Phase separation in low-density neutron matter

0.17 MB·

by  Alexandros Gezerlis

Phase separation in low-density neutron matter
Alexandros Gezerlis1 and Rishi Sharma2
1 Department of Physics, University of Washington, Seattle, Washington, 98195-1560, USA and
2 Theory Group, TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada
(Dated: January 18, 2012)
Low-density neutron matter has been studied extensively for many decades, with a view to bet-
ter understanding the properties of neutron-star crusts and neutron-rich nuclei. Neutron matter
is beyond experimental control, but in the past decade it has become possible to create systems
of fermionic ultracold atomic gases in a regime close to low-density neutron matter. In both these
contextspairingissignificant,makingsimpleperturbativeapproachesimpossibletoapplyandneces-
sitating ab initio microscopic simulations. Atomic experiments have also probed polarized matter.
In this work, we study population-imbalanced neutron matter (possibly relevant to magnetars and
2 to density functionals of nuclei) arriving at the lowest-energy configuration to date. For small to
1 intermediaterelativefractionsthesystemturnsouttobefullynormal,whilebeyondacriticalpolar-
0 ization we find phase coexistence between a partially polarized normal neutron gas and a balanced
2 superfluid gas. As in cold atoms, a homogeneous polarized superfluid is close to stability but not
stable with respect to phase separation. We also study the dependence of the critical polarization
n
on the density.
a
J
PACSnumbers: 21.65.-f,03.75.Ss,05.30.Fk,26.60.-c
7
1

I. INTRODUCTION are paired and the number of up and down species is
]
h equal. Even at densities much smaller than the nuclear
t saturationdensity,aismuchlargerthantheinterparticle
- The phase structure of polarized (or population-
l spacing and obtaining quantitatively reliable results re-
c imbalanced) Fermi gases is very rich. The energetically
quires non-perturbative techniques. Detailed Quantum
u favoredphasedependsonthemassesandrelativepopula-
n Monte Carlo (QMC) simulations have calculated the en-
tionsofthespecies,theinteractionstrength,andwhether
[ ergydensity upto densitiesequaltoorlargerthanthe
the interactions are long ranged. Polarizedgasescan ex- E
nuclear saturation density [12, 13]. At low densities, the
2 ist in a variety of physical systems, including ultracold
s-wave pairing gap obtained peaks at nearly 2 MeV.
v trapped atomic Fermi gases [1], dense hadronic phases
8
near the cores of neutron stars [2–4], and possibly quark Magnetic fields tend to split the up and down Fermi
9
matter [5]. surfacesthusdisruptingthepairingandcreatingapolar-
2
0 Inthis paperwe focus onthe possibility thatneutrons ization. For magneticfields (B) smallerthan1016 1017
−
. in the inner crust of neutron stars are spin-polarized [6]. Gauss,pairingwins (assumingnoeffectcomingfromthe
1
Theoreticalcalculationssuggestthatthisregion,whichis latticeofnuclei),whilelargerfieldscancreateapolarized
1
1 aboutakilometerthick,featuresunboundneutronswhile phase. In BCS theory, the critical field, or the resulting
1 the protons cluster in neutron rich nuclei. While inter- chemical potential split (δµ = gNµNB where gNµN is
: actions with nuclei affect the properties of the unbound the neutron magnetic moment), is called the Clogston-
v
i neutrons [7, 8], in this work we consider an isolated neu- Chandrasekhar point. At this point, the BCS superfluid
X tronfluidandcomputetheconditionsforthecoexistence cancoexistinequilibriumwitha partiallypolarizednor-
r ofanunpolarizedphaseandapolarizedphaseatdensities malphasewhichisessentiallyagasoffreeneutrons. Nu-
a roughly 30 250 times smaller than the nuclear satura- merically δµc = ∆/√2 where ∆ is the gap in the BCS
tion density−. A polarizedphase in the neutron star crust phase. A weakly polarized gas of neutrons confined in a
will have a larger specific heat and therefore its pres- volume forms a mixed phase [14] consisting of the BCS
ence may affect the time scales associatedwith observed and normal phases with volume fractions depending on
crustal transient phenomena [9]. Neutron superfluidity the polarization. Hartree corrections substantially mod-
also affects thermal transport properties [10, 11]. If the ify the Clogston-Chandrasekhar point even for modest
polarizationislargeenoughalleffectsofsuperfluidityare couplings kFa 0.5,wherekF =(3π2n)1/3 istheFermi
| |∼
expected to disappear. wave vector [15].
