+ t (1 − x ) ̺ β + 3 t (1 + x ) + 3 t (1 + x ) ̺ γ ] k ,U n ( k ) = − t (1 − x ) − t ̺ α (1 − x ) + 2 ~ [ t (1 + x )(19)+ t (1 + x ) ̺ γ − t (1 − x ) − t (1 − x ) ̺ β ] k ≡ a n + b n k . It follows from Eqs. (12) and (18) that ξ = p m n − µ, (20)where the effective neutron mass m n is defined by theformula ~ m n = ~ m + ̺ t (1 − x ) + t (1 − x ) ̺ β (21)+ 3 t (1 + x ) + 3 t (1 + x ) ̺ γ ] , and the renormalized chemical potential µ should be de-termined from Eq. (7).Taking into account the explicit form of the FL ampli-tude U in Eq. (19), solution of Eq. (13) for the energygap ∆ should be sought in the form∆( p ) = b n νp, (22)where ν is some unknown quantity satisfying the equa-tion ν = Z ∞ q π ~ f || ( q ) d q. (23)This equation can be obtained from Eqs. (11),(13) af-ter passing from summation to integration, V P . . . → R d q (2 π ~ ) . . . , and performing then the angle integration.Thus, with account of Eqs. (5) for the distributionfunctions f , we obtain the self–consistent equations (7)and (23) for the renormalized chemical potential µ andthe unknown ν , determining the splitting ∆ in the en-ergy spectrum (3) of neutrons with different helicities.Note that the energy spectrum (3) is invariant under thetime reversion but not under the parity transformation.Hence, the state with ∆ = 0 is characterized by a spon-taneously broken P -symmetry.Now we present the solutions of the self-consistentequations at zero temperature for BSk18 Skyrmeforce [17]. Note that the self-consistent equations havealways the trivial solution ∆ = 0 (or ν = 0), corre-sponding to the normal neutron matter. Besides, theself-consistent equations are invariant with respect to thechange ∆ → − ∆, and, hence, nontrivial solutions for ∆enter in a pair with the same magnitude and oppositesign. The majority of neutrons will have positive helic-ity, if ∆ >
0, and negative helicity, if ∆ <
0. Aftersolving the self-consistent equations, the average helic-ity λ , playing the role of the order parameter of a phasetransition, can be obtained from Eq. (8). According toEq. (8), both signs of the helicity of the given magni-tude are possible because of the two possible signs of theenergy splitting ∆. Note that in order to preserve therealistic EoS of neutron matter obtained in Ref. [18], thefollowing constraints on the additional parameters of theBSk18 parametrization were set β = γ, t (1 − x ) = − t (1 + x ) . (24) BSk18 ∆ ( p F ) / ε F ρ [fm -3 ] FIG. 1. The energy splitting ∆( p F ) between the neutron spec-tra with different helicities normalized to the neutron Fermienergy as a function of density at zero temperature for BSk18interaction. BSk18 λ ρ [fm -3 ] FIG. 2. Same as in Fig. 1 but for the average helicity ofneutron matter.
Because of these constraints, the t and t terms cancelcompletely in the FL amplitude U and in the effectiveneutron mass m n , and only the FL amplitude U is af-fected by the new terms.Fig. 1 shows the energy splitting ∆( p = ~ k F ) be-tween the branches of the neutron spectra with differ-ent helicities normalized to the Fermi energy ε F = ~ k F m n of the normal neutron matter as a function of den-sity. A spontaneous phase transition to the state with anonzero helicity occurs at the critical density ̺ ≈ . ̺ ( ̺ = 0 . − for BSk18 force). The energy split-ting continuously increases with the density and becomescomparable with the neutron Fermi energy ε F . Note thatonly the branch with the positive energy gap is shown inFig. 1 while the symmetric branch (∆ → − ∆) with thenegative energy gap is not presented there.Fig. 2 shows the average helicity of neutron matter asa function of density obtained with the BSk18 Skyrme BSk18 δ E / V [ M e V /f m ] ρ [fm -3 ] FIG. 3. Same as in Fig. 1 but for the difference between theenergy densities of the state with a nonzero average helicityand the normal state ( λ = 0) of neutron matter. interaction. The average helicity monotonously increasesfrom zero till it is saturated and reaches the value λ = 1at ̺ ≈ . ̺ . Beginning from that density, all neutronspins will be aligned along their momenta (or opposite tothem for the branch with the negative helicity, not shownin Fig. 2).In order