1/N expansion for strongly
correlated quantum Fermi gas
and its application to quark matter
Hiroaki Abuki
(Tokyo University of Science)
Tomas Brauner
(Frankfurt University)
Based on PRD78, 125010 (2008)
27 May 2009, @Komaba
Outline

Introduction

Nonrelativistic Fermi gas



Dense relativistic Fermi gas




Formulation
Results
Nambu-Jona Lasinio (NJL) description
High density approximation
Results
Summary
Introduction

Cold atom system in the Feshbach
resonance attracts renewed interests on
the BCS/BEC crossover:
Leggett(80), Nozieres Schmitt-Rink(85)

Interaction tunable via Magnetic field!!

K40 , Li6 atomic system in the laser trap
Regal et al., Nature 424, 47 (2003): JILA grop
Strecker et al., PRL91, (2003): Rice group
Zwierlen et al., PRL91 (2003): MIT group
Chin et al., Science 305, 1128 (2004): Austrian group
…etc, etc…
From: Regal, cond-mat/0601054
Naïve application of BCS
leads power law blow up
naive
c
T
Tc
1
/ EF ~ 2 2
k F as log  2 / k F2 as2 
Unitary
Smooth crossover
regimeBCS/BEC:
BEC
no small
expansion
parameter
1924
no reliable
theoretical
broken
framework
symmetry
4
phase

s
k2
 strong attraction
k F  n , EF  n
1957
Eagles (1969), Leggett (1980)
Nozieres & Schmitt-Rink (1985)
Tc / EF ~ 0.2314..
1/ 3
EF
2/ 3
+1
0
-1
Unitarity limit
BCS
Tc / EF ~ e - / 2 kF |as |
1
weak attraction 
k F as
Introduction

Nonperturbative, but universal
thermodynamics at the unitarity
 Theoretical challenges to describe such
strongly correlated Fermi gas
Gas in Unitary limit: nonperturbative but with universality
X At T=0, thermodynamic quantities would have the form:
 (n, r0 , as )
n5/ 3 3
 f  r0 / as ,1/ k F as  ;  F (n) 
 nEF
 F ( n)
m 5


 x : the universal dimensionless constant
d
2 d  
 (n)  x F (n), P(n)  n
 x EF
   x PF (n),  
dn  n 
dn
X Universal, does not depend on microscopic details of the 2-body force
ex. Cold atoms, Neutron gas with n-1/3 |as(1S0)| =18 fm
X Non-perturbative information condenses in the universal parameter x
1.
Green’s function Monte Carlo simulation:
Carlson-Chang-Pandharipande-Schmidt, PRL91, 050401 (’03), x 0.441
Astrakharchik-Boronat-Casulleras-Giorgini, PRL93, 200404 (’04), x 0.421
2.
Extrapolation of infinite ladder sum in the NSR split:
H. Heiselberg, PRA 63, 043606 (‘01); T. Schafer et al, NPA762, 82 (‘05), x 0.32
3.
-expansion around 4-space dimension:
Nishida, Son, PRL97 (2006) 050403: Next-to-leading order, x 0.475
4.
Experiment:
Bourdel et al., PRL91, 020402 (’03); x 0.7 but for T/TF > 0.5 and also in a finite trap
1/N expansion applied to Fermi gas

fluctuation effects are important!

systematic, controlled expansion possible
when spin SU(2) generalized to SP(2N)
X Nikolic, Sachidev, PRA75 (2007) 033608 (NS)
1. TC at unitarity
X Veillette, Sheehy, Radzihovsky, PRA75 (2007) 043614 (VSR)
1. TC at unitarity
2. T=0, x parameter at and off the unitality
1/N expansion

In this work,
X Tc at and off the unitarity
and analytic asymptotic
behavior in the BCS limit
X Apply 1/N spirit to the
relativistic fermion system,
Possible impacts on QCD?
1/N expansion, philosophy (1)

Euclidian lagrangian
2 2
æy ­ ÷
ö
æ
ö
h
Ñ
G
ç
+
*
T
L = y + çç¶ t ­
­ m1÷
y
­
y
s
y
y
s 2y ), y = çç ÷
÷
(
(
)
÷
2
÷
÷
çè
y
ç
2m
4
ø
è ¯÷
ø

