Phase Diagram of Neutron-Proton Condensate in Asymmetric Nuclear Matter

0.12 MB·English

by  Meng Jin

Phase Diagram of Neutron-Proton Condensate in Asymmetric Nuclear Matter
Meng Jin1,2, Lianyi He1, and Pengfei Zhuang1
1Physics Department, Tsinghua University, Beijing 100084, China
2Institute of Particle Physics, Central China Normal University, Wuhan 430070, China
We investigate the phase structure of homogeneous and inhomogeneous neutron-proton conden-
sate in isospin asymmetric nuclearmatter. At extremely low nuclear density thecondensed matter
isinhomogeneousphaseatanytemperature,whileingeneralcaseitisinLarkin-Ovchinnikov-Fulde
-Ferrell phase at low temperature. In comparison with thehomogeneous superfluid, the inhomoge-
neoussuperfluid can surviveat higher nuclear density and higher isospin asymmetry.
PACSnumbers: 21.65.+f,21.30.Fe,26.60.+c
7 It is well-known that the neutron-proton (np) pair- Fulde -Ferrell(LOFF) phase[18] is taken into account?
0 ing plays an important role in nuclear physics and as- How will the LOFF phase change the strange tempera-
0
trophysics, such as the structure of medium-mass nu- ture behavior of np condensate found in [15]? In fact,
2
clei produced in radioactive nuclear beam facilities[1], there should exist a rich phase structure in asymmet-

n the deuteronformationinintermediateenergyheavy-ion ric nuclear matter, since the isospin asymmetry essen-
a collisions[2], the pion and kaon condensation[3], the r- tially plays the same roleas the populationimbalance in
J
process[4, 5], and the cooling of neutron stars. The mi- the two-component resonantly interacting atomic Fermi
3
croscopic calculations show that the nuclear matter sup- gas[17]. Different to the cold atoms , in nuclear matter
2
3 tphoerttsennsporCcooompepropnaenirtinogf tinhethneuc3lSe1a−r3foDr1cec,haanndnetlhedupeaitro- tdheenpahnadsethseepLarOaFtiFonpahtalsaergmealyenbgethensceargleetmicaayllybefafvoorbreidd-.
v ing gap is of the order of 10 MeV[6, 7, 8, 9, 10, 11, 12] Although the LOFF phase was discussed in asymmet-
5 at the saturation nuclear density. At low enough den- ric nuclear matter at saturation density[19], the phase
6 sity the np Cooper pairs would go over to Bose-Einstein structure including the LOFF phase in the whole den-
0 condensation(BEC) of deuterons in symmetric nuclear sity, temperature and isospin asymmetry space and the
9
matter[2, 9]. effectoftheLOFFphaseonthestrangetemperaturebe-
0
havior of the np condensate are still unknown. We will
6 The emergence of isospin asymmetry will generally
0 suppress the np pairing, and the condensate will dis- present in this paper the phase diagrams in the density,
/ temperature and isospin asymmetry space.
h appear when the asymmetry becomes sufficiently large.
The oftenusedformulaeforsuperfluidin nuclearmat-
t Near the saturation density, the np pairing correlation
- ter are discussed in detail in [20] where the superfluid
l depends crucially on the mismatch between the two
c state is described by a normal and an anomalous nu-
Fermisurfaces,andasmallisospinasymmetrycanbreak
u cleon distribution functions and . Generally, they
n the condensate due to the Pauli blocking effect. At F G
are functions of momentum p and matrices in spin and
: very low density, when neutrons and protons start to
v isospin space. The formulae can be easily generalized to
formdeuteronsandwhen the spatialseparationbetween
i study the isospin asymmetric superfluid with total pair
X deuterons and between deuterons and neutrons is large,
momentum 2q. We willstartwith the LOFF phase,and
r the Pauli blocking loses its efficiency in destroying a np
a condensate. Insuchsituation,theisospinasymmetrycan thehomogeneousphasecanberecoveredbytakingq=0.
