1N expansion for strongly correlated quantum Fermi gas and its application to quark matter

1
1/N expansion for strongly correlated quantum
Fermi gasand its application to quark matter

• Hiroaki Abuki
• (Tokyo Rikadaigaku)
• Tomas Brauner
• (Frankfurt University)
• Based on PRD78, 125010 (2008)

2
Outline

• Introduction
• Nonrelativistic Fermi gas
• Formulation
• Results
• Dense relativistic Fermi gas
• Nambu-Jona Lasinio (NJL) description
• High density approximation
• Results
• Summary

3
Introduction

• Cold atom system in the Feshbach resonance
  attracts renewed interests
• on the BCS/BEC crossover
• Eagles (69), Leggett (80), Nozieres Schmitt-Rink
  (85)
• Interaction is tunable via Magnetic field!
• K40 , Li6 atom system in laser trap
• K40 Regal et al., Nature 424, 47 (03), PRL92
  (04) JILA grop
• Li6 Strecker et al., PRL91, (03) Rice group
• Li6 Zwierlen et al., PRL91 (03) MIT group
• Chin et al., Science 305, 1128 (04) Austrian
  group
• etc, etc

4
Naïve application of BCS leads power law blow up
Unitary regime no small expansion parameter no
reliable theoretical framework
BEC
1957
Smooth crossover BCS/BEC Eagles (1969), Leggett
(1980) Nozieres Schmitt-Rink (1985)
BCS
broken symmetry phase
1 0 -1 ? strong attraction
weak attraction ?
5
Gas in Unitary limit nonperturbative but
universal
X At T0, energy density takes the form
X not depending on microscopic details of the
2-body force ex. Cold atoms, Neutron gas
n-1/3 ?as(1S0) ?18 fm
X Nonperturbative information condenses in the
parameter x

• Greens function Monte Carlo simulation
• Self-consistent T-matrix (F-derivable approach)
• e-expansion about 4-space dimension
• Experiment

Carlson-Chang-Pandharipande-Schmidt, PRL91,
050401 (03), x 0.44(1) Astrakharchik-Boronat-Cas
ulleras-Giorgini, PRL93, 200404 (04), x 0.42(1)
Haussmann et al, PRA75 023610 (07) x 0.36
Nishida, Son, PRL97 (2006) 050403
Next-to-leading order, x 0.475
Bourdel et al., PRL91, 020402 (03) x ?0.7 but
T/TF gt 0.5 and also in a finite trap
6
1/N expansion applied to Fermi gas

• fluctuations about MF 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. T0, x parameter at and off the unitality

7
1/N expansion (This work)

• In this work,
• X Tc at and off the unitarity
• and analytic asymptotic behavior
• in the BCS limit
• X Apply 1/N technique to
• relativistic fermion system,
• Possible impacts on QCD phases?

8
1/N expansion, philosophy (1)

• Euclidian lagrangian
• Extend SU(2) ? Sp(2N) by introducing N copies of
  spin doublet flavor

Sp(2N,R) ? SU(2N)
9
1/N expansion, philosophy (2)

• SU(2) singlet Cooper pair
• ? Sp(2N) singlet pairing field
• No additional symmetry breaking, no unwanted NG
  bosons other than the Anderson-Bogoliubov
  associated with correct U(1) (total number)
  breaking

10
Counting by factor of N (1)

• Bosonized action
• Enables us to perform expansion in 1/N
• Each boson f-propagator contributes 1/N and
  fermion loop counts N from the trace
• Equivalent to expansion in of bosonic loops

11
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 N1
• 1/1 is not really small,
• but at least gives a systematic ordering
• of corrections beyond MFA

12
Pressure up to NLO (VSR)

• Thermodynamic potential at NLO
• At NLO, bosons contribute
• Anderson-Bogoliubov (phason-dominant)
• Sigma mode (ampliton-dominant)

LO ? O(1) NLO ? O(1/N)
Fermion one loop
Boson one loop quantum effect
D?f?
13
Coupled equations to be solved

• Equations that have to be solved
• For T0
• For Tc

?
?
14
Gapless-Conserving dichotomy

• Self-consistent solutions? Dangerous!
• Violation of Goldstone theorem
• Artifact in common with conserving approx.
• (Luttinger-Ward, Kadanoff-Bayms F-derivable)
• Linked to longstanding problem in interacting
  Bose system called
• 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)
15
The way to bypass the problem order by order
expansion

