Transcript Unitary Fermi gas in the e expansion Yusuke Nishida (Univ. of Tokyo & INT) in collaboration with D.

Unitary Fermi gas
in the e expansion
Yusuke Nishida (Univ. of Tokyo & INT)
in collaboration with D. T. Son
[Ref: Phys. Rev. Lett. 97, 050403 (2006),
cond-mat/0607835, cond-mat/0608321]
26 December, 2006 @ Univ. of Tokyo
Unitary Fermi gas
in the e expansion
Contents of this talk
1. Fermi gas at infinite scattering length
2. Formulation of expansions
in terms of 4-d and d-2
3. Results at zero/finite temperature
4. Summary and outlook
Contents of thesis
1. Introduction
2. Two-body scattering in vacuum
3. Unitary Fermi gas around d=4
4. Phase structure of polarized Fermi gas
5. Fermions with unequal masses
6. Expansions around d=2
7. Matching of expansions at d=4 and d=2
8. Thermodynamics below Tc
9. Thermodynamics above Tc
10. Summary and concluding remarks
4/30
Introduction :
Fermi gas at infinite scattering length
Interacting Fermion systems
Attraction

Superconductivity / Superfluidity
Metallic superconductivity (electrons)
Onnes (1911), Tc = ~9.2 K
BCS
theory
 Liquid 3He
Lee, Osheroff, Richardson (1972), Tc = 1~2.6 mK(1957)

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)]
5/30
Feshbach resonance
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90, (2003)
6/30
Attraction is arbitrarily tunable by magnetic field
S-wave scattering length :  [0, ]
a (rBohr)
a>0
Feshbach resonance
Strong coupling
|a|
Bound state
formation
a<0
40K
Weak coupling
|a|0
No bound state
BCS-BEC crossover
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
7/30
Strong interaction
Superfluid
phase
1/(akF)=-
BCS state of atoms
weak interaction: akF-0
-B
1/(akF)=+
1/(akF)=0
BEC of molecules
weak interaction: akF+0
Strong coupling limit : |akF|
• Maximal S-wave cross section
• Threshold: Ebound = 1/(2ma2)  0
Unitarity limit
Fermi gas in the strong coupling limit akF= : Unitary Fermi gas
Unitary Fermi gas
George Bertsch (1999),
“Many-Body X Challenge”
8/30
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 neutron matter
|aNN|~18.5 fm >> r0 ~1.4 fm
Universal parameter x
• Simplicity of system
x is universal parameter
9/30
• 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).
No systematic & analytic treatment of unitary Fermi gas
10/30
Unitary Fermi gas at d≠3
g
d=4
BCS
-
Strong coupling
Unitary regime
g
d=2
• d4 : Weakly-interacting
system of fermions & bosons,
their coupling is g~(4-d)1/2
BEC
+
• d2 : Weakly-interacting
system of fermions,
their coupling is g~(d-2)
Systematic expansions for x and other
observables (D, Tc, …) in terms of “4-d” or “d-2”
11/30
Formulation of
e expansion
e=4-d <<1 : d=spatial dimensions
12/30
Specialty of d=4 and d=2
2-component fermions
local 4-Fermi interaction :
2-body scattering in vacuum (m=0)

