Unitary Fermi gas in the e expansion Yusuke Nishida (Univ. of Tokyo & INT) in collaboration with D.
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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]
16 January, 2007 @ T.I.Tech
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
3/27
Introduction :
Fermi gas at infinite scattering length
Interacting Fermion systems
Attraction
Superconductivity / Superfluidity
Metallic superconductivity (electrons)
Kamerlingh Onnes (1911), Tc = ~9.2 K
BCS
Liquid 3He
theory
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)]
4/27
Feshbach resonance
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90, (2003)
5/27
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)
6/27
Strong interaction
?
Superfluid
phase
-
+
0
BCS state of atoms
weak attraction: akF-0
-B
BEC of molecules
weak repulsion: 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”
7/27
Atomic gas : r0 =10Å << kF-1=100Å << |a|=1000Å
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
8/27
• 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=0.440.5
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
9/27
Unitary Fermi gas at d≠3
g
d=4
BCS
-
Strong coupling
Unitary regime
g
d=2
• d4 : Weakly-interacting
system of fermions & bosons,
their coupling is g~(4-d)1/2
BEC
+
• d2 : 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”
10/27
Formulation of
e expansion
e=4-d <<1 : d=spatial dimensions
11/27
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
12/27
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
13/27
• 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 (m ~ e)
14/27
15/27
Feynman rules 2
• L2 :
“Counter vertices” to
cancel 1/e singularities
in boson self-energies
1.
k
p
O(e)
2.
p
+
= O(e)
p
+
= O(em)
p+k
k
p
O(em)
p+k
Power counting rule of
1. Assume
and consider
e
16/27
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
Number of m insertions
Number of couplings “g ~ e1/2”
= O(1)
O(e)
Expansion over e = d-2
17/27
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
18/27
Results at zero/finite temperature
Leading and next-to-leading orders
19/27
Thermodynamic functions at T=0
• Effective potential : Veff = vacuum diagrams
p
Veff (0,m) =
k
+
k
+
p-q
+ O(e2)
q
O(1)
O(e)
• Gap equation of 0
C=0.14424…
Assumption
• Pressure :
is OK !
with the solution 0(m)
Universal parameter x
20/27
• Universal equation of state
• Universal parameter x around d=4 and 2
Arnold, Drut, Son (’06)
Systematic expansion of x in terms of e !
21/27
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
22/27
• 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
23/27
• Borel transformation + Padé approximants
Expansion around 4d
x
♦=0.42
2d boundary condition
2d
• Interpolated results to 3d
4d
d
24/27
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
25/27
Matching of two expansions (Tc)
Tc / eF
• Borel + Padé approx.
4d
• Interpolated results to 3d
2d
d
NLO e1
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
26/27
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 quasiparticles for unitary Fermi gas
may be a good starting point even at d=3
Summary 2
e expansion for unitary Fermi gas
27/27
Matching of two expansions around d=4 and d=2
• NLO 4d + NLO 2d
• Borel transformation and Padé approximants
Results are not too far from MC simulations
Future Problems
More understanding on e expansion
• Large order behavior + NN…LO corrections
• Analytic structure of x in “d” space
Precise determination of universal parameters
Other observables, e.g., Dynamical properties
28/27
Back up slides
Specialty of d=4 and 2
Z.Nussinov and S.Nussinov, 29/27
cond-mat/0410597
2-body wave function
Normalization at unitarity a
diverges at r0 for d4
Pair wave function is concentrated near its origin
Unitary Fermi gas for d4 is free “Bose” gas
At d2, any attractive potential leads to bound states
“a” corresponds to zero interaction
Unitary Fermi gas for d2 is free Fermi gas
30/27
Feynman rules 2
• L2 :
“Counter vertices”
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
31/27
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
32/27
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 !
NNLO correction for x
Arnold, Drut, and Son,
cond-mat/0608477
33/27
• O(e7/2) correction for x
• Borel transformation + Padé approximants
x
Interpolation to 3d
• NNLO 4d + NLO 2d
NLO 4d
NLO 2d
cf. NLO 4d + NLO 2d
NNLO 4d
d
34/27
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)
35/27
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=1.23550.0050
• Boundary condition (exact value at d=2)
g=1.23800.0050
e expansion is asymptotic series but works well !
e expansion in critical phenomena
36/27
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=1.23550.0050 & =0.03600.0050
asymptotic series
but works well ! • Boundary condition (exact value at d=2)
g=1.23800.0050 & =0.03650.0050
How about our case???