BCS-BEC
BCS
BEC Crossover inAsymmetric Nuclear Matterwith nn
pp np Pairings
nn,pp,np

Shijun Mao，Xuguang
Mao Xuguang Huang and Pengfei Zhuang
Tsinghua University

Content• Introductiond• Basic Formalism• Numerical Results• Summary1. Introduction

BECBCSWork has been done…
★Transition from BCS pairing to BEc in low-density
asymmetrict i nuclearlmattertt
U. Lombardo, P. Nozières:PRC64, 064314 (2001)★Spatial structure of neutron Cooper pair in lowdensity uniform matter
Masayuki Matsuo:PRC73, 044309 (2006)★BCS-BEC crossover of neutron pairs in symmetricand asymmetric nuclear matters
J. Margueron: arXiv:0710.4241vl [nucl-th] 23 Oct
2007★ What we will do: consideringboth nn, pp, and np pairings

2.Basic Formalism
The Lagrangian:
gg
L=22
∂∇
∂
∇vvvv++−+++−++[p(x)(μ)p(x)n(x)(μ)
n(x)]
∑σp σ
σn σ
∂τ 2m
∂τ 2mσ =↑,↓v v' + v + v'v'v
3 v'−∫ d xVnn (x − x )[n↑ (x)n↓ (x )n↓ (x )n↑ (x)]v' v v' + v + v'v'v−∫ d3xV(x−x)[p(x)p(x)p(x)p(x)]
pp
↑↓
↓
↑

1 3 v' v v' + v + v'v'vv'v+ v + v'
− ∫ d xVnp (x − x )[n↑ (x) p↓ (x ) − p↑ (x)n↓ (x )][ p↓ (x )n↑ (x) − n↓ (x ) p↑ (x)]2Paris Potential and effective mass
for uniform nuclear system:v v'v v'V ( x − x ) = g I δ ( x − x );

ρ γg I = v0 [1 − η I ( ) ]ρ0I

np Pairing (I=0): η= 0. v0= - 530MeVfm3;
nn,, pp Pairingg (I=1):η=( ) η 0.45.γ=
γ 0.47,, v0= - 481Mevfm3;

Cutoff energy
energy:ε c = 60
60MeVMeV

Gap Equation and Density Equation:
Partition function :v
Z = ∏σ ∫ [dnσ ][dpσ ][dnσ ][dpσ ]exp( ∫ dτ ∫ d 3 xL)++

β0

Thermodynamic potential in mean
field approximation:v
iΔ+Δ
Ed k −
2Δ− E ij / Tj−
Ω=−−+∫[ε n + ε p − ∑ ( + T ln(1 + e))]3g0g1(2π )i , j =± 222n2p

3condensate

quasi-particle
quasiparticle

gap equation:∂Ω∂Ω∂Ω
= 0,= 0,= 0,∂Δ n∂Δ p∂Δdensity equation:
μn , μ p → μ = ( μn + μ p ) / 22, δμ = ( μn − μ p ) / 2∂Ω∂Ωρn , ρ p → ρ = ρn + ρ p = −, δρ = ρ n − ρ p = −,∂μ∂δμδ

and define relative density asymmetry α =δρ.
ρ3.Numerical Results ofBCS BEC Crossover
BCS-BECCatt ZeroZTemperature
Tt

Ⅰ qualitative descriptionⅠ.
☆ wave function:

K + K K
ψijj ( r ) = C BCS a i ↑ ( x )a j ↓ ( x + r ) BCS+K
K ikK⋅rKd k'=C ∫ψ ( k )e ,3 ij(2π )3☆ probability density: r ψij ( r )22Friedel Oscillation
Hi h density
Highdit :(n-n. p-p, n-p)large spatial extension
and strong oscillationt i l BCS behavior
typicalb h i

CrossoverLow density:oscillation weakens;w;

np pair is stronger
correlatedl t d thanthnn, pp pairs.
iwave function shrinks

possibleibl BEC

Ⅱ. more quantitative description☆BCS boundary and BCS limit:ξij / d ij = 1 ,( ξ ij =∞∫ r ψ dr )042ij

μ ij / ε F → 1
I=0,
I=1,

ρ >0.5ρ0 → BCS
nn, ρ >0.17ρ0 → BCS
pp, ρ >0.14ρ0 → BCS

I=0,1:

ρ ≈ 0 .8 7 ρ 0 ,BCS limit☆ BEC boundary and BEC limit(i). BEC boundary:s-waveswave scattering length:aijm11=+∑KK
4πaij
gij
k 2εkboundary condition:0.004ρ01=1kF aij

(ii). BEC limit
Probability Pij (d) -----partners of the ij Cooper pair tocome close to each other within the average distance d ( d = ρ−1/ 3 )∫P (d ) =
∫
ijj

d0∞

02

ψij ( r ) r 2 dr2≈1

ψij ( r ) r 2 dr

att lowldensity:dit
Pnp (d ) ≈ 1, ⇒deuteron BEC

Pnn (d ), Ppp (d ) ≈ 0.8

no true BEC

☆ BCS-BEC Crossovernp BEC0np crossovernn, pp relatively strong
correlated

np BCSρnn, pp BCSⅢ. ρ − α phase diagram

np BEC
np BEC4. Summary:☆ For asymmetric nuclear matter, weconsistently consider both n-pn p and n-n,n n p-p
ppCooper pairs and obtain ρ-α phase diagram atT=0;☆ A possible signature of BCS-BEC Crossoveris Friedel Oscillation,
Oscillation which weakens asdensity decreases;☆ AsA ddensity
i decreases,
dnp-BCS-BEC
BCS BEC
Crossover occurs and deuteron BEC isreached,h d while
hil no BEC statet t forf nn,pp pair.
i