Like the studies in [13, 14, 15, 16, 19], we discuss the np
beverylarge,andthenpcondensatesurvivesintheform
ofdeuteron-neutronmixtureinmomentumspace[13,14]. pairing in the 3S1−3D1 channel with total spin S = 1,
isospin T =0 and their projections S =T =0. In this
Different from the symmetric nuclear matter where the z z
case the distribution functions take the structure
thermal motion destroys the np condensate, for asym-
metric nuclear matter the temperature effect will melt (p) = 00(p)σ0τ0+ 03(p)σ0τ3,
thecondensateononehandandincreasetheoverlapping F F F
(p) = 30(p)σ3σ2τ2, (1)
between the two effective Fermi surfaces on the other G G
hand. As a result of the competition, in a wide density where σ and τ are the Pauli matrices in spin and
i i
regime the temperature dependence of the superfluidity isospin spaces. Using the minimum principle of the
is very strange[15, 16]: The maximum condensate is not thermodynamic potential and the procedure of block
locatedatzerotemperature,andthe pairingevenoccurs diagonalization[21], we can express the elements as
only atintermediate temperature forlargeisospinasym-
metry. F00(p) = 1/2−ξp 1−f(Ep+)−f(Ep−) /(2Ep),
The above results are obtained by assuming the con- F03(p) = f(Ep−)−(cid:2)f(Ep+) /2, (cid:3)
dtreunesaptheaissehsotmruocgtuenreeooufsnipnctohnedgernosuantedwstiathtei.soWsphiantaissytmhe- G30(p) = −(cid:2) ∆p 1−f(Ep+)(cid:3)−f(Ep−) /(2Ep) (2)
metry when the inhomogeneous Larkin-Ovchinnikov- with the notations(cid:2)ξ = p2+q2 /(2(cid:3)m) µ, E =
p p
−
(cid:0) (cid:1)
2
ξp2 +∆2p and Ep± = Ep ±(δµ+p·q/m), where m is ie.ne.e,rgayntehgaantivteheρshommeoagnesntehoeusLsOuFpFerflpuhiadsephhaassel.ower free
qthe effective nucleon mass in the medium and f(x) =
1/(ex/T + 1) is the Fermi-Dirac function with T be- The momentum dependence of the gap function ∆p is
ing the temperature. We have introduced the average normally rather weak in a wide momentum region[13],
chemical potential µ = (µ +µ )/2 and the mismatch and we can approximately treat it as a constant ∆ for
n p
δµ = (µ µ )/2 instead of the neutron and proton a qualitative analysis. At zero temperature, once the
n p
chemical po−tentials µ and µ . We have also neglected isospinasymmetryisturnedon,theremustexistasharp
thepossibleneutron-pnrotonmapsssplittinginducedbythe breached region where the quasiparticle energy Ep− < 0
isospinasymmetrywhichisbelievedtobe small. The np withthenecessaryconditionδµ>∆,andthemomentum
condensate ∆ is generally momentum dependent and integration in (6) can be analytically carried out,
p
satisfies the gap equation
d3k ρ =ρ 1 p3+Θ(µ+)+p3−Θ(µ−) δµ , (7)
∆p = (2π)3V(p,k)G30(k), (3) s " − 3π2ρ δµ2 ∆2#
Z −
p
where V is the nucleon-nucleon (NN) interaction poten-
tial. The LOFF momentum q should be determined via where p± = √2mµ± are possible gapless nodes with
minimizingthethefreeenergy ,whichensuresthetotal µ± = µ δµ2 ∆2. At high density the matter is in
± −
E BCSregimewhereδµ,∆ µ,andthebreachedregionis
current j in the ground state to be zero,
s p ≪
p− < p <p+. Since p± are close to the Fermi momen-
| |
d3p p q tum kF, the superfluid density should be negative since
ρ|q|−4 (2π)3 q· F03(p)=0, (4) ρs ρ(1 δµ/ δµ2 ∆2). On the other hand, at low
Z | | eno≃ughde−nsitythema−tterisinBECregime,thechemical
p
wherewehaveusedthetotalnucleondensityρ=ρn+ρp potentialµbecomesnegativewhichleadstoµ− <0anda
and the isospin density asymmetry δρ=ρn ρp instead reduced breached region 0< p <p+. In this case p+ is
− | |
of the neutron and proton densities ρ and ρ , muchsmallerthank andthesuperfluiddensitybecomes
n p F
positive. Therefore,atzerotemperaturethesuperfluidis
d3p d3p expected to evolve from an inhomogeneous phase to the
ρ=4 (2π)3F00(p), δρ=4 (2π)3F03(p). (5) homogeneous phase when the nuclear density decreases,
Z Z
which is a general phenomenon for BCS-BEC crossover
Once the NN potential V is known, we can solve the with population imbalance[17].