Atthelowdensitiesweconsider,thedominanttermin At strong coupling (mathematically given by k a >
F
the interaction between spin-up and spin-down neutrons 1) the energies of both the superfluid and the|norm| a∼l
isans-wavepotentialwithascatteringlengtha 18.6 phase need to be computed non-perturbatively. In this
∼−
fm and range r 2.7 fm. In a balanced system this at- paper,weperformthesecalculationsforthenormalphase
e
∼
traction drives the formation of Cooper pairs of spin-up and find the critical polarization at which it can coexist
and spin-down neutrons of opposite momenta resulting withthesuperfluidphase. Wealsocomparetheenergyof
in a neutron superfluid whose properties can be under- aphaseseparatedstate withthe previouslyevaluated[6]
stood qualitatively using BCS theory. All the neutrons homogeneous polarized superfluid phase. In this paper
2
we only consider competition between homogeneous and theexperimentsforbothunpolarized[13,37,38]andpo-
isotropic phases and ignore the possibility of promising larized[39–42]Fermigasesandthisgivesusconfidencein
but still unconfirmed scenarios(like p wave[16] and in- applying the same techniques to the problem with finite
−
homogeneous phases [17]). a and r . More directly, by tuning the scattering length
e
Asmentionedabove,ourmicroscopicab initiocalcula- andtheeffectiverange[43]itmaybepossibletoobtaina
tions may impactthe observedbehaviorofneutron stars system with similar values of k a and k r as the inner
F F e
with extremely largeinternal magnetic fields. It is inter- crust and test our results.
esting to concurrently examine possible connections of
such simulations with terrestrial experiments. Our re-
sults may be useful for constraining Density Functional II. CONSTRUCTION OF THE MIXED PHASE
Theories (DFTs) that are employed to study properties
of heavy nuclei. Equation of state results at densities
Theconditionsforapolarizedphase(P)tobeinphase
closetothenuclearsaturationdensityhavebeenusedfor
equilibrium with an unpolarized superfluid (SF) phase
some time to constrain Skyrme and other other density
are as follows:
functional approaches to heavy nuclei [18]. The poten-
tial significance of such calculations has led to a series
1. The pressures of the two phases should be equal.
of publications on the equation of state of low-density
2. The average chemical potential of the two species
neutron matter over the last few decades [12, 19–29].
in the phases, µ=(µ +µ )/2, should be equal.
The density-dependence of the 1S gap in low-density a b
0
neutron matter has also been used to constrain Skyrme- 3. The chemicalpotentialsplitting, δµ=(µ µ )/2,
a b
Hartree-Fock-Bogoliubovtreatments in their description should be less than the gap ∆ in SF. −
of neutron-rich nuclei [30]. Similarly, an ongoing project
is studying the limit of extreme polarization in neutron We construct the mixed phase between the superfluid
matter [31]. This limit is known in condensed-matter and the partially polarized normal phase (NP). The
physics as the “polaron” problem: one impurity embed- pressurein a polarizedphase depends onthe net density
ded in a sea of fermions. Quantities of interest here in- of fermions (n +n =n) and their relative fraction x=
a b
clude the polaron binding energy and its effective mass, n /n where a (or ) is the majority species.
b a
↑
which can be used to constrain Skyrme and other func- Motivated by the unitary Fermi gas [44, 45], we write
tionals. In this line of study, DFTs can be constrained down the energy density as:
by the equation of state of polarized unpaired neutron
matter that we report here. 3(6π2)2/3
(n ,n )= (n g(x,n))5/3 . (1)
Another connection of our calculations with experi- a b a
E 5 2m
ment may be possible through the field of atomic ultra-
cold Fermi gases. There, the scattering length of the in- For convenience we denote the energy per particle by E
teraction between two hyperfine states near a Feshbach and the corresponding energy for a free gas by E =
FG
resonance can be tuned by changing the magnetic field. 3k2/(10m). In terms of E, g(x,n)=(E/E )3/5.
