スライド 1 - University of Tokyo
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Transcript スライド 1 - University of Tokyo
e expansion in cold atoms
Yusuke Nishida (Univ. of Tokyo & INT)
in collaboration with D. T. Son (INT)
Ref: Phys. Rev. Lett. 97, 050403 (2006)
1. BCS-BEC crossover and unitarity limit
2. Formulation of e (=4-d) expansion
3. LO & NLO results
4. Summary and outlook
21COE WS “Strongly correlated many-body systems” 19/Jan/07
Interacting Fermion systems
Attraction
2/16
Superconductivity / Superfluidity
Metallic superconductivity (electrons)
Kamerlingh Onnes (1911), Tc = ~9.2 K
BCS
theory
Liquid 3He
Lee, Osheroff, Richardson (1972), Tc = 1~2.6(1957)
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
Feshbach resonance
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90, (2003)
3/16
Attraction is arbitrarily tunable by magnetic field
S-wave scattering length : [0, ]
a (rBohr)
Feshbach resonance
a>0
Bound state
formation
molecules
Strong coupling
|a|
a<0
40K
Weak coupling
|a|0
No bound state
atoms
BCS-BEC crossover
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
4/16
Superfluid phase
-
+
0
BCS state of atoms
weak attraction: akF-0
BEC of molecules
weak repulsion: akF+0
Strong interaction : |akF|
• Maximal S-wave cross section
• Threshold: Ebound = 1/(ma2) 0
Unitarity limit
Fermi gas in the strong coupling limit : akF=
Unitary Fermi gas
Unitary Fermi gas
George Bertsch (1999),
“Many-Body X Challenge”
5/16
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
• Strong coupling limit
Perturbation akF=
6/16
• 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).
This talk
Systematic expansion for x and other
observables (D,Tc,…) in terms of e (=4-d)
7/16
2-body scattering around d=4
2-component fermions
local 4-Fermi interaction :
2-body scattering at vacuum (m=0)
(p0,p)
iT
=
1
n
T-matrix at d=4-e (e<<1)
ig
iT
=
ig
iD(p0,p)
Coupling with boson
g = (8p2 e)1/2/m
is SMALL !!!
Lagrangian for e expansion
8/16
• 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)
9/16
10/16
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
11/16
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)
12/16
Thermodynamic functions at T=0
• Universal equation of state
• Effective potential and gap equation for 0
Veff (0,m) =
+
+ O(e2)
+
O(1)
• Universal number x around d=4
O(e)
Systematic
expansion !
13/16
Quasiparticle spectrum
• O(e) fermion self-energy
p-k
k-p
- i S(p) =
+
p
k
p
p
• Fermion dispersion relation : w(p)
Around
minimum
Expansion over 4-d
Energy gap :
0
Location of min. :
k
p
Extrapolation to d=3 from d=4-e
14/16
• 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
15/16
• Borel transformation + Padé approximants
Expansion around 4d
x
♦=0.42
2d boundary condition
2d
• Interpolated results to 3d
4d
d
Summary
16/16
1. Systematic expansions over e=4-d
• Unitary Fermi gas around d=4 becomes
weakly-interacting system of fermions & bosons
2. LO+NLO results on x, D, e0
• NLO corrections around d=4 are small
• Extrapolations to d=3 agree 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
3. Future problems
• Large order behavior + NN…LO corrections
More understanding
Precise determination
17/16
Back up slides
Specialty of d=4 and 2
Z.Nussinov and S.Nussinov, 18/16
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
19/16
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
20/16
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
21/16
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”
Expansion over e = d-2
22/16
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
NNLO correction for x
Arnold, Drut, and Son,
cond-mat/0608477
23/16
• 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
24/16
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
25/16
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)
e expansion in critical phenomena
26/16
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???