• Want to solve
• Expand also
• LO equation
• NLO equation

no dangerous propagator inside!
16
Order by order expansion

• Detailed form of NLO equations
• for T0
• for Tc

17
Relation to other approaches (1)

• Nozieres-Schmitt-Rink theory
• 1/N correction to Thouless criterion missing
• Solve the number equation in (m, T)
• 1/N (NLO) term in number eq. dominates in the
  strong coupling and recovers TC in the BEC limit
• The phase diagram in (m, T)-plane unaffected
  Only the equal density contours in the (m,
  T)-plane affected

18
Relation to other approaches (2)

• Haussmanns self-consistent theory
• 1/N correction to thouless criterion included
• Solve the coupled equations self-consistently
• Leads problems related to gapless-conserving
  dichotomy LO pair propagator gets negative
  mass even above Tc ? Negative weight in
  partition function!

X Haussmann et al, PRA75 (2007) 023610
19
The results Unitarity

• NS VSR
• 1/N corrections to (TC, mC), formally equivalent,
• but they are large!
• Corrections are a bit smaller at T0

T0
20
The results Off the unitarity at T0

• from VSR

Monte Calro results at unitarity are
located between MF(LO) and the NLO result 1/N
corrections seem to work at least in the
correct direction But the obtained value x0.28
not satisfactory
x(MF)0.5906 (Leggett) x(MC)0.44(1)
(Carlson) x(1/N)0.28 (VSR)
MF 0.6864 MC 0.54 1/N 0.49
Monte Calro Carlson et al, PRL91 (2003)
BCS
BEC
21
Results TC off the unitarity
2nd
LO (MFA)
1st
NSR
0.23
1/N to bC (1/TC)
1/N to TC
BCS
BEC

• 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

c.f. Fetter, Waleckas textbook
22
Why 1/N reproduces perturbative m in the deep BCS
regime?
g0, O(N)
g2, O(1)
g, O(1) Hartree term
g2, O(1/N)

• is LO in 1/N
• (c) included in RPA (NLO in 1/N)
• (d) is NNLO not included here, but vanishes

23
Origin of asymptotic offset in TC ?
Weak coupling analytical evaluation possible in
the deep BCS
The BCS limit kFas ? -0 Pair propagator extremel
y sensitive to variation of m
Singularity in ?mD2W and slow convergence of mC
to EF!
24
Mid-Summary

• Extrapolation to N1 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
• On the other hand, b1/TC-based extrapolation
  yields TC/EF0.14, close to MC result 0.152(7)
  E. Burovski et al., PRL96 (2006) 160402
• b is natural parameter? Needs more convincing
  justification!
• Expansion about MF fails in molecular BEC regime
• 1/N expansion may still give useful prediction in
  the BCS region

25
A conjectured Phase diagram of QCD

• Quark matter in NS
• Relevant scales
• Tc?100MeV?1012K
• r?1fm-3 (m ? 1GeV)
• Relativistic matter
• ?kF/mc
• 10-9 in atom system
• ?10-7 in 3He
• ?10-2 nuclear matter
• ?10 quark matter?

300milion tons/cm3
Source CBM (Compressed Baryonic Matter)
Experiment at FAIR _at_ GSI, Germany http//www.gs
i.de/fair/experiments/CBM/1intro.html
26
Phase diagram froma Nambu-Jona Lasinio Model
Low energy model for QCD Has the same symmetry as
QCD
Abuki, Kunihiro, NPA768 (06)
27
1/N expansion in dense, relativistic Fermi
system Color superconductivity

• Motivation
• What is BCS/BEC in a relativistic matter?
• ?kF/mc relativity parameter (new mass
  scale!)
• ?10-9 in atom system
• ?10-7 in 3He
• ?10-2 nuclear matter
• Impact of fluctuation on the (m, T)-plane?
• Strong coupling effects discussed
• Abuki-Hatsuda-Itakura (02), Kitazawa et al. (02)
  
• NSR scheme also applied no change in
• (m, T)-phase diagram Nishida-Abuki (05), Abuki
  (07)
• Are fluctuation effects different for several
  pairing patterns? (2SC, CFL, etc)

28
1/N expansion in dense, relativistic Fermi
system Color superconductivity

• take the simplest NJL (4-Fermi) model
• Several species with equal mass,
• equal chemical potential
• qq pairing in total spin zero,
• Arbitrary color-flavor structure
• Different fluctuation channels
• 2SC
• CFL