F FG(↑)
Most experiments have been performed in the region For the unitary gas, the discussion of mixed phases is
where the scattering length is infinite (unitarity limit) simplified greatly since the function g is only a function
andthe effective rangeis negligible [32–36]. QMC calcu- of x. In neutron matter, the interaction potential has
lations performed with the same techniques that we use inherent scales (these can be thought of simplistically as
here in the same limit give results that agree well with a and r ).
e
Tosatisfythe conditions ofphaseequilibrium,we needto calculatethe chemicalpotentialsandthe pressure,which
include the functional dependence of g on both x and n. The chemical potential is:
2/3
32/3π4/3 n g(x,n)2/3 2n ∂g(x,n)+(1 x)∂g(x,n)+g(x,n)
∂ (cid:16)x+1(cid:17) (cid:16)x+1∂n − ∂x (cid:17)
µ= E = . (2)
∂n(cid:12)(cid:12)δn 24/3m
(cid:12)
The chemical potential splitting is:
2/3
32/3π4/3 n g(x,n)2/3(( x 1)∂g(x,n)+g(x,n))
∂ (cid:16)x+1(cid:17) − − ∂x
δµ= E = . (3)
∂δn(cid:12)(cid:12)n 24/3m
(cid:12)
The pressure is simply P = (n,δn)+µn+δµδn, where δn=n n .
a b
−E −
For the unpolarized superfluid (x=1), the function g depends only on the density (δn=0) and we have:
32/3π4/3 n 2/3g (n)2/3 n∂gSF(n)+g (n)
(cid:0)2(cid:1) SF (cid:16) ∂n SF (cid:17)
µ = . (4)
SF 24/3m
3
Toorganizethediscussionitisconvenienttostartfrom III. GREEN’S FUNCTION MONTE CARLO
the completely polarized (x = 0) gas. Here the inter- SIMULATIONS
actions between the only species present are weak and
the ground state is well described by a normal phase. The Hamiltonian for low-density neutron matter is:
(p wavepairing and interactions will modify the energy
of−the fully polarized neutron gas from a free Fermi gas. N ¯h2
AswediscussinSec.III,weincludetheeffectofp−wave H=X(−2m∇2k)+Xv(rij′) . (7)
interactions exactly, but ignore p wave pairing.) k=1 i<j′
−
Nowconsiderthefollowingthoughtexperiment. Keep- where N is the total number of particles. The neutron-
ing the density n constant, change a few a fermions to b neutron interaction is not purely s-wave but still some-
fermions so that x changes. For small x we expect that what simple if one considers the AV4’ formulation: [46]
the phase obtained will be a partially polarized normal
phase. As we increase the fraction of b fermions, it is v4(r)=vc(r)+vσ(r)σ1 σ2, (8)
·
possiblethatsomepolarizedsuperfluidphase(SFP)be-
In the case of S=0 (singlet) pairs this gives:
comes the favoredstate ofmatter. Another possibility is
that at some fractional density xc, it becomes favorable vS(r)=vc(r) 3vσ(r) . (9)
to form a mixed phase between the partially polarized −
normalphase(NP)andtheunpolarizedsuperfluidphase However, it also implies an interaction for S=1 (triplet)
(SF). In the present section, we will focus on the mixed pairs:
phase between NP and SF. In Sec. III we will calculate
the energy for the polarizednormalgas(NP) phase and vP(r)=vc(r)+vσ(r) . (10)
in Sec. IV we will look at the competition between the
Ref. [47] explicitly included such p-wave interactions in
homogeneouspolarizedsuperfluid (SFP)of Ref. [6] and
the same-spinpairs (the contributionof whichwas small
the mixed phase.
even at the highest density considered), and perturba-
Equating the chemical potentials and the pressures tively corrected the S = 1,M = 0 pairs to the correct
S
gives two equations in two variables, xc and nSF. nSF p-wave interaction.
