bec-bcs crossover in njl model; towards qcd phase diagram ... · bcs-bec crossover zchiral field...

18 Mar 2010 NFQCD10

BEC-BCS crossover in NJL model; towards QCD phase diagram & role of fluctuationsHiroaki Abuki

(Tokyo Science University)

Many thanks to:

Gordon Baym, Tetsuo Hatsuda, Naoki Yamamoto& Tomas Brauner

Plan

BEC-BCS crossover with axial anomaly

Notion of BEC-BCS crossoverIn general relativistic systems?Axial anomaly induced BEC-BCS transition(1)

How can we deal with fluctuations?

1/N expansion vs. Nozieres-Schmitt-RinkApplied to Nonrelativistic Fermi gas(2)

CFL/2SC models of color superconductor(2)

(1) HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408(2) HA, T. Brauner, PRD78, 125010 (2008)

What is the BEC-BCS crossover?

(1) D. M. Eagles, Phys. Rev. 186, 456 (1969) (2) A. J. Leggett, J. Phys. C 41, 7 (1980)(3) Nozières and Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985)

What is the BEC-BCS crossover?

1F sk a

c

F

TE

0Weak Strong

BCS

depends exponentiallyon coupling

/ 2 | |/ ~ π− F sk ac FT E e

1957

1924

Naïve application of BCSleads power law blow up

( )naive

2 2 2 2

1/ ~log 2 /c F F s F s

T Ek a k a

Eagles (1969), Leggett (1

980)

Nozieres & Schmitt-Rink

(1985) …

BEC

“universal value”not depending

on coupling

/ ~ 0.2314..c FT E/ 1dξ

1/3d N −=( ) 1/32 / 2

2T Bd N

mTπλ −= ≈ =

1/3 2 /3,F Fk n E n∝ ∝ Unitarity limit

Crossover observed in UCA

Note: interaction is tunable via Magnetic field!K40: Regal et al., Nature 424, 47 (03), PRL92 (04): JILA groupLi6: Strecker et al., PRL91 (03): Rice group; Zwierlen et al., PRL91 (03): MIT group; Chin et al., Science 305, 1128 (04): Austrian group

K40 (JILA)

Relevant scales:

Tc ∼ 100nKTF ∼ 1µKρ ∼ 1014cm-3λ ∼ 100nm

N0/N

BCSBEC

In general relativistic systems?

What is a relativistic matter?

kF/mc∼λc/d (:=relativity parameter)

∼10−9 in cold atom systems∼10−7 in 3He∼10−2 nuclear matter (deuteron gas;

Lombardo, Nozieres, Schuck, Schulze, Sedrakian,PRC64, 064314 (2001)

In cold quark matter with CSC

kF/mc 100 relativistic enough!

.

A schematic phase diagram of QCD

∼300milion tons/cm3

CP

χSBCSC

µq

T

∼300MeV

mq ≠ 0

100

101

102

103

104

105

ξ c/d

q

103 104 105 106

qc d/ξ

Cooper pair sizevs.

inter-quark spacing

HA, T. Hatsuda, K. Itakura, PRD 65 (02)See also M. Matsuzaki, PRD62 (00) for pair size

]MeV[μ

BEC-like?

BCS-like

NJL model with axial anomaly (1)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408

〈φ〉≠0〈d〉=0 〈φ〉≠0

〈d〉≠0

〈φ〉=0〈d〉=0


Model setup (chiral limit)

(6)(4)0( )q i q LLL µγ= + +∂ +

( )( )

(4)

(4)

8 tr

2 tr R Rd L L

G

H d d d d

L

L

χ φ φ+

+ +

⎧⎪ =⎪⎪⎨⎪ = +⎪⎪⎩

Chiral fields: Diquark fields:

L

H

L

L L

L

G

L

R R

( )ij Rj Liq qφ = ( )a abc b c

L ijk Lj Lkid q Cqε ε=

For general construction of GL free energy (→ Talk: T. Hatsuda, 3/10)see also Yamamoto, Tachibana, Baym & T.H., PRL97 (06)

(r,g,b)(u,d,s)


Model setup (chiral limit)

KMT term for U(1)A breaking

K responsible for η’ mass decoupling

(6)(4)0( )q i q LLL µγ= +∂ + +

( )

( )KMT

KMT'

(6)

(6)

8 det h.c.

tr' h.c.R LK

L

L d d

K φ

φ+

⎧⎪ = − +⎪⎪⎨⎪ = +⎪⎪⎩ K

uL uR

sL sR

dL dR

/36 ( ) ( )

