Transcript スライド 1 - University of Tokyo

Unitary Fermi gas
in the e expansion
Yusuke Nishida
March 3, 2009 @ CMT informal seminar
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Fermions at infinite scattering length
Interacting Fermion systems
Attraction

3/34
Superconductivity / Superfluidity
Metallic superconductivity (electrons)
Kamerlingh Onnes (1911), Tc ~4.2 K
BCS
 Liquid 3He
theory
Lee, Osheroff, Richardson (1972), Tc ~2 mK (1957)

High-Tc superconductivity (electrons or holes)
Bednorz and Müller (1986), Tc ~100 K

Cold atomic gases (40K, 6Li)
Regal, Greiner, Jin (2003), Tc ~ 50 nK
• Nuclear matter (neutron stars): Tc ~ 1 MeV ?
• Color superconductivity (cold QGP): Tc ~ 100 MeV ??
4/34
Feshbach resonance
S-wave scattering length :

m
m+Dm
m

E
DE=DmB
r
40K
bound level
interatomic potential
C.A.Regal and D.S.Jin, Phys.Rev.Lett. (2003)
S-wave scattering length : a
5/34
Strong
attraction
a>0
zero binding
energy : |a|
bound
molecule
Weak attraction
a<0
a
40K
(Gauss)
ry(r)
a>0
|a|
a<0
r0
V0
Attraction is arbitrarily tunable by magnetic field
r
BCS-BEC crossover
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
6/34
superfluid phase
+
-
0
strong attraction
weak attraction
BEC of molecules
BCS state of atoms
scattering length : a
kF = (3pn)1/3
Fermi momentum
B (gauss)
BCS-BEC crossover
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
7/34
superfluid phase
+
strong attraction
880 mm
BEC of molecules
0
-
weak attraction
BCS state of atoms
Vortex lattices
throughout BCSBEC crossover
M. Zwierlein et al.
Nature (2005)
BCS-BEC crossover
add=0.6a
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
8/34
Strong interaction
|akF| >> 1
+
strong attraction
0
-
weak attraction
BEC of molecules
BCS state of atoms
Bose gas
with weak repulsion
akF << 1
Fermi gas
with weak attraction
|akF| << 1
9/34
Unitary Fermi gas
strong interaction
weak BEC
+
weak BCS
-
0
|akF|
S-wave scattering length : a
40K
B (Gauss)
10/34
Unitary Fermi gas
weak BEC
+
strong interaction
weak BCS
-
0
 Strong coupling limit : |akF|
• Maximal s-wave cross section
• No perturbative expansion
Unitarity limit
Challenge for theorists
e expansion !
 Scale invariant interaction
• a & zero range r00
Nonrelativistic CFT
 Universality
• Atomic gas @ Feshbach resonance
• Dilute neutron matter : |aNN|~19 fm >> r0 ~1 fm
11/34
New approach from d≠3
g
d=4
weak BEC
+
Strong coupling
Unitary regime
0
g
d=2
• d4 : Weakly-interacting
fermions & bosons with
small coupling g2~(4-d)
weak BCS
-
• d2 : Weakly-interacting
fermions with
small coupling g~(d-2)
Systematic expansions for various physical
observables in terms of “4-d” or “d-2”
12/34
e expansion
13/34
Scale invariant interaction
Atomic gas @ Feshbach resonance : 0 r0 << kF-1 << a 
spin-1/2 fermions interacting via a zero-range
& infinite scattering length contact interaction


ry(r)
r0
ry(r)
r
ry(r)
r
r
V0
V0 ~ 1/(mr02)

Specialty of d=2 & 4
d
Z.Nussinov and S.Nussinov, 14/34
cond-mat/0410597
2-body wave function in general dimensions
4
Wave function y(r) becomes smooth at r0 for d=2
3
“a” corresponds to zero interaction
( Any attractive potential in d=2 leads to bound states )
2
Fermions at unitarity in d2 are free fermions
Specialty of d=2 & 4
d
Z.Nussinov and S.Nussinov, 15/34
cond-mat/0410597
2-body wave function in general dimensions
4
Normalization
3
diverges at r0 for d=4
Pair wave function is concentrated near its origin
2
Fermions at unitarity in d4 are free bosons
16/34
Ground state energy
• Ground state energy of unitary Fermi gas at T=0
Density “N” is the only scale
x : fundamental quantity of unitary Fermi gas
Mean field approx., Engelbrecht et al. (1996):
Simulations • Carlson et al., Phys.Rev.Lett. (2003):
• Astrakharchik et al., Phys.Rev.Lett. (2004):
• Carlson and Reddy, Phys.Rev.Lett. (2005):
x<0.59
x=0.44(1)
x=0.42(1)
x=0.42(1)
Experiments Innsbruck(’04): 0.32(13), Duke(’05): 0.51(4),
Rice(’05): 0.46(5), JILA(’06): 0.46(12), ENS(’07): 0.41(15)
Ground state energy in d=2 & 4
17/34
• Ground state energy of unitary Fermi gas
d
4
 Unitary Fermi gas in d4 is a free Bose gas
Cf. MC simulation in 3d
3
in d=3 !?
J.Carlson and S.Reddy (2005)
 Unitary Fermi gas in d2 is a free Fermi gas
2
Ground state energy in d=2 & 4
18/34
• Ground state energy of unitary Fermi gas
d
4
 Unitary Fermi gas in d4 is a free Bose gas
3
d=4 & d=2 are starting points
for systematic expansions of x
 Unitary Fermi gas in d2 is a free Fermi gas
2
Field theoretical approach
Y.N. and D.T.Son
19/34
PRL(’06) & PRA(’07)
Spin-1/2 fermions
with contact interaction :
2-body scattering at vacuum (m=0)