is the total superfluid density and nNP is the total den- In these calculations it is customary to first employ a
sity of the normal phase. For x > xc one can find the standardVariationalMonteCarlosimulation,whichmin-
coexistence curve in (x,n) space. Suppose that a vol- imizes the expectation value of the Hamiltonian given a
umefractionv isoccupiedbytheNP phaseandthe rest variationalwavefunctionΨ . Atasecondstage,theout-
V
(1 v) by the superfluid phase. Then: putofthe VariationalMonteCarlocalculationisusedas
−
inputinafixed-nodeGreen’sFunctionMonteCarlosim-
ulation, which projects out the lowest-energy eigenstate
n(v)=vn +(1 v)n (5)
NP SF Ψ fromthetrial(variational)wavefunctionΨ . Thisis
− 0 V
accomplished by treating the Schr¨odinger equation as a
diffusion equation in imaginary time τ and evolving the
variationalwavefunctionuptolargeτ. Thegroundstate
is evaluated from:
( xc n +(1 v)nSF)
x(v)= (1+1xcnNP +(1−v)nS2F) , (6) Ψ0 = exp[−(H −ET)τ]ΨV (11)
1+xc NP − 2 = Yexp[−(H −ET)∆τ]ΨV,
evaluated as a branching random walk. The fixed-node
where v [0,1].
∈ approximation gives a wave function Ψ0 that is the
The parametric dependence on v can be eliminated to lowest-energystate with the sames nodes (surface where
find g(x,n ) along the coexistence curve. The coex- Ψ = 0) as the trial state Ψ . The resulting energy E is
NP V 0
istence curve (for example, see dashed curve in Fig. 2) an upper bound to the true ground-state energy.
should not be seen as the continuation of the NP curve The ground-state energy E can be obtained from:
0
atconstantdensity,sincethedensitychangesalongit. It
Ψ H Ψ Ψ H Ψ
issimplyaprojectionofthecoexistencecurvein(g,x,n) V 0 0 0
E = h | | i = h | | i. (12)
0
space, on the (g,x) plane. ΨV Ψ0 Ψ0 Ψ0
h | i h | i
To proceed with the calculation of the coexistence The variational wave function is taken to be of the
curve, we need to calculate g(x,n) for the NP phase following form:
and g (n) for the SF phase. We use Quantum Monte
SF
Carlo techniques to calculate the energies at a few val- ΨV(R)=YfP(rij) Y fP(ri′j′)Yf(rij′)Φ(R) .
ues of (x,n) and interpolate to determine the functional i6=j i′6=j′ i,j′
dependence. This is discussed next. (13)
4
Inourearlierworkswestudiedsuperfluidneutronmatter
and therefore used a trial wave function of the Jastrow- 1.20
BCS form with fixed particle number. [6, 13, 47] In the GFMC
present work we are attempting to determine the sta- 1.15 GFMC unpolarized superfluid
bility of different phases. In order to do this, we have Cubic fit
completelymappedoutthe energyofanormal(i.e. non-
1.10
(sa1ut3pd)eirdfffleeusricedrn)itbnedeseuntthsrioetnipesag.ratsiTcalhseusasa,fsutnbhceetiniΦogn(Rion)fatwhfeerepueosFeladerrimiznaitEgioaqns. 3/5)E/E↑)FG(1.05
(i.e., lets all the correlations lie within the Jastrowfunc- (
1.00
tions). In other words, the wave function is composed
oftwoSlaterdeterminants (oneforspin-upparticlesand
0.95
one for spin-down ones):
ΦS(R)=ΦS(R)N↑ΦS(R)N↓′ (14) 0 0.2 0.4x = N↓/Ν0↑.6 0.8 1
where
FIG. 1: (color online) Ground-state energy per particle (in
Φ (R) = [φ (r )φ (r )...φ (r )] (15) unitsofthefreeFermigasenergy)scaled tothepowerof3/5
S N↑ A n 1 n 2 n N↑ for normal spin-polarized neutron matter. Shown are QMC
results at three different total densities n1,n2,n3 (as points:
and
blackcircles,redsquares,bluediamonds,respectively),along
withcubicfitstotheMonteCarloresults(aslines: blacksolid,
ΦS(R)N↓′ =A[φn(r1′)φn(r2′)...φn(rN↓′)] (16) reddashed,bluedotted,respectively). Alsoshownusinghol-
low symbols are theresults for an unpolarized superfluid.
The primed (unprimed) indices correspond to spin-up
(spin-down) neutrons and N +N′ = N. is the anti-
↑ ↓ A
symmetrizer and φn(rk)=eikn·rk/L3/2.
