Transcript An e expansion for Fermi gas at infinite scattering length Yusuke Nishida (Univ.

An e expansion for Fermi gas
at infinite scattering length
Yusuke Nishida (Univ. of Tokyo & INT)
Y. N. and D. T. Son, arXiv: cond-mat/0604500
15 May, 2006 @ INT Seminar
An e expansion for Fermi gas
at infinite scattering length
Contents of this talk
1. Fermi gas at infinite scattering length
2. Formulation of
e (=4-d) expansion
3. Results at zero temperature
4. Results at finite temperature
5. Summary
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Strongly interacting Fermi gas
Interacting Fermion systems
Attraction

Superconductivity / Superfluidity
Metallic superconductivity (electrons)
Onnes (1911), Tc = ~9.2 K
BCS
theory
 Liquid 3He
(1957)
Lee, Osheroff, Richardson (1972), Tc = 1~2.6 mK

High-Tc superconductivity (electrons or holes)
Bednorz and Müller (1986), Tc = ~160 K

Atomic gases (40K, 6Li)
Regal, Greiner, Jin (2003), Tc ~ 50 nK
• Nuclear matter (neutron stars): ?, Tc ~ 1 MeV
• Color superconductivity (quarks): ??, Tc ~ 100 MeV
• Neutrino superfluidity ??? [Kapusta, PRL(’04)]
4/28
Feshbach resonance
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90, (2003)
5/28
Interaction is arbitrarily tunable by magnetic field
scattering length :  [0, ]
a (rBohr)
Repulsive
(a>0)
Bound state
formation
Strong coupling
|a|
40K
Weak coupling
|a|0
Feshbach resonance
Attractive
(a<0)
No bound state
BEC-BCS crossover
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
6/28
Superfluid
phase
1/(akF)=+
B
1/(akF)=-
1/(akF)=0
BEC of molecules
weak interaction: akF+0
BCS state of atoms
weak interaction: akF-0
Strong interaction : |akF|
• Maximal S-wave cross section
• Threshold: Ebound = 1/(2ma2)  0
Unitarity limit
Unitary Fermi gas
George Bertsch (1999),
“Many-Body X Challenge”
7/28
Atomic gas : |a|=1000Å >> kF-1=100Å >> r0 =10Å
What are the ground state properties of
the many-body system composed of
spin-1/2 fermions interacting via a zero-range,
infinite scattering length contact interaction?
0 r0 << kF-1 << a 
kF-1
kF is the only scale !
Energy
per particle
r0
x is independent of systems
V0(a)
cf. dilute nuclear matter
|aNN|~19 fm >> r0 ~1 fm
Universal parameter x
• Simplicity of system
x is universal parameter
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• Difficulty for theory
No expansion parameter
Models • Mean field approx., Engelbrecht et al. (1996): x<0.59
• Linked cluster expansion, Baker (1999):
• Galitskii approx., Heiselberg (2001):
• LOCV approx., Heiselberg (2004):
• Large d limit, Steel (’00)Schäfer et al. (’05):
Simulations • Carlson et al., Phys.Rev.Lett. (2003):
• Astrakharchik et al., Phys.Rev.Lett. (2004):
• Carlson and Reddy, Phys.Rev.Lett. (2005):
x=0.3~0.6
x=0.33
x=0.46
x=0.440.5
x=0.44(1)
x=0.42(1)
x=0.42(1)
Experiments Duke(’03): 0.74(7), ENS(’03): 0.7(1), JILA(’03): 0.5(1),
Innsbruck(’04): 0.32(1), Duke(’05): 0.51(4), Rice(’06): 0.46(5).
Systematic expansion for x and
other observables in terms of e (=4-d)
9/28
Formulation of
e expansion
e=4-d <<1 : d=spatial dimensions
Specialty at d  4
Z.Nussinov and S.Nussinov,
cond-mat/0410597
10/28
2-body scattering at zero energy limit
Tune the attractive potential V0 at threshold (a/r0 )
r d-2R(r)
R(r) : pair wave function
r : separation of 2 particles
r0 : range of potential
a
C
r00
V0
Normalization of wave func.
r
diverges at r0 for d4
Pair wave function is concentrated near its origin r0
Finite density system at unitarity for d4
is weakly-interacting Bose gas
But they never developed an expansion around d=4
11/28
Field theoretical approach
Universality allows
Local 4-Fermi interaction :
2-body scattering at vacuum (m=0)