(1) ( ( 1) )nA L R L R

U Z q q±→ → −


Favorable mean fields

Chiral: 〈φij〉 = δij (χ/2) (V-singlet, 0+)

CFL: 〈dLaj〉 = −−〈dRaj〉 = δaj (s/2)(for K’ >0 ) (V+C-singlet, 0+)

Spontaneous symmetry breaking in CFLAlford, Rajagopal, Wilczek, NPB99

(3) (3) (3) (1) (1)C L R B ASU SU SU U U× × × ×

2(3)L R CSU Z+ +→ × Z6


Dirac/Mojorana mass formations

.

Lots of work done:

(with variety of CSC phases)Ruester et al, (05)Blaschke et al., PRD72 (05)

(+vector interaction)Zhang, Fukushima, Kunihiro, PRD79 (09)Zhang, Kunihiro, PRD80 (09)

etc…See for a review:Buballa, Phys. Rept. 407 (05)


〈φ〉=0〈d〉=0

Pisarski-Wilczek (‘84)

〈φ〉≠0〈d〉=0 〈φ〉=0

〈d〉≠0

K’=0 (no interplay b/w chiral and super)

(G,K,H):M. Buballa, Phys. Rept. 407 (05)


〈φ〉≠0〈d〉=0 〈φ〉≠0

〈d〉≠0

K’ = 4.2K0 (turning on the interplay)

〈φ〉=0〈d〉=0

Talk; T. Hatsuda 3/10,Hatsuda, Tachibana, Yamamoto, Baym (06)

For another possible realization:Talk; T. Kunihiro 3/10,Kitazawa, Kunihiro, Koide, Nemoto (02)Zhang, Kunihiro (09, w Fukushima 08)


〈φ〉≠0〈d〉=0 〈φ〉≠0

〈d〉≠0

m = 5.5 MeV (turning on quark mass)

〈φ〉≈0〈d〉=0

Critical Point (CEP)Asakawa, Yazaki (89)Chiral sym. is onlyapproximate one


Chiral transition; dependence on (K’,K)

Effect of K’ :• weaken the chiral

transition

Effect of K :• strengthen 1st order

chiral transition

Effect of mq :• helps to weken the

chiral transition

K/K0

K’/

K0

5

4

3

2

1

03210

1st order

crossover

K'c

m q=0

5

4

3

2

1

03210

1st order

crossover

m q=

5 MeV

mq= 1

40 Me

V


〈d〉≠0

BEC-BCS crossover in the CFL (COE)?

Transitions to theCFL (COE) phase :

• Both are 2nd order

• Both singal U(1)Bbreaking and arecharacterized byThouless condition

• However, differentphysical origins


Development of precursory to CSC:Kitazawa, Kunihiro, Koide, Nemoto (02)

BEC of bound diquarks:Nishida,Abuki (PRD05, 07)Kitazawa,Rischke,Shovkovy (08)

Mass of diquark hitschemical potential!

ω =ΜD−2µ = 0µ=Mq


Effect of axial anomaly on (qq) attraction

Neglecting fluctuation in chiral field

KMT’ term increases (qq)-attraction in the positive parity channel

' (1/ 4) 'H HH K Hχ→ = + >

( )KMT'(6) t' r R L R LL dK d d dφ φ+ + += +

( )( )( )( )( )( )

KMT'(6) (1/4) ' tr

(1/4) ' tr

R L R L

R L R L

L K d d d d

K d d d d

χ

χ

+ +

+ +

− − −

+ + +


µ [MeV]

K’/

K0

How robust is theBEC-like CFL?

• K’ increases, thenBEC sets in at somecritical K’ (Q)

• This happens beforethe chiral transitiongets crossover (P)

• Between (Q) and (P)transition from BECto BCS is 1st order!

Q

P

Summary of 1st partRich phase diagram due to axial anomalyand its coupling to chiral/diquark fields

Chiral crossover

Diquark field contributes as if it is an external field for chiral transitionLow-T critical point appears!

BCS-BEC crossover

Chiral field behaves as if it is a mediating fieldthat increases (qq)-attraction

GL prediction: Hatsuda, Tachibana, Yamamoto, Baym (06)

c.f. Crossover by increasing H: Nishida, Abuki (05,07)NJL phase diagram: Kitazawa,Rischke,Shovkovy (08)

Plan

BEC-BCS crossover with axial anomaly

Notion of BEC-BCS crossoverIn general relativistic systems?Axial anomaly induced BEC-BCS transition(1)

How can we deal with fluctuations?