TheJastrowpartisusuallytakenfromalowest-order- theground-stateenergieswiththeenergiesofcorrespond-
constrained-variational method calculation described by ing free spin-up Fermi gases,andraisedthe result to the
a Schr¨odinger-like equation: 3/5 power. When x = 0 we see that the values are very
close to 1, though still above it: this is due to the same-
¯h2 spin p-wave interactions. Similarly, as we increase the
2f(r)+v(r)f(r) =λf(r) (17)
− m∇ density the energy is decreased, a fact that, as we see in
Fig. 1, holds at any relative fraction x. The statistical
for the opposite-spin f(r) and by a corresponding equa- errors of the Quantum Monte Carlo results are smaller
tion for the same-spin fP(r). Since the f(r) and fP(r) thanthe symbolsshowninthe figure. Also shownto the
arenodeless,theydonotaffectthefinalresultapartfrom rightofthe figure are the QMC results for anupolarized
reducing the statistical error. superfluid at the three densities of interest (hollow cir-
We calculate ground-state energies at different total cles, squares, and diamonds, respectively). The results
numberdensities[n=(N↑+N↓)/L3],morespecificallyat for n1 and n2 were taken from Ref. [47]; the value at n3
n1 =6.65 10−4,n2 =2.16 10−3,andn3 =5.32 10−3 was re-optimized for the purposes of the present work.
fm−3. To×put these densit×ies into perspective w×e can
comparethemtonuclearmattersaturationdensity: they
are0.41,1.35,and3.32percent,respectively,ofn =0.16
0
fm−3. These simulations were performed at values of IV. PHASE COMPETITION
the relative fractions chosen specifically in order to en-
sure a full coverage of the x-axis. More specifically, sim- Havingarrivedatthefunctionaldependenceoftheen-
ulations were carried out at relative fractions of x = ergy of a normal neutron gas on the relative fraction
0, 0.333, 0.579, 0.818, 1,correspondingtoparticlenum- (thereby getting information on the g(x,n) function de-
bers of 33+0, 57+19, 57+33, 33+27, 33+33, re- fined in Eq. (1) and appearing in the rest of Sec. II), we
spectively. These systems are quite large (and therefore can now use the simple cubic fits for the three densities
computationallydemanding)soastoensureaminimiza- n ,n ,n todeterminethecompetitionbetweendifferent
1 2 3
tion of finite-size effects (see also Refs. [13, 47, 48]). phases. More specifically, Fig. 1 shows the behavior of
The results are shown in Fig. 1 as points (circles, the NP (normal polarized) equation of state. From the
squares, and diamonds, respectively), along with cubic hollow points on the same figure we also have access to
fits to the microscopic data. The latter will be useful to the energyofthe SF(unpolarizedsuperfluid) phase. Us-
us in the following section, when we try to check the rel- ing these two dependences, we can find out the critical
ativestability ofdifferentphases. To facilitate the useof values of the relative fraction (x ,x ,x , respectively)
c1 c2 c3
theseresultsinconnectionwithEq. (1),wehavedivided at the three densities we are studying. Above a critical
5
Thisanalysisrequiresthatforx<x thenormalphase
c
is thermodynamically stable. That is, the eigenvalues
Normal of the matrix d2E are positive. For n = n we can
Superfluid dnidnj 2
1.15 check and have checked explicitly that this is the case.
Phase coexistence
Finally, we find δµ at x is given by δµ /E =0.16(2)
c2 c2 F
where E = (k2/2m). Comparing with ∆ /E
3/5)↑)1.10 0.23 obtaFined froFm Ref. [47], we see that ∆SSFF22> Fδµc∼2
EFG( satisfying the third condition for coexistence.