Extend SU(2)  Sp(2N) by introducing N
copies of spin doublet: “flavor”
(
y = (y ­ , y ¯ ) ® y a = y 1­ , y 1¯, y 2­ , y 2¯,..., y N ­ , y N ¯
T
T
)
2 2
æ
ö
h
Ñ
G
+ ç
+
*
T
L = y a ç¶ t ­
­ m1÷
y
­
y
s
y
y
s 2y b
÷
(
)
a
a 2 a
b
÷
çè
2m
4N
ø
(
)
1/N expansion, philosophy (2)

SU(2) singlet Cooper pair
 Sp(2N) singlet pairing field
G
f (x ) :
2N

N
å
y aT (x )s 2 y a (x )
a=1
No additional symmetry breaking, no
unwanted NG bosons other than the
Anderson-Bogoliubov associated with
correct U(1) (total number) breaking
Counting by factor of N (1)




Bosonized action
2
é 1/ T
ù
f
(
x
,
t
)
S = N êêò d t ò dx
- TrLogD - 1 [f (x , t )]ú
ú
0
G
êë
ú
û
Enables us to perform formal expansion
in 1/N
Each boson f-propagator contributes 1/N
and fermion loop counts N from the trace
factor
Equivalent to expansion in # of bosonic
loops
Counting by factor of N (2)

LO in 1/N  equivalent to MFA

NLO in 1/N  one boson loop corrections

At the end, we set N=1:
1/1 is not really small,
but at least gives a systematic ordering
of corrections beyond MFA
Pressure up to NLO (VSR)

Thermodynamic potential at NLO
1
1 (1)
(0)
WD
( , m,T ,1/ kF as ) = W +
W + ...
N
N
Fermion one loop
Boson one loop
Df

At NLO, bosons contribute


Anderson-Bogoliubov (phason), and
Sigma mode (ampliton), they are mixed
Coupled equations to be solved



Equations that have to be solved:
For T=0
For Tc
ìï 1 ¶ W
ïï
= 0
ïï N ¶ D
í
3
ïï 1 ¶ W k F
=
ïï 2
N
¶
m
3
p
ïî
ìï 1 ¶ W
ïï
= 0
2
ïï N ¶ D D = 0
í
ïï 1 ¶ W k F3
=
ïï 2
N
¶
m
3
p
ïî

ìï D 0 (1/ kF as , N )
ï
í
ïï m0 (1/ kF as , N )
ïî

ìï TC (1/ kF as , N )
ï
í
ïï mC (1/ kF as , N )
ïî
Gapless-Conserving dichotomy

Self-consistent solutions to these coupled
equations? … Dangerous!
Violation of Goldstone theorem
Universal artifact in common with “conserving”
approximation (Luttinger-Ward, Kadanoff-Baym’s
F-derivable): Well-known longstanding problem:
Gapless-conserving dichotomy
X Haussmann et al, PRA75 (2007) 023610
X Strinati and Pieri, Europphys. Lett. 71 359 (2005)
X T. Kita, J. Phys. Soc. Jpn. 75, 044603 (2006)
The way to bypass the problem:
order by order expansion



ur ur
What to be solved is of type: F ( a ) = 0
ur uur
1 uur
F = F0 +
F1 + ...
N ur uur
1 uur
a 1 + ...
We also expand … a = a 0 +
N
… to find solution order by order
uru uur
O(1):
F0 (a 0 ) = 0 (MFA)
uur
uur uur
uu
r
æ
1
ö
1
O(1/N): ç a ×Ñ ÷F
+
F1(a 0 ) = 0
a÷ 0
çèN 1
ø aur = auur N
0
Order by order expansion

Detailed form of NLO equations …
for T=0: D 0 = D
æ¶ W(0)
çç DD
çç
(0)
çè¶ D mW
(0)
0
1 (1)
1 (1)
(0)
+
D 0 , m0 = m0 +
m0
N
N
(1) ö
(1) ö
æ
æ
¶ mD W(0) ö
D
¶
W
÷
÷
çç 0 ÷
çç D
÷
÷
÷
÷
÷
÷
=
ç
ç
÷
÷
(0) ÷
(1)
(1)
ç
ç
÷
÷
÷
¶ mmW ø
m
¶
W
ç
ç
÷
÷
÷
m
è
ø
è
ø
0
T = 0,(m( 0) ,D ( 0) )
0
for Tc: TC = TC(0) +
æ¶ 2 W(0)
çç T D
çç
(0)
çè ¶ T mW
0
1 (1)
1 (1)
TC , mC = mC(0) +
mC
N
N
(0) ö
(1) ö
(1) ö
æ
æ
W
T
¶
W
÷
÷
çç C ÷
çç D 2
mD 2
÷
÷
÷
÷
÷
÷
=
çç (1) ÷
çç
÷
( 0) ÷
(
1
)
÷
÷
÷
¶ mmW ø
m
¶
W
÷
÷
÷
ç
ç
m
è
ø
( 0) ( 0) è C ø
D= 0,(T , m )
¶
C
C
Relation to other approaches (1)