to clarify whether the state with a nonzeroaverage helicity is thermodynamically preferable over thenormal state of neutron matter, we should compare thecorresponding energies (at zero temperature). Fig. 3shows the difference between the energy densities of thestate with a nonzero average helicity and the normal stateof neutron matter. It is seen that for all densities where nontrivial solutions (with ∆ = 0) exist, this difference isnegative and, hence, the state with the majority of neu-tron spins directed along (or opposite) to their momentais preferable at that density range.In summary, we have considered the states with a spon-taneous nonzero average helicity in neutron matter withthe BSk18 Skyrme NN interaction, which are character-ized by broken parity. The self-consistent equations forthe parameter, determining the energy splitting betweenthe neutron spectra with different helicities, and the ef-fective chemical potential of neutrons have been obtainedand analyzed at zero temperature. It has been shownthat the self-consistent equations have solutions corre-sponding to a nonzero average helicity beginning fromthe critical density ̺ ≈ . ̺ . Under increasing density,the magnitude of the average helicity increases and is sat-urated at the density ̺ ≈ . ̺ , when all neutron spinsare aligned along ( λ = 1), or opposite ( λ = −
1) to theirmomenta. The comparison of the respective energies atzero temperature shows that the state with a nonzero av-erage helicity is preferable over the normal state at thedensities beyond the critical one. The possible existenceof the state with a nonzero average helicity in the densecore of a neutron star will affect the neutrino opacities,and, hence, may be of importance for the adequate de-scription of the thermal evolution of pulsars. In this re-spect, it is interesting to study also the impact of a strongmagnetic field, characteristic for pulsars (magnetars), onthe average helicity of dense neutron matter [27].J. Y. was supported by grant 2010-0011378 from BasicScience Research Program through NRF of Korea fundedby MEST and by grant R32-2009-000-10130-0 from WCUproject of MEST and NRF through Ewha Womans Uni-versity. [1] M.J. Rice, Phys. Lett.
A29 , 637 (1969).[2] S.D. Silverstein, Phys. Rev. Lett. , 139 (1969).[3] E. Østgaard, Nucl. Phys. A154 , 202 (1970).[4] A. Viduarre, J. Navarro, and J. Bernabeu, Astron. As-trophys. , 361 (1984).[5] S. Reddy, M. Prakash, J.M. Lattimer, and J.A. Pons,Phys. Rev. C , 2888 (1999).[6] A.I. Akhiezer, N.V. Laskin, and S.V. Peletminsky, Phys.Lett. B383 , 444 (1996); JETP , 1066 (1996).[7] M. Kutschera, and W. Wojcik, Phys. Lett. , 271(1994).[8] A.A. Isayev, JETP Letters , 251 (2003).[9] A.A. Isayev, and J. Yang, Phys. Rev. C , 025801(2004); A.A. Isayev, Phys. Rev. C , 057301 (2006).[10] V.R. Pandharipande, V.K. Garde, and J.K. Srivastava,Phys. Lett. B38 , 485 (1972).[11] S.O. B¨ackmann and C.G. K¨allman, Phys. Lett.
B43 , 263(1973).[12] P. Haensel, Phys. Rev. C , 1822 (1975).[13] I. Vida˜na, A. Polls, and A. Ramos, Phys. Rev. C ,035804 (2002).[14] S. Fantoni, A. Sarsa, and E. Schmidt, Phys. Rev. Lett. , 181101 (2001).[15] F. Sammarruca, and P. G. Krastev, Phys. Rev. C ,034315 (2007).[16] G.H. Bordbar, and M. Bigdeli, Phys. Rev. C , 045804(2007).[17] N. Chamel, S. Goriely, and J. M. Pearson, Phys. Rev. C , 065804 (2009).[18] B. Friedman and V. R. Pandharipande, Nucl. Phys. A361 , 502 (1981).[19] L. G. Cao, U. Lombardo, and P. Schuck, Phys. Rev. C , 064301 (2006).[20] J. Rikovska Stone, J. C. Miller, R. Koncewicz, P. D.Stevenson, and M. R. Strayer, Phys. Rev. C , 034324(2003).[21] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part2 (Pergamon, Oxford, 1980).[22] I. A. Akhiezer, and E. M. Chudnovskii, Phys. Lett. ,433 (1978).[23] E. M. Chudnovsky, and A. Vilenkin, Phys. Rev. B ,4301 (1982).[24] A. I. Akhiezer, A. A. Isayev, S. V. Peletminsky, A. P.Rekalo, and A. A. Yatsenko, JETP , 1 (1997). [25] A.A. Isayev, and J. Yang, in Progress in FerromagnetismResearch , edited by V.N. Murray (Nova Science Publish-ers, New York, 2006), p. 325.[26] D. Vautherin and D. M. Brink, Phys. Rev. C , 626 (1972).[27] A.A. Isayev, and J. Yang, Phys. Rev. C80