(E/ 1.05 In Fig. 2 we have plotted as a black solid line the
n results from Fig. 1; at this density the perturbative
1
1.00 S = 1,M = 0 correction in the Hamiltonian men-
S
tioned in Sec. III is the smallest, leading to the high-
est degree of confidence in the accuracy of our micro-
0.95
0.7 0.75 0.8 0.85 0.9 0.95 1 scopicresults. We focus onlargevalues onthe x-axis for
x = N↓/Ν↑ reasons that will soon become clear. We also show as
red square points the unpolarized superfluid result from
FIG.2: (coloronline)Theconstructionofthemixedphaseat Ref. [47] (at x = 1) and the homogeneous polarized su-
densityn1. Shownistheenergyofanormalneutrongasfrom perfluid results of Ref. [6] (at smaller x). In addition,
Fig. 1 of the present work (black solid line), along with the
we show a coexistence curve which is arrived at by a
homogeneous superfluid results of Refs. [6, 47] (red square
tangent construction from the unpolarized superfluid to
points), and a tangent construction showing the phase co-
the normal polarized curve when the y-axis is chosen as
existence curve (green dashed line). For large values of the
we have [45]. This coexistence line meets the NP curve
relative fraction, the phase separated state is energetically
favored with respect to a homogeneous polarized superfluid. at xc1 = 0.74 (δµc1/EF = 0.2). As can be clearly seen
in this Figure, the coexistence line lies below the homo-
geneous polarized superfluid Quantum Monte Carlo re-
sults, implying that these are not energetically favored
(foreseeing this possibility, Ref. [6] included a critical
relative fraction the system phase separates into a mix-
assumption explicitly excluding the possibility of phase
tureofSFandNP,whereasbelowthatvaluetheneutron
separation). This situation followsthe behaviorof ultra-
gas is normal, and therefore does not exhibit any of the
coldfermionicgasesatunitarity: [1,40,41]therethecrit-
widelystudiedfeaturesofasuperfluidintheneutronstar
icalvalue of the relative fractionwascloserto x =0.44.
c
crust.
The value we find is different for a variety of reasons:
To determine xc we needthe functionaldependence of first, n1 corresponds to kFa = 5, which is somewhat
−
g(x,n) on both x and n, which is given by the interpo- smaller than k a = . More importantly, the neutron-
F
∞
lations for n1, n2 and n3. Let us first focus on n2. In neutron interaction is characterized by a finite effective
Eq.(2)wealsoneedthederivativewithrespectton. We range and the presence of higher partial waves. The lat-
estimate this derivative by finite differences: (g(x,n2) ter point explains why when we repeat this exercise for
−
g(x,n1))/(n2−n1)and(g(x,n3)−g(x,n2))/(n3−n2). We n2 (kFa = −7.4) and n3 (kFa = −10) the correspond-
estimatethederivativesforSF usedinEq.4withalinear ing results (from tangent constructions) for the critical
form in 1/(kFa) and kFre. In practice, ∂∂ng is small and fraction are xc2 =0.78 (δµc2/EF =0.16) and xc3 =0.88
doesn’t affect the value of xc and nSF significantly. We (δµc3/EF = 0.07), respectively. The trend appears to
can then solve the conditions for coexistence and find xc be that at larger density (stronger coupling) the critical
and nSF. The numerical values show slight dependence fraction is moving toward 1, implying that farther down
on the interpolation method used. A simple cubic fit as toward the core of the neutron star polarized neutron
a function of x gives xc2 = 0.78 and nSF = 2.17 10−3 matter quickly becomes normal for the vast majority of
fm−3. A three parameter fit in (1,x,x5/3) mo×tivated possible polarizations.
by [1, 40] gives xc2 =0.77 and nSF =2.18×10−3 fm−3. The value of δµc is relevant for neutron star phe-
This gives some idea about the errorsmade in the inter- nomenology since whether δµ is greater or less than δµ
c
polation: we estimate our error bar on xc to be 0.02 for determines the phase. This is seen more simply in the
the three densities. This detailed proceduregives results grand canonical ensemble where we can compare the
verysimilarto asimpler procedure,namelyconstructing pressureasafunctionofthechemicalpotentials. InFig.3
a tangent to the constant density NP curve for g(x,n) weplotthepressureasafunctionofδµforfixedµ=1.94
passingthroughthe value ofgSF(n) atthe samedensity, MeV. For this value of µ the density of the superfluid
which gives xc2 = 0.78. The close agreement between phase at the critical δµ is n2. For δµ<δµc the pressure
the two approaches stems from the fact that ∂g is small of the SF phase is larger and the system is unpolarised.
∂n
— for ∂g = 0, the coexistence curve is given exactly by Evenifthe magneticfield isnotlargeenoughto polarize
∂n
the tangent construction [45]. Therefore, we will use the the system, certainthermalobservables,for example the
tangent construction to find x for n and n below. specific heat, are sensitive to the difference between δµ
c 1 3
6
density the Hamiltonian becomes quite simple and we
5 include its dominant well-known terms. We have calcu-
lated the energy as a function of polarization using the
AV4′ interaction at three different densities.