Nozieres-Schmitt-Rink theory
ìï ¶ W(0) ( m,T )
= 0
ïï D 2
C D= 0
ï
3
í
k
1
ïï
( 0)
(1)
F
­
¶
W
(
m
,
T
)
­
¶
W
(
m
,
T
)
=
m
m
2
ïï
C
C
N
3
p
D= 0
ïî




1/N correction to Thouless criterion missing
Not really systematic expansion about MF:
Solve the number equation in , T) nonperturbatively in 1/N
1/N (NLO) term in # eq. dominates in the strong
coupling and recovers the BEC limit
The phase diagram in , T)-plane unaffected: Only
affects the equal density contours in the , T)-plane
Relation to other approaches (2)

Haussmann’s self-consistent theory
besed on Luttinger-Ward formalism
ìï
1
ïï ¶ 2 W(0) ( m,T C ) +
¶ 2 W(1) ( m,T C )
= 0
D
D
ïï
N
D= 0
í
3
ïï
k
1
¶ mW(1) ( m,T C )
= ­ F2
ïï ¶ mW( 0) ( m,T C ) +
N
3p
ïî
D= 0



1/N correction to thouless criterion included
Solve the coupled equations self-consistently
Leads several problems related to “gaplessconserving dichotomy”: LO pair propagator
gets negative “mass” even above Tc
 Negative weight to partition function!
X Haussmann et al, PRA75 (2007) 023610
The results: Unitarity
NS
ìï E F
5.317
ïï
= 2.014 +
ï TC
N
ïí
ïï mC
2.785
=
1.504
+
ïï
N
ïî T C


VSR
ìï T C
ïï
ï EF
ïí
ïï mC
ïï
ïî E F
ìï D 0
ïï
ïï E F
T=0 í
ïï m0
ïï
ïî E F
= 0.496 -
1.310
N
= 1.747 -
0.580
N
= 0.686 -
0.163
N
= 0.591 -
0.312
N
1/N corrections to (TC, C), formally equivalent,
but they are large!
Corrections are a bit smaller at T=0
The results: Off the unitarity at T=0
from VSR
m0
EF
D0
Monte Calro results at
unitarity are located
between MF(LO) and
the NLO result
x(MF)=0.5906 (Leggett)
x(MC)=0.44(1) (Carlson)
x(1/N)=0.28 (VSR)
1/N corrections seem
to work at least
in the correct direction
MF : 0.6864
MC : 0.54
1/N : 0.49
But the obtained value
x0.28 not satisfactory
EF
BCS
BEC
Monte Calro:
Carlson et al, PRL91 (2003)
1/ kF as
Mid-Summary