(p0,p) 
iT
=
1
n

T-matrix at arbitrary spatial dimension d
“a”
Scattering amplitude has zeros at d=2,4,…
Non-interacting limits
T-matrix around d=4 and 2
13/30
T-matrix at d=4-e (e<<1)
ig
=
iT
ig
iD(p0,p)
Small coupling
b/w fermion-boson
g = (8p2 e)1/2/m
T-matrix at d=2+e (e<<1)
ig2
iT
=
Small coupling
b/w fermion-fermion
g = (2p e/m)1/2
Lagrangian for e expansion
14/30
• 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
15/30
16/30
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 ~ e1/2”
But exceptions
Fermion loop integrals produce 1/e in 4 diagrams
Exceptions of power counting 1
1. Boson self-energy
17/30
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
18/30
3. Tadpole diagram with m insertion
p
k
= O(e1/2)
Sum of tadpoles = 0
naïve O(e3/2)
Gap equation for 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 :
4. Its powers of
or
e will be Ng/2 + Nm
5. The only exception is
= O(1)
19/30
Expansion over e = d-2
20/30
Lagrangian
Power counting rule of e
1. Assume
and consider
justified later
to be O(1)
2. Draw Feynman diagrams using only L0 and L1
3. If there are subdiagrams of type
add vertices from L2 :
4. Its powers of
e will be Ng/2
21/30
Results at zero/finite temperature
Leading and next-to-leading orders
22/30
Thermodynamic functions at T=0
• Effective potential : Veff = vacuum diagrams
p
Veff =
k
+
k
+
p-q
+ O(e2)
q
O(1)
O(e)
C=0.14424…
• Gap equation of 0
Assumption
• Pressure : P=-Veff with solution of gap eq. 0
is OK !
Universal parameter x
23/30
• Universal relations
• Universal parameter x around d=4 and 2
Arnold, Drut, Son (’06)
Systematic expansion of x in terms of e !
24/30
Quasiparticle spectrum
• Fermion dispersion relation : w(p)
O(e)
p-k
Self-energy
- i S(p) =
diagrams
k-p
+
p
k
p
p
Expansion over 4-d
Energy gap :
Location of min. :
Expansion over d-2
0
k
p
Extrapolation to d=3 from d=4-e
25/30
• 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)
cf. extrapolations from d=2+e
NLO are 100 %
Matching of two expansions in x
26/30
• Borel transformation + Padé approximation
x
♦=0.42
2d boundary condition
2d
• Interpolated results to 3d
4d
d
27/30
Critical temperature
• Gap equation at finite T
Veff =
+
+
+ m insertions
• Critical temperature from d=4 and 2
NLO correction
is small ~4 %
Simulations :
• Bulgac et al. (’05):
Tc/eF = 0.23(2)
• Lee and Schäfer (’05): Tc/eF < 0.14
• Burovski et al. (’06): Tc/eF = 0.152(7)
• Akkineni et al. (’06): Tc/eF  0.25
28/30
Matching of two expansions (Tc)
Tc / eF
• Borel + Padé approx.
4d
• Interpolated results to 3d
2d
d
NLO e1
2d + 4d
Bulgac et al.
Burovski et al.
Tc / eF
P / eFN
E / eFN
m / eF
S/N
0.249
0.135
0. 212
0.180
0.698
0.183
0.172
0.270
0.294
0.642
0.23(2)
0.27
0.41
0.45
0.99
0.152(7)
0.207
0.31(1)
0.493(14)
0.16(2)
Summary 1
e expansion for unitary Fermi gas
Systematic expansions over 4-d and d-2
• Unitary Fermi gas around d=4 becomes
weakly-interacting system of fermions & bosons
• Weakly-interacting system of fermions around d=2
LO+NLO results on x, D, e0, Tc (P,E,m,S)
• NLO corrections around d=4 are small
• Naïve extrapolation from d=4 to d=3 gives
good agreement with recent MC data
Picture of weakly-interacting fermionic &
bosonic quasiparticle for unitary Fermi gas
may be a good starting point even at d=3
29/30
Summary 2
e expansion for unitary Fermi gas
30/30
Matching of two expansions around d=4 and d=2
• NLO 4d + NLO 2d
• Borel transformation and Padé approximation
Results are not too far from MC simulations
Future Problems
• Large order behavior + NN…LO corrections
• Analytic structure of x in “d” space
Precise determination of universal parameters
• Other observables, e.g., Dynamical properties
31/30
Back up slides
Specialty of d=4 and 2
Z.Nussinov and S.Nussinov, 32/30
cond-mat/0410597
2-body wave function
Normalization at unitarity a
diverges at r0 for d4
Pair wave function is concentrated near its origin
Unitary Fermi gas for d4 is free “Bose” gas
At d2, any attractive potential leads to bound states
“a” corresponds to zero interaction
Unitary Fermi gas for d2 is free Fermi gas
Effective potential Veff (0)
• Leading order O(1)
33/30
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
34/30
• 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 !
NNLO correction for x
Arnold, Drut, and Son,
cond-mat/0608477
35/30
• O(e7/2) correction for x
• Borel transformation + Padé approximation
x
Interpolation to 3d
• NNLO 4d + NLO 2d
NLO 4d
NLO 2d
cf. NLO 4d + NLO 2d
NNLO 4d
d
36/30
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
37/30
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)
Comparisons of Tc
38/30
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)
39/30
Large order behavior
• d=2 and 4 are critical points
free gas 2
3
4
r0≠0
• Critical exponents of O(n=1) 4 theory (e=4-d  1)
g
O(1)
+e1
+e2
+e3
+e4
+e5
Lattice
1
1.167
1.244
1.195
1.338
0.892
1.239(3)
• Borel transform with conformal mapping
g=1.23550.0050
• Boundary condition (exact value at d=2)
g=1.23800.0050
e expansion is asymptotic series but works well !
e expansion in critical phenomena
40/30
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???