3 diquark flavor 651 (TltTC)
9 diquark flavor 18 8811 (TltTC)
29
Introduce taste of quarks

• q ? qA (A1,2,3,,N)
• Lagrangian has SU(3)C?SO(N)?(flavor group)
• Assume SO(N)-singlet Cooper pair, then
• No unwanted NG bosons other than AB mode
• We make a systematic expansion in 1/N and set N1
  at the end of calculation
• Construction is general, can be applied to any
  pattern of Cooper pairing

Then we repeat the same procedure in the
nonrelativistic case. First we introduce a new
quantum number taste for example. We denote the
index for this new quantum number by capital A
which runs from 1 to N. Then the largangian is
now invariant under larger group, color SU(3)
times flavor group times SO(N) associated with a
new quantum number. In order to perform the
bosonization, we then assume the boson field
which transforms as singlet with respect to new
group SO(N). Then we can avoid unwanted Goldstone
bosons alter the thermodynamics of the
system. Then we make the 1/N expansion to the
bosonized action as before. N will be set to
unity at the end of calculation.
30
1/N expansion to shift of TC

• Interested in shift of TC in (m,T)-plane
• Not interested in (m, r)-relation since density
  can not be controlled
• (m is fundamental in equilibrium)
• Approach from normal phase T?TC 0
• Then consider Thouless criterion alone

Pair fluctuation becomes massless at TC
Let me describe a bit more detail of
formulation. First, in this study we are
interested in the shift of critical
temperature in the (mu,T)-plane, and not
interested in the bulk relation between mu and
rho. Then we only consider the Thouless
criterion at Next-to-leading order. We approach
from above critical temperature. Then what we
should solve is the 1/N corrected Thouless
criterion here. This condition is nothing but the
condition at which temperature the fluctuation
propagator at zero momentum becomes gapless. So
in terms of Boson propagator, this criterion can
be written as this. The first term is the leading
order boson propagator, while 1/N correction can
be regarded as the Self energy correction to it.
31
1/N expansion to inverse boson propagator, NLO
Thouless criterion

• Boson propagator at LO
• NLO correction Boson self energy

LO ? O(N) NLO ? O(1)
? O(1/N)
cpair Cooperon
vertex
Pm0
? O(1)
32
NLO correction to boson self energy

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

Color-flavor-structure gives
g
d
a
b
33
Phase dependent algebraic factor

• Information of flavor structure of pairing
  pattern condenses in simple algebraic factor
  NB/NF
• NLO fluctuation effect in CFL is twice as large
  as 2SC
• Mean field Tcs split at NLO

34
High density approximation

• NLO integral badly divergent
• Take advantage of HDET (m?D)
• 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 can renormalize the bare coupling G in favor
  of mean field Tc(0)
• TC(0)/m parameterize the strength of coupling

35
1/N correction to TC, final result

• In this framework
• In the weak coupling limit TC(0)0.567D0(0)
• Use TC(0)/m as parameter for strength of coupling
• gives

• O(N) textbook
• material
• ? O(1)

at T TC
36
Numerical results for universal function
Fluctuation suppresses TC significantly Phenomen
ologically interesting coupling gives
fNLO

• 15-30
• supression
• of critical
• temperature

weak
strong
TC(0)/m
37
Implication to QCD phase diagram

• Suppression of TC is phase dependent CFL TC is
  more suppressed than 2SC one
• Schematic phase diagram

There is quantumthermal-fluctuation driven 2SC
window even if Ms0 is assumed. Suppression of
Tc is order of 10 Non-negligible
38
Summary

• General remarks on 1/N expansion
• Perturbative extrapolation based on MF values of
  D, m, T,
• Avoids problems with self-consistency,
  technically very easy
• Only reliable when the NLO corrections are small
  (in BCS, not in molecular BEC region)
• Many body bound state? Efimov state like N-body
  (singlet) bound state? If yes, at which order of
  N do these bound state contribute to
  thermodynamics?
• Color superconducting quark matter
• Fluctuation corrections non-negligible
• Different suppressions in TC according to pairing
  pattern ? competition of various phases
• Improvement necessary Imbalance, Color
  neutrality, etc.
• Generalization below the critical temperature
• Application to pion superfluid, of color is
  useful

39
The way to bypass the problemorder by order
expansion

• What to be solved is of type
• We also expand
• to find solution order by order
• O(1) (MFA)
• O(1/N)

40
Economical way to introduce expansion parameter N
possible?

• What about extending NC3 to NCN?
• 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!