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 lower TC
1/TC-based extrapolation yields TC/EF=0.14, close
to MC result 0.152(7): E.Burovski et al., PRL96 (2006) 160402
b is natural parameter? Needs convincing
justification!
Expansion about MF fails in BEC
We may, however, expect that 1/N expansion still
gives useful prediction in the BCS region
Result for TC : Off the unitarity
2nd
0.218
NSR
LO (MFA)
1st
1/N to bC (1/TC)
1/N to TC
• TC reduced by a constant factor in the BCS limit!
• Chemical potential in the BCS limit governed by perturbative
corrections: Reproduces second-order analytic formula
2
m
4
4(11 - 2 log 2)
= 1+
k F as +
kF a )
(
2
EF
3p
15p
c.f. Fetter, Walecka’s textbook
Why 1/N reproduces perturbative ?
g0, O(N)
g, O(1)
g2, O(1)
g2, O(1/N)
(a) is LO in 1/N
(b)(c) included in RPA (NLO in 1/N)
(d) is NNLO not included here, but this is zero
What is the origin of asymptotic offset
in TC then?
æ¶ 2 W(0)
çç bD
çç
(0)
¶
W
çè bm
(0) ö
(1) ö
(1) ö
æ
æ
W
b
¶
W
÷
÷
çç C ÷
çç D 2
mD 2
÷
÷
÷
÷
÷
÷
=
çç (1) ÷
çç
÷
( 0) ÷
(
1
)
÷
÷
÷
¶ mmW ø
m
¶
W
÷
÷
÷
ç
ç
m
è
ø
( 0) ( 0) è C ø
D= 0,(T , m )
¶
C
C
Weak coupling analytical evaluation possible in the deep BCS
(0)
æ
æ p
öö
3
nT
3
n
÷
çç
÷
ç
C
÷
ç
- 1÷÷
÷
çç - 4E
2 ç
÷
÷
ç
4
E
4
k
a
÷
è
ø
s
çç
F
F
F
÷
-
÷
çç 2
÷
÷
(0) 3
÷
ççp n (T C )
3n
÷
÷
çç
»
0
­
÷
2
÷
çèç 4E F
2E F
÷
ø
÷
(1)
(1)
(0)
(1)
¶
W
+
m
¶
W
2
2
bC
4E F
C
D
mD
(1)
=
­
=
¶
W
+
2
(0)
(0)
D
bC
¶ 2W
3n
bD
The BCS limit:
kFas  -0
Pair (fluctuation)
propagator extremely
sensitive to variation
of 
æ1 9 + 2 log 2
ö
çç ­
k F as ÷
÷
÷
è3
ø
5p
Singularity in D2W and slow convergence of C to EF responsible!
1/N expansion in dense, relativistic
Fermi system, Color superconductivity

Motivation

What is the impact of pair fluctuation on , Tphase diagram?
In the NSR scheme, only the , r-relation gets
modified: No change in , T-phase diagram
see, Nishida-Abuki, PRD (05), Abuki, NPA (07)

Are fluctuation effects different for several
pairing patterns?
1/N expansion in dense, relativistic
Fermi system: Color superconductivity

take NJL (4-Fermi) model
G
L = q (i ¶ + mg 0 ­ m )q + å qC g5Qaq (qg5Qa+ qC )
4 a
(



)
Several species with equal mass, equal
chemical potential
qq pairing in total spin zero,
Arbitrary color-flavor structure:
Different fluctuation channels
1 ij
2SC: [Qa ]bc = e eabc (a = r , g, b) 3 diquark “flavor”
2
jk
1 ijk
é
ù
CFL: ëQai û = e eabc ((a, i ) = (u, r ),...,(s, b))
bc
2
ij


9 diquark “flavor”
Economical way to introduce expansion
parameter N possible?

What about extending NC=3 to NC=N?


However, diquark is not color singlet 
Full RPA series not resummed at any finite
order in 1/N unless coupling  O(1)
If coupling scales as O(1), the expansion in
1/N will not be under control
This type of planer (ladder) graph
will have growing power of N
With # of loops!
No way but to introduce new “flavor”,
taste of quarks

q  qi (i=1,2,3,…,N)
G
L = qi (i ¶ + mg 0 ­ m )qi +
4N
å (
a

Lagrangian has SU(3)CSO(N)(flavor group)

Assume SO(N)-singlet Cooper pair, then
fa =


G
2N
å
)
qiC g5Qaqi (qj g5Qa+ qCj )
qiC g5Qaqi
i
No unwanted NG bosons other than AB mode
We make a systematic expansion in 1/N and set N=1 at
the end of calculation: Expansion in bosonic loops:
Construction is general, can be applied to any pattern
of Cooper pairing
1/N expansion to shift of TC



Only interested in shift of TC in (,T)phase diagram
Not interested in (, r)-relation here since
the density can not be controlled:  is
more fundamental quantity in equilibrium
Then consider Thouless criterion alone
1
(0)
¶ 2 W ( m,T ) +
¶ 2 W(1) ( m,T ) = 0
D
N D
Pair fluctuation becomes massless at TC
(0)
B
- 1
G (Pm = 0)
1
+
dG B(1) (Pm = 0)- 1 = 0
N
1/N expansion to inverse boson
propagator, NLO Thouless criterion
(0)
B
- 1
G (Pm = 0)
LO  O(N)