4 Oneofourmainresultsisadeterminationofthe criti-
calrelativefractionvaluesabovewhichthesystemphase
4) separates into an unpolarized superfluid and a polarized
V
e normalneutron gas. Above these values, a homogeneous
M3
4 polarized superfluid is found not to be energetically fa-
0
1
P( vored. Belowthese values, the neutrongas is completely
normal. Both these facts can lead to astronomically im-
2 portant consequences. Importantly, the trend of our re-
P sults seems to imply that at slightly larger densities xc
SF
wouldbeveryclosetoone: thiswouldmeanthatevenat
1 zero temperature a small polarization would be enough
0 δµ 1 2 3 4
c δµ(MeV) to close the superfluid pairing gap, and thus easily lead
to larger polarization. In a non-astrophysical context, it
FIG. 3: (color online) Pressure as a function of δµ for fixed isconceivablethatourresultscouldbetesteddirectlyby
µ=1.94 MeV.Forδµ>δµc =0.53 MeVtheunpairedphase using ultracold fermionic atom gases with unequal spin
(bold black curve) has a higher pressure. At δµc [marked by populations and a finite effective range. In cold atoms,
the vertical line (brown online)], the SF and the NP phases QuantumMonteCarlosimulationsofspin-polarizedmat-
can coexist. For smaller δµ, the pressure of the SF phase terhaveinrecenttimes beenrepeatedlyusedasinputto
markedbythehorizontalline(blueonline) islarger than the computationally less demanding density-functional the-
pressure of the NP phase marked by the dashed line (red
ory approaches. Similarly, we expect that the results
online), and the SF phase is favored. On the other extreme
presented in this work can also be used as input to self-
for δµ = 4.18 MeV, x = 0.0005. We choose µ so that the
consistent mean-field models of nuclei.
density of the normal phase at δµc is n2.
A possible direction for future work, that would be
interesting to both nuclear astrophysicists and Skyrme
practitioners, would be a study of the behavior of po-
and δµ . larized and unpolarized superfluid and normal neutron
c
Determining the critical relative fractions is also rel- matterinanexternalperiodicpotential. Suchastudyof
evant: if a magnetic field is strong enough to polarize the static response of neutron matter could also in prin-
neutronmattersufficiently(x x ),the gasceasestobe ciple be guided (or guide) analogous research related to
c
superfluid even at zero temper≤ature. At such large mag- optical lattice experiments with cold atoms.
netic fields, processes that rely on the presence of a bulk Intheneutronstarcontext,itwillbeusefultoanalyze
superfluidin the inner coreofneutronstars,for example the finite temperaturephasediagramto understandhow
a recently proposed heat-conduction mechanism which the specific heat is modified as a function of the chemi-
requires superfluid phonons [10], would no longer be op- cal potential splitting, and how this modification affects
erational. On a different note, a precise determination transient phenomena in the neutron star crust.
of these fractions (along with the rest of the equation
of state) is also significant for Skyrme functional prac-
titioners: for the region where the NP phase is the true Acknowledgments
groundstateofthesystem,theresultsshowninFig.1are
a microscopic constraint to nuclear energy-density func- A.G. thanks George Bertsch, Kai Hebeler, and Achim
tional theory, which can help improve its predictions on Schwenk for stimulating discussions. R.S. acknowledges
neutron-rich heavy nuclei. discussions with Jeremy Heyl. The authors thank Mark
AlfordandSanjayReddyforvaluablecommentsandsug-
gestions. The authorsacknowledgethe hospitalityofthe
V. SUMMARY & CONCLUSIONS PerimeterInstitute andthe organizersofthe Micra2011
workshop where this work was initiated. This work was
In summary, we have studied spin-polarized low- supportedbyDOEGrantNo. DE-FG02-97ER41014and
density neutron matter at many values of the polariza- the National Sciences and Engineering ResearchCouncil
tion. OurQuantumMonteCarloapproachprovidestight of Canada (NSERC). Computations were performed at
upper bounds which are expected to be quite close to the US DoE’s National Energy ResearchScientific Com-
the true ground state-energy of the system. At very-low puting Center (NERSC).
7
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