(p0,p) 
T
=
1
n

T-matrix at d=4-e (e<<1)
g
T
=
g
D(p0,p)
Coupling with boson
g = (8p2 e)1/2/m
is SMALL !!!
12/28
Lagrangian
• Hubbard-Stratonovish trans. & Nambu-Gor’kov field :
=0 in dimensional regularization
Ground state at finite density is superfluid :
Expand
with
• Rewrite Lagrangian as a sum : L = L0+ L1+ L2
Boson’s kinetic
term is added,
and subtracted here.
Feynman rules 1
• L0 :
Free fermion quasiparticle  and boson 
• L1 :
Small coupling “g”
between  and 
(g ~ e1/2)
Chemical potential
insertions
13/28
14/28
Feynman rules 2
• L2 :
“Counter terms”
of boson 
“Naïve” power counting of
1. Assume
and consider
e
justified later
to be O(1)
2. Draw Feynman diagrams using only L0 and L1 (not L2)
3. Its powers of
e will be Ng/2 + Nm
Number of m insertions
Number of couplings “g”
(g ~ e1/2)
But exceptions
Fermion loop integrals produce 1/e in 4 diagrams
Exceptions of power counting 1
1. Boson self-energy
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naïve O(e)
k
p
p
p+k
= O(e)
+
Cancellation with L2 vertices to restore naïve counting
2. Boson self-energy with m insertion
k
p
p
p+k
+
= O(e2)
naïve O(e2)
Exceptions of power counting 2
16/28
3. Tadpole diagram with m insertion
p
k
= O(e1/2)
Gap equation of 0
naïve O(e3/2)
Sum of tadpoles = 0
+ ··· = 0
+
O(e1/2)
O(e1/2)
4. Vacuum diagram with m insertion
k
= O(1)
O(e)
Only exception !
“Revised” power counting of
1. Assume
and consider
e
justified later
to be O(1)
2. Draw Feynman diagrams using only L0 and L1
3. If there are subdiagrams of type
or
add vertices from L2 :
5. Its powers of
or
e will be Ng/2 + Nm
6. The only exception is
= O(1)
17/28
18/28
Results at zero temperature
Leading and next-to-leading orders
Effective potential Veff (0)
• Leading order O(1)
19/28
Boson’s 1-loops
k
and
k
vanish at T=0
• Next-to-leading order O(e)
p
p-q
q
C=0.14424…
Universal parameter x
20/28
• Gap equation of 0
Assumption
is OK !
• Fermion density and Fermi energy
• Universal parameter : x = m/eF
Systematic expansion of x in terms of e !
21/28
Quasiparticle spectrum
• Fermion 1-loop self-energy O(e)
p-k
k-p
- i S(p) =
+
p
k
p
p
• Fermion dispersion relation : w(p)
k
p
LO :
Around minimum :
Energy gap :
Location of min. :
NLO
22/28
Extrapolation to d=3
• Keep LO & NLO results and extrapolate to e=1
NLO corrections are small: 5 ~ 35 %
Good agreement with recent Monte Carlo data
J.Carlson and S.Reddy,
Phys.Rev.Lett.95, (2005)
Experiments : Duke(’05): x = 0.51(4), Rice(’06): x = 0.46(5)
23/28
Results at finite temperature
preliminary
24/28
Hierarchy in temperature
At T=0, D(T=0) ~ m/e >> m
2 energy scales
(i) Low : T ~ m << DT ~ m/e
D(T)
• Fermion excitations are suppressed
• Phonon excitations are dominant
(i)
(ii)
(iii)
T
0
(ii) Intermediate : m < T < m/e
(iii) High : T ~ m/e >> m ~ DT
• Condensate vanishes at Tc ~ m/e
• Fermions and bosons are excited
~m
Tc ~ m/e
Similar power counting
• m/T ~ O(e)
• Consider T to be O(1)
25/28
Critical temperature
• Leading & next-to-leading orders
Veff =
+
+
+ m insertions
• Gap equation :
• Fermion density :
Critical temperature & energy density
NLO
corrections
are small
4~9 %
Comparisons of Tc
26/28
e expansion
(LO+NLO)
Simulations : • Wingate (’05):
• Lee and Schäfer (’05):
• Bulgac et al. (’05):
• Burovski et al. (’06):
Tc/eF = 0.04
Tc/eF < 0.14
Tc/eF = 0.23(2)
Tc/eF = 0.152(7)
Tc/eF = 0.27(2)
cf. BEC limit : TBEC/eF = 0.218…
Experiment : • Kinast et al. (’05):
Ideal BEC at d=4-e (convergent if |e| 2)
e expansion in critical phenomena
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Critical exponents of O(n=1) 4 theory (e=4-d  1)
O(1)
g

1
0
+e1
1.167
0
+e2
+e3
+e4
+e5
Lattice
Exper.
1.239(3)
1.240(7)
1.22(3)
1.24(2)
0.0185 0.0372 0.0289 0.0545 0.027(5)
0.016(7)
0.04(2)
1.244
e expansion is
1.195
1.338
0.892
• Borel summation with conformal mapping
g=1.23550.0050 & =0.03600.0050
asymptotic series
but works well. • Boundary condition (exact value at d=2)
g=1.23800.0050 & =0.03650.0050
How about our case???
Summary
e expansion for unitary Fermi gas
28/28
• The only systematic expansion at T=0 for now
• LO+NLO results on x, D, e0, Tc, E
• NLO corrections are small compared to LO
• Extrapolations to d=3 give good agreement
with MC simulations and experiments
(infinite) Future Problems
•
•
•
•
NNLO corrections + Resummation
Thermodynamics
• Vortex structure
• Dynamical properties
Finite polarization
BCS-BEC crossover (finite a)
29/28
Back up slides
Feshbach resonance
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90, (2003)
30/28
Interaction is arbitrarily tunable by magnetic field
scattering length :  [0, ]
repulsive
S-wave
E
m
m+Dm
m
strong
coupling
DE=DmB
r
40K
bound level
interatomic potential
attractive