1
+
dG B(1) (Pm = 0)- 1 = 0
N
NLO  O(1)
Boson propagator at LO:
G
1
G (Pm) =
º
N 1 - G c pair (Pm)
(0)
B
cpair : Cooperon

fa
P
NLO correction to boson self energy
GB(0) (Qm) : G / N
vertex: : N
1
dG B(1) (Pm = 0)- 1 =
N
P=0
 O(1)
NLO correction to boson self energy
G B(0) (Qm)dcd
flavor-structure of the graph gives
1
dG B(1) (Pm = 0)- 1 =
N
=
NB
NF
ek =
å
Q0
d
c
a
ur
( 0)
dQ
NG
(Q ) å
ò
B
e, f = ±
b
dk
ò (2p )3
k 2 + m 2 , xke = ek + e m, I (a, b;Q 0 ) =

: dcd Tr éëQaQb+QcQd+ ù
û=
NB
NF
dab
é
ù
2
m
+
k
×
(
k
+
q
)
ê1 + ef
úI (e xe , f xe ;Q )
ê
ú k k+q 0
e
e
êë
ú
k k+q
û
1
8a 2
t anh
ba
bb
ba
+ t anh
- b a cosh - 2
2
2
2 + ..
Q0 + b + a
Information of color/flavor structure of pairing
pattern condenses in simple algebraic factor NB/NF
Pairing pattern dependent algebraic
factor

Information of flavor structure of pairing pattern
condenses in simple algebraic factor NB/NF
pairing
NB
NF
NB/NF
“BCS”
1
1
1
2SC
3
6
1/2
CFL
9
9
1

NLO fluctuation effect in CFL is twice as large as 2SC

Mean field Tc’s split at NLO
High density approximation

NLO integral badly divergent

Then take advantage of HDET




In the far BCS region, the pairing and Fermi
energy scales are well separated
Only degrees of freedom close to Fermi surface
are relevant for pairing physics
We want to avoid interference with irrelevant
scales, in particular all vacuum divergences
We can renormalize the bare coupling G in
favor of mean field gap D0 or mean field Tc(0)
1/N correction to Tc, final result

In this framework
(0)
B
- 1
G (0)
æmk F ÷
ö
TC
= N çç 2 ÷
log (0)
÷
çè 2p ø T
C
NB
1
(1)
- 1
dG B (0) =
N
NF


In the weak coupling limit TC(0)=0.567D0
Use TC(0)/ as parameter for coupling strength
1
dG (0) +
dG B(1) (0)- 1 = 0
N
(0) öö
æ 1 N
æ
T
÷
ç
(0)
ç
B
C ÷
÷
÷
ç
gives T C = T C exp çfNLO çç
÷
÷
÷
çè m ø÷
÷
çè N N F
ø
(0)
B

æT C ö÷
æmk F ÷
ö
çç ÷
çç
f
÷
çè 2p 2 ÷
÷
ø NLO çè m ø÷
Numerical results for universal function
TC » T
(0)
C
(0) öö
æ
T
çç1 ­ N B f æ
÷÷
çç C ÷
÷
÷
÷
çç
NLO ç
÷
ç
N
N
m
÷
è
ø÷
è
ø
F
fNLO
( 0)
æT ( 0) ÷
ö
T
fNLO ççç C ÷
» 2.8 C
÷
çè m ÷
m
ø
strong
weak
TC(0)/
Fluctuation suppresses
TC significantly
Suppression of order of
30% at
phenomenologically
interesting coupling
strength
TC(0)
m
: 0.1
Implication to QCD phase diagram

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Suppression of TC is phase dependent:
CFL TC is more suppressed than 2SC one
Schematic phase diagram:
There is quantum-fluctuation
driven 2SC window even if
Ms=0 is assumed.
Suppression of Tc is order
of 10% : Non-negligible
Summary
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General remarks on 1/N expansion
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Perturbative extrapolation based on MF values of D, , T, …
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Avoids problems with self-consistency, technically very easy
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Only reliable when the NLO corrections are small (in BCS, not
in molecular BEC region)
Efimov-like N-body (singlet) bound state can contribute? If
yes, at which order of N?
Color superconducting quark matter
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Fluctuation corrections non-negligible
Different suppressions in TC according to pairing pattern 
competition of various phases
Improvement necessary: Fermi surface mismatch, Color
neutrality, etc.
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Generalization below the critical temperature
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Application to pion superfluid, # of color is useful