coupled set of gap equations (3) and (4) together with
At finite temperature, the breached region is smeared
thedensityequations(5)atgiventemperatureT,baryon
due to the thermal excitation, and δµ > ∆ is not nec-
density ρ or equivalently the Fermi momentum k =
F essary. At the critical temperature T there should be
(1.5π2ρ)1/3 andisospinasymmetryα=δρ/ρ,andobtain c
a second order phase transition from the superfluid to
allpossiblephases,namelythenormalphase∆ =0,the
p normal state, and for temperature T . T the pairing
homogeneous superfluid phase ∆ = 0, q = 0 and the c
LenOeFrgFiesphwaeseca∆npde6=ter0m, iqne6=th0e.trBupye 6cgoromupnadrisntgatteh.eir free gthaeprbeeghualavresbaehsa∆vi(oTr)of∝th(e1s−upTe/rTfluc)i1d/2dewnhsiitcyh ρlesa(Tds) t∝o
(1 T/T ) > 0. This means the temperature tends to
The details of the phase diagram depend on the NN − c
stabilize the homogeneous phase. Combining with the
potential V we will chose in the numerical calculations,
behavior of the superfluid density at zero temperature,
however, the qualitative topology structure of the phase
the homogeneous phase at sufficiently low density will
diagram does not depend on that. To show this, we an-
keep stable at any temperature below T , while at high
alyze the stability of the homogeneous superfluid phase c
density theremust existaturning temperatureT where
against the formation of a nonzero Cooper pair momen- s
ρ changessign,andthe superfluidshouldbe ininhomo-
tum. Forthispurpose,weinvestigatetheresponseofthe s
geneous phase at low temperature T <T and in homo-
free energy to a small pair momentum q via the small s
q expansionE, (q)= (0)+j q/m+ρ q2/(2m)+ , geneous phase at high temperature Ts < T < Tc. Since
E E s· s ··· the homogeneous state is unstable at low temperature,
where j = m∂ /∂q is the total current which is pro-
s E combining with the fact that the critical isospin asym-
portional to the left hand side of (4), and ρ is just the
s metry for the LOFF phase is much larger than the ho-
superfluid density defined by ρ = m∂2 /∂q2 of which
s E mogeneous phase[19], the strange temperature behavior
the explicit form reads
of the pairing gap found in [15] is probably unrealistic.
2 d3p p2 Wenowmovetonumericalcalculations. TheParisNN
ρ =ρ+ f′(E+)+f′(E−) (6)
s m (2π)3 3 p p q=0 potential is often used to describe the nuclear structure
Z (cid:2) (cid:3)(cid:12) and nucleon superfluidity, and describes well the BCS-
with the definition f′(x) = df(x)/dx. The c(cid:12)urrent j BEC crossover of np condensate[2, 13]. Since the quali-
s
vanishes due to the gap equation for q, and the sign of tative topology structure of the phase diagram does not
ρ controls the stability of the homogeneous superfluid, depend on specific models, for the sake of simplicity, we
s
3
1
3 L T=0.1MeV
ρ=0.16fm−3 0.8
2.5
α=0.05
2 0.6 Normal
Α
V) HS
Me1.5 0.4
∆( 1 α=0.07 0.2 LOFF
0.5 0
0 0.2 0.4 0.6 0.8 1 1.2
0 kFHfm-1L
0 0.5 1 1.5 2 2.5 4
T(MeV)
Α=0.3
10
3
ρ=0.16fm−3
8 α=0.07 LV Normal
Me2 HS L
α=0.05 H
T
V) 6
Me 1
LOFF
q( 4
0
2 0 0.2 0.4 0.6 0.8 1
k Hfm-1L
F
3
0
0 0.2 0.4 0.6 0.8 1 1.2 Ρ=0.16fm-3
T(MeV) 2.5
FIG. 1: The pairing gap ∆ for homogeneous (dashed lines) 2 Normal
and inhomogeneous (solid lines) condensates and LOFF mo- LMeV1.5 HS
mentum q as functions of temperature at normal density ρ0 H L
and for two values of isospin asymmetry. T
1
LOFF
0.5
employ a separable form of the Paris NN potential
0
0 0.02 0.04 0.06 0.08 0.1
ρ r1+r2 γ Α
V(r1,r2)=v0 1 η 2 δ(r1 r2), (8)
" − (cid:0) ρ0 (cid:1)! # − FIG. 2: The phase diagrams in kF −α, kF −T and α−T
planes. In each plane, the labels HS, LOFF and Normal in-
whichwasdevelopedin[22]toreproducethepairinggap dicatehomogeneous superfluid,LOFFsuperfluidandnormal
in S = 1,T = 0 channel and the bound state between phase, the dashed line is the border of the unstable HS, and
zeroenergyanddeuteronbindingenergy,whereρ0 isthe thethree phases meet at a Lifshitz point L.
normal nuclear density and the parameters v0,η,γ and
anenergycutoffε to regulatethe modelaredetermined
c
by recovering the pairing gap in the realistic Paris NN critical temperatures T and T for large isospin asym-
o c
potential. This separable form can also describe well metry,whichisshownasdashedlines inthe upperpanel
the BCS-BEC crossover of np condensate[14]. In our of Fig.1 for saturationdensity without loss of generality.