1/N expansion vs. Nozieres-Schmitt-RinkApplied to Nonrelativistic Fermi gas(2)

CFL/2SC models of color superconductor(2)

(1) HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408(2) HA, T. Brauner, PRD78, 125010 (2008)

Introduction of 1/N expansionFluctuations are obviously important in strong coupling (unitary) regime

1/N is a method to take into account fluctuations about MF in a systematic, controlled way

X Nikolic, Sachidev, PRA75 (2007) 033608 (NS)1. TC at unitarity

X Veillette, Sheehy, Radzihovsky, PRA75 (2007) 043614 (VSR)2. TC at unitarity3. T=0 off the unitarity

X Abuki, Brauner, PRD78, 125010 (2008) (AB)4. TC off the unitarity

1/N expansion (1)Euclidian lagrangian for ψ = (ψ↑, ψ↓)T

Introduce N “flavors” of spin doublet:

( )( )2 2

*2 2 ,2 4

TGLmτ

ψ µ ψ ψ σ ψ ψ σ ψ+ +⎛ ⎞∇ ⎟⎜= ∂ − − −⎟⎜ ⎟⎟⎜⎜⎝ ⎠

( )( )2 2

*2 22 4

TN N

GLNmτα α

ψ µ ψ ψ σ ψ ψ σ ψ+ +⎛ ⎞∇ ⎟⎜= ∂ − − −⎟⎜ ⎟⎟⎜⎜⎝ ⎠

( )1 1 2 2( , ) , , , ,..., ,NT

TNαψ ψ ψ ψ ψ ψ ψ ψ ψ ψ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓= → =

2 2 2, 1T

N N NU U U Uσ σ+= = : Sp(2N,R) ⊂ SU(2N)

1/N expansion (2)

SU(2) singlet Cooper pairSp(2N,R) singlet pairing field

No additional symmetry breaking No unwanted NG bosons There is only physical one, the Anderson-Bogoliubov mode associated with correct U(1) (total number) breaking

21

( ) ( ) ( )2

NTGx x

Nxα α

αφ ψ σ ψ

=∑∼

1/N ordering of correctionsBosonized action

For systematic handling, setting

Thermodynamic potential at NLO

[ ]TrLog2

1/1

0

( , )( , )

TN

xS d dx D x

Gφ τ

τ φ τ−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦∫ ∫

, 0

( , ) ( , )1 i ipxp

x p eN

ωτ

ωφ τ δφ ω − +

≠

= ∆ + ∑ Bosonic fluctuation

Fermion one loop:Mean Field app.

Boson one loop:quantum effect

LO ∼ O(1) NLO ∼ O(1/N)

(1/(0) )1 ( , , ,1/ ) 1 ...NsF NT k a

Nµ δΩ ∆ = Ω +Ω+

Coupled equations to be solved

Equations that have to be solved:

And similar equations of (∆0 , µ0) at T=0Self-consistent solution? … Dangerous!

Gapless-conserving dichotomy (Violation ofGoldstone theorem or Conservation law)

(1/ )

(1/

(0)

2 20 0

3(0)

2

)

1 0

13

N

NF

N

kNµ

δπ

δ

µ

∆= ∆=

⎧⎪∂Ω ∂⎪ + =⎪⎪ ∂Ω

Ω

∆ ∂∆⎪⎪⎨⎪∂Ω ∂⎪⎪ + = −⎪⎪ ∂ ∂⎪⎩

(TC , µC)

X Haussmann et al, PRA75 (2007) 023610X Strinati and Pieri, Europphys. Lett. 71 359 (2005)X T. Kita, J. Phys. Soc. Jpn. 75, 044603 (2006)

Order by order expansion (NS)

Want to solve:

Expand also

LO equation …

… NLO equation

(0) (1/ )1( ) ( ) ... 0NF x F xN

+ + =

(0(0) )( ) 0F x =

0

(0)(1/ )( )

01/1 1 ( ) 0

x

N

x

N dF xFN dx

xN=

+ =

(1 )(0) /1 ...NxxN

x= + +

no dangerous propagator inside!