(p0,p) 
iT
=
1
n

T-matrix in general dimensions
“a”
Scattering amplitude has zeros at d=2,4,…
Non-interacting limits
Y.N. and D.T.Son
20/34
PRL(’06) & PRA(’07)
Field theoretical approach
d
T-matrix in general dimensions
4
When d=4-e (e<<1)
3
iT
2
=
ig
ig
iD(p0,p)
Small coupling between fermions & boson
g = (8p2 e)1/2/m
Field theoretical approach
d
Y.N. and D.T.Son
21/34
PRL(’06) & PRA(’07)
T-matrix in general dimensions
4
When d=2+e (e<<1)
3
ig
iT
2
=
Small coupling between fermion & fermion
g = 2p e/m
22/34
Results to next-to-leading order
Calculation of pressure (NLO)
• Pressure and gap equation around d=4
P(0,m) =
+
+ O(e2)
+
O(e)
O(1)
• Pressure and gap equation around d=2
P(0,m) =
+ O(e2)
+
O(1)
O(e)
23/34
Equation of state at T=0
24/34
• Universal equation of state
Density “N” is the only scale !
• Universal parameter x around d=4 and 2
Systematic expansion of x in terms of e !
x0 (d4 : free Bose gas)
x1 (d2 : free Fermi gas)
25/34
Quasiparticle spectrum
• Fermion dispersion relation : w(p)
LO
self-energy - i S(p) =
diagrams
or
O(e)
O(e)
Expansion over 4-d
Energy gap :
Location of min. :
Expansion over d-2
0
Extrapolation to d=3 from d=4-e
26/34
• 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
27/34
• Padé approximants (+ Borel transformation)
x = Eunitary /Efree
♦=0.42
2d
4d
d
free Fermi gas
Interpolations to 3d
free Bose gas
Critical temperature
Y.N., Phys. Rev. A (2007)
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• Critical temperature from d=4 and 2
Tc / eF
• Interpolated results to d=3
4d
Monte Carlo 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)
d
• Akkineni et al. (’06): Tc/eF  0.25
2d
free Fermi gas
free Bose gas
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Few body aspects
Correspondence
S.Tan, cond-mat/0412764
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F.Werner & Y.Castin, PRA (2006)
• Schrödinger equation in free space with E=0
Scaling solution
• Schrödinger equation in a harmonic potential
g = anomalous dimension of operator in nonrelativistic CFT
Y.N. & D.T.Son, PRD (2007)
31/34
3 fermions in a harmonic potential
Angular momentum l =0
Angular momentum l =1
2d
4d
4d
2d
32/34
3 fermions in a harmonic potential
Angular momentum l =0
Angular momentum l =1
2d
4d
4d
2d
Summary
33/34
Fermion gas at infinite scattering length
= New strongly interacting matter in cold atoms
• Unitary Fermi gas around d=4 becomes
weakly-interacting system of fermions & bosons
• Weakly-interacting system of fermions around d=2
• Thermodynamics & Quasiparticle spectrum (Y.N. & D.T.Son 2006)
• Atom-dimer & dimer-dimer scatterings (G.Rupak 2006)
• Phase structure of polarized Fermi gas with (un)equal masses
(Y.N. 2007, G.Rupak & T.Schafer & A.Kryjevski 2007)
• BCS-BEC crossover (J.W.Chen & E.Nakano 2007)
• Momentum distribution & condensate fraction (Y.N. 2007)
• Energy of a few atoms in a harmonic potential (Y.N. & D.T.Son 2007)
• Low-energy dynamics (A.Kryjevski 2008)
• Energy-density functional (G.Rupak & T.Schafer 2009)
• …
Summary
34/34
Fermion gas at infinite scattering length
= New strongly interacting matter in cold atoms
• Unitary Fermi gas around d=4 becomes
weakly-interacting system of fermions & bosons
• Weakly-interacting system of fermions around d=2
• Thermodynamics & Quasiparticle spectrum (Y.N. & D.T.Son 2006)
Very
simple and
useful
starting
points to
• Atom-dimer
& dimer-dimer
scatterings
(G.Rupak
2006)
• Phase structure
of polarized
gas with
(un)equal
understand
theFermi
unitary
Fermi
gasmasses
in d=3 !
(Y.N. 2007, G.Rupak & T.Schafer & A.Kryjevski 2007)
• BCS-BEC crossover (J.W.Chen & E.Nakano 2007)
• Momentum distribution & condensate fraction (Y.N. 2007)
• Energy of a few atoms in a harmonic potential (Y.N. & D.T.Son 2007)
• Low-energy dynamics (A.Kryjevski 2008)
• Energy-density functional (G.Rupak & T.Schafer 2009)
• …
NNLO correction for x
36/34
• NNLO correction for x
Arnold, Drut, Son, Phys.Rev.A (2006)
Nishida, Ph.D. thesis (2007)
Fit two expansions
using Padé approximants
x
♦=0.40
Interpolations to 3d
• NNLO 4d + NNLO 2d
cf. NLO 4d + NLO 2d
d
37/34
Polarized Fermi gas around d=4
• Rich phase structure near unitarity point
in the plane of
and
: binding energy
Polarized normal state
Gapless superfluid
1-plane wave
FFLO : O(e6)
Gapped superfluid
BCS
BEC
unitarity
Stable gapless phases (with/without spatially varying
condensate) exist on the BEC side of unitarity point
e expansion in critical phenomena
38/34
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???
2 fermions in a harmonic potential
T.Busch et.al., Found. Phys. (1998)
T.Stoferle et al., Phys.Rev.Lett. (2006)
39/34
2 fermions in a harmonic potential
|a|
40/34