numerical calculation, we choose v0 = −530 MeV·fm3, However, the superfluid density ρs for the homogeneous
η =0,εc =60MeVandtakemasthedensity-dependent phaseisnegativeatT <Ts andpositiveatTs <T <Tc,
nucleon mass corresponding to the Gogny force D1S[23]. wheretheturningtemperatureT islargerthanthelower
s
Wehavecheckedthatdifferentparametersets[22]leadto critical temperature T . This tells us that the homoge-
o
only a slight change in the numerical results. neous phase is stable at high temperature T > T and
s
InthelowdensityBECregime,thehomogeneousphase unstable to formation of LOFF condensate at low tem-
is stable at any temperature below T , and the conden- perature T < T . By calculating the LOFF pairing gap
c s
sate is a regulardecreasingfunction of temperature. Be- ∆ and momentum q = q which are shownas solid lines
| |
yond this regime the strange phenomenon of the homo- in the upper and lower panels of Fig.1 and comparing
geneous condensate arises: The maximum pairing gap the free energies for the homogeneous and LOFF states,
is located at non-zero temperature, and the condensate the LOFF phase is energetically more favored than the
evenappears only at intermediate temperature with two homogeneous phase at T < T . Especially, different to
s
4
the homogeneous condensate, the LOFF condensate al- we consider HS only, the HS-Normal boundary (dashed
ways starts at zero temperature. Therefore, after con- line) is below the LOFF-Normal boundary, this shows
sidering both homogeneous and inhomogeneous conden- thatthe introductionofLOFFphase enlargesthe super-
sates,thestrangetemperaturedependenceofthepairing fluidregion. Inthek T andα T planes,the HSand
F
− −
gap[15,16]disappearsandthecondensatebecomesareg- LOFF phases are separated by the turning temperature
ulardecreasingfunction oftemperature. The LOFFmo- T . Note that T starts at k =0 in k T plane which
s s F F
6 −
mentumq,showninthelowerpanelofFig.1,dropsdown corresponds to the stable HS at extremely low density
withincreasingtemperatureandapproachestozerocon- butstartsatα=0inα T planeforhighdensitywhich
−
tinuouslyatT ,whichindicatesacontinuousphasetran- means that the HS with smallisospinasymmetry is easy
s
sition from homogeneous phase to LOFF phase. The to be stabilized. Again, in comparison with the calcula-
continuity can be proven analytically. The gap equation tion with only HS, the superfluid is extended to higher
(4) for q can be written as qW(q) = 0 with a trivial so- density or higher asymmetry regiondue to the introduc-
lution q = 0 for the homogeneous phase and a non-zero tion of the LOFF phase, and the unstable HS-Normal
solution from W(q) = 0 for the LOFF phase. Using the boundary (dashed lines) which reflects the strange “in-
expansion for we find ρ = W(0). Therefore, we must termediate temperature superfluidity” is replacedby the
s
E
have q =0 at T =T , providing that the LOFF solution LOFF-Normal boundary. The phase transitions in the
s
is unique. three planes are all of second order, and in any case the
The phase diagrams of the np pairing are shown in three phases meet at a Lifshitz point L[24]. The α T
−
Fig.2. We first discuss the one in the k α plane at a phase diagramwe obtained is very similar to the generic
F
−
verylowtemperatureT =0.1MeV.Whenρ 0wefind phase diagram of two-component ultracold Fermi gas in
→
µ ε /2 at α = 0 where ε is the deuteron binding a potential trap[25].
b b
→ −
energy, which means that the np condensate survives in
In summary, we have qualitatively investigated the
the form of deuteron BEC. Consistent with the findings
phase structure of np condensate in isospin asymmetric
in atomic Fermi gas[17], the homogeneous phase (HS)
nuclear matter and confirmed our analysis with a model
is stable only at very low density BEC regime. In this
NNpotential. Theimportantfindingsare: 1)TheLOFF
regime,the criticalisospinasymmetry canbe verylarge,
and even approaches to 1 for k < 0.23 fm−1. Beyond phaseisthegroundstateinawideregionofnuclearden-
F sity,temperatureandisospinasymmetry,exceptforvery
this extremely lowdensity regime,the superfluid density
low density and high temperature. 2) The strange tem-
of the homogeneous phase becomes negative which indi-
peraturebehaviorofthe np condensate is washedoutby
cates that the LOFF phase is energetically favored. By
the LOFF phase at low temperatures. 3)The superfluid
calculating the superfluid density of the HS phase and
region is expanded to high density and high asymmetry
the LOFF solution, we can determine the phase bound-
due to the introduction of LOFF phase.
aries between HS and LOFF and between LOFF and
normal phase. The LOFF momentum is large at high Acknowledgement: The work was supported by
ρ and high α and approaches to zero at the HS-LOFF the grants NSFC10575058, 10428510, 10435080 and
boundary which means a continuous phase transition. If 10447122.
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