Comparison with NSR

Nozieres-Schmitt-Rink theory

Missing 1/N correction to Thouless criterionSo no problem in solving two equations selfconsistently in (µ(kF), T(kF))1/N term in number eq. dominates in the strong coupling and recovers TC in BEC limitThe phase diagram in (µ, T)-plane unaffected: Only the bulk (µ, kF)-relations affected

2

(0)

(0)

03

20

(1/ )

( , ) 0

1( , ) ( , )3

C

FC C

N

T

kT T

Nµ µ

µ

µ µπ

δ

∆ ∆=

∆=

⎧⎪∂ =⎪⎪⎪⎪⎨⎪⎪∂ + ∂ = −⎪⎪⎪Ω

⎩

Ω

Ω

2(1/

0

)1 ( , )N CTNδ µ

∆ ∆=Ω+ ∂

The results: UnitarityNS VSR

Corrections are large! 1/N corrections to (βC, βCµC) and (TC, µC) formally equivalent at large N, but not at N=1!

5.3172.014

2.7851.504CC

F

C

EN

N

T

Tµ

⎧⎪⎪ = +⎪⎪⎪⎪⎨⎪⎪ = +⎪⎪⎪⎪⎩

1.3100.496

0.5801.747

C

C

F

F

E N

E N

T

µ

⎧⎪⎪ = −⎪⎪⎪⎪⎨⎪⎪ = −⎪⎪⎪⎪⎩

T=00

0 0.1630.686

0.3120.591

F

F

E N

E Nµ

⎧⎪⎪ = −⎪⎪⎪⎪⎨⎪⎪ = −⎪⎪⎪⎪⎩

∆

Results: TC, µC off the unitarity

1/N to βC (1/TC)

1/N to TC

NSR LO (MFA)

• Chemical potential in the BCS limit coincides with perturbativeresult: Reproduces second-order analytic formula

c.f. Fetter, Walecka’s textbook

1st

2nd

0.23

BEC BCS

( )224(11 2 log2)

15 Fk a

π−+

413 sFF

k aEµ

π= +

BEC BCS

Summary of 1/N to Nonrelativistic Fermi gas

Extrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion

TC useless at unitarity, even negative!Only qualitative conclusion: fluctuation lowers TC

On the other hand, β=1/TC-based extrapolation yields TC/EF=0.14, close to MC result 0.152(7):E. Burovski et al., PRL96 (2006) 160402

Expansion about MF fails in molecular BEC regime

1/N expansion may still give useful prediction in the BCS region

1/N expansion in dense, relativistic Fermi system: Color superconductivity

Impact of fluctuation on the (µ, Τ)-plane?Several strong coupling effects discussed:

M. Matsuzaki (00), Abuki-Hatsuda-Itakura (02), Kitazawa, Koide, Kunihiro, Nemoto (02)Kitazawa, Rischke, Shovkovy (08)

NSR scheme also applied: no change in (µ, Τ)-phase diagram:

Nishida-Abuki (05), Abuki (07)

Are fluctuation effects different for several pairing patterns? (2SC, CFL, etc…)

Lets try 1/N expansion!

1/N expansion in dense, relativistic Fermi system: Color superconductivity

take the simplest NJL (4-Fermi) model

Several species but with equal mass, equal chemical potentialqq pairing in total spin zero, positive parityFor color-flavor structure:

2SC:

CFL:

( ) ( )( )5 5014

BNC CGL q i m q q F q q F qα α

αµγ γ γ +

=

= ∂ + − + ∑

[ ] 1 ( , , )2

ij ija bc abcF ra g bε ε= =

1 (( ( , ),...,( , ))2

, )jk ijk

ai abcbcF u r sa i bε ε⎡ ⎤ = =⎣ ⎦

3 diquark “flavors”

9 diquark “flavors”

Introduce a new flavor “taste” of quarks q qA (A=1,2,3,…,N)

SU(3)C×(flavor group)×SO(N)Assume SO(N)-singlet pair field, then

No unwanted NG bosons. We make a systematic expansion in 1/N and set N=1 at the end of calculation

5( , ) 2a AC

AA

aGx q qN

Fφ τ γ= ∑

( ) ( )( )5 50 4 B BA A AC C

a aa

A

GL q i m q q F q q F qN

µγ γ γ += ∂ + − + ∑

1/N expansion to shift of TCInterested in shift of TC in (µ,T)-plane:Not interested in bulk (µ, ρ)-relation since density can not be controlled(µ is fundamental in equilibrium)

Approach from normal phase T TC +0

Then consider Thouless criterion alone

(0) 1 (1/ ) 11( 0) ( 0) 0NB BG P G PNµ µδ− −= + = =

Pair fluctuation becomes massless at TC

2 2

( 1/ )0) (( , ) ( , ) 0NN T Tδ µµ∆ ∆

Ω∂ Ω + ∂ =

1/N expansion to inverse boson propagator, NLO Thouless criterion

Boson propagator at LO:

NLO correction: Boson self energy

(0) 1 (1/ ) 11( 0) ( 0) 0NB BG P G PNµ µδ− −= + = =

Pµ

φaχpair : Cooperon

(1/ ) 11 ( 0)NBG PN µδ −= =

(0) 1( )1 ( )B

GN

G PG Pµ µχ

= ≡− pair

(0)( ) 1/BG Q Nµ ∼

Pµ=0

LO ∼ O(N) NLO ∼ O(1)

∼ O(1)

N∼vertex:

∼ O(1/N)

Color-Flavor structure of vertexes in loop gives

Information of color/flavor structure of pairing pattern condenses in simple algebraic factor NB/NF

NLO correction to boson self energy

0

(1/ ) 1 (0)

2

3 0+,

1 ( 0) ( ) ( )

( ) ( , , , ; )(2 )

NB B

Q nT

e e

B

efe f

F

G P dQNG Q F Q

dF Q I e f Q

NNN µ π

δ

ξ ξπ

−

=

=±

= =

≡ +

∑ ∫

∑ ∫ k k qk k k q

Tr BF

NF F F F

Nα γγδ β δ αβδ δ+ +⎡ ⎤ =⎢ ⎥⎣ ⎦∼γδ

α

(0)( )BG Qµ γδδ

β

Qµ

NLO fluctuation effect in CFL is twice as large as 2SC one

Mean field Tc’s split at NLO

Phase dependent algebraic factor

199CFL

1/2632SC

111“BCS”

NB/NFNFNBpairing

Caveat: NLO integral badly divergentTake the advantage of HDET (µ ∆)

Only degrees of freedom close to Fermi surface are relevant for pairing physicsRenormalize the bare coupling G in favor of mean field Tc(0) : TC(0)/µ parameterize the strength of coupling

In the weak coupling limit TC(0)=0.567∆0(0)

High density approximation

(1)2

(0)11 (0)

2B

F

CNLO

FB

NN N

fk

GTµ

δπ µ

− ⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠

(0) 12 (0)(0) log2F

BC

kG N T

Tµπ

− ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠∼O(N): textbook

material

∼ O(1)

f NLO

TC(0) /µ

Numerical result for NLO function(0)

(0)(0) (0) 1

CN

B

FLO

NN B

Tf

CC C C NL

N

FO

TT T e T

Nf

NNµ

µ

⎛ ⎞⎟⎜ ⎟⎜ ⎟− ⎜ ⎟⎜ ⎟⎟⎜⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜= ≈ − ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎟⎜⎜ ⎟⎜ ⎝ ⎠⎝ ⎠

strong

Fluctuation suppressesTC significantly

Phenomenologicallyinteresting couplingregion

15-30% supressionof critical TCweak

Implication to QCD phase diagram

Suppression of TC is phase dependent: CFL TC is more suppressed than 2SC one

There may be afluctuation driven2SC windoweven when Ms=0

Suppression of TC is order of 15-30% :

Non-negligible!

T

µ

mu,d,s = 0

CFLNG

NOR

2SC

Tc(MF)

Summary of 2nd part

1/N expansion to TC at/off the unitarityAvoids problems with self-consistency

Only reliable when the NLO corrections are small (in BCS, not in molecular BEC region)

Color superconducting quark matterFluctuation corrections non-negligible

Suppressions of TC different according to pairing patterns Various phases!

OutlookImprovement: Color neutrality, etc.

Outlook (for 1st part of talk)General Ginzburg-Landau (GL) analysis

What if mu,d ms ? Neutrality condition?

Coupling with the density

Full NJL analysiswith flavor asymmetric matter with vector interaction & KMT term & neutralityPolyakov loop? PNJL

Nucleon (b-f) formation due to diquark-quark residual interaction

c.f. talk: T. Kunihiro 3/10, Zhang, Kunihiro (09)Zhang, Kunihiro, Fukushima (08)

c.f. Fukushima, PLB04, Ratti, Thaler, Weise, PRD06Roessner, Ratti, Weise, PRD07, Abuki, Ruggieri et al., PRD08

c.f. talk: T. Hatsuda 3/10, Maeda,, Baym, Hatsuda, PRL103 (09)

bec-bcs crossover in njl model; towards qcd phase diagram ... · bcs-bec crossover zchiral field...

Documents