e expansion in cold atoms Yusuke Nishida (INT, U.of Washington MIT) in collaboration with D.
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Transcript e expansion in cold atoms Yusuke Nishida (INT, U.of Washington MIT) in collaboration with D.
e expansion in cold atoms
Yusuke Nishida (INT, U.of Washington MIT)
in collaboration with D. T. Son (INT)
1. Fermi gas at infinite scattering length
2. New approach : e (=4-d, d-2) expansions
Idea and how to use e expansion
LO & NLO results and interpolations
3. Summary and outlook
September 16, 2008 @ Center for Ultracold Atoms
2/20
Fermions at infinite scattering length
Feshbach resonance
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90 (2003)
3/20
Attraction is arbitrarily tunable by magnetic field
S-wave scattering length : a (rBohr)
Strong
attraction
a>0
Feshbach resonance
zero binding energy
|a|
Weak attraction
a<0
bound
molecule
a
40K
B (Gauss)
V0(a)
r0
BCS-BEC crossover
Eagles (1969), Leggett (1980)
Nozières and Schmitt-Rink (1985)
4/20
Strong interaction
add=0.6a
?
Superfluid
phase
+
BEC of molecules
weak repulsion: akF+0
-
0
BCS state of atoms
weak attraction: akF-0
Unitary Fermi gas
• Strong coupling limit : |akF|
• Atomic gas @ Feshbach resonance
George Bertsch (1999),
“Many-Body X Challenge”
Unitary Fermi gas
5/20
Atomic gas @ Feshbach resonance: 0 r0 << kF-1 << a
spin-1/2 fermions interacting via a zero-range,
infinite scattering length contact interaction
kF
Universal properties, but
-1
• Strong coupling limit
Perturbation akF=
r0
V0(a)
Previous approaches …
• Mean field approximations
• Monte Carlo simulations
• Difficulty for theory
No expansion parameter
e expansion !
• Use spatial dimensions
as a small parameter
6/20
Our approach from d≠3
g
d=4
BEC
+
Strong coupling
Unitary regime
g
d=2
• d4 : Weakly-interacting
system of fermions & bosons,
their coupling is g2~(4-d)
BCS
-
• d2 : Weakly-interacting
system of fermions,
their coupling is g~(d-2)
Systematic expansions for various physical
observables in terms of “4-d” or “d-2”
7/20
e expansion
Specialty of d=4 and 2
Z.Nussinov and S.Nussinov,
cond-mat/0410597
8/20
2-body wave function
Normalization at unitarity a
diverges at r0
for d4
Pair wave function is concentrated near its origin
Fermions at unitarity in d4 are free bosons
At d2, any attractive potential leads to bound states
“a” corresponds to zero interaction
Fermions at unitarity in d2 are free fermions
Simple example
9/20
• Energy of 2 fermions at unitarity in a harmonic potential
Exact result is known :
: fermions at unitarity become free fermions
: fermions at unitarity forms pointlike bosons
Ground state energy in d=2 & 4
10/20
• Ground state energy of unitary Fermi gas
Unitary Fermi gas in d2 is a free Fermi gas
Unitary Fermi gas in d4 is a free Bose gas
in d=3 !?
Starting points for systematic
expansions of x around d=4 & 2
Cf. MC simulation in 3d
J.Carlson and S.Reddy,
Phys.Rev.Lett. 95, (2005)
11/20
Field theoretical approach
Spin-1/2 fermions with
local 4-Fermi interaction :
2-body scattering at 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
12/20
T-matrix at d=4-e (e<<1)
=
iT
ig
ig
iD(p0,p)
Small coupling
b/w fermion-boson
g = (8p2 e)1/2/m
T-matrix at d=2+e (e<<1)
ig
iT
=
Small coupling
b/w fermion-fermion
g = 2p e/m
13/20
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)
14/20
Equation of state at T=0
15/20
• 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 !
x0 (d4 : free Bose gas)
x1 (d2 : free Fermi gas)
16/20
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
17/20
• 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
18/20
• Borel transformation + Padé approximants
x = Eunitary /Efree
♦=0.42
Expansion around 4d
2d boundary condition
2d
4d
• Interpolated results to 3d
d
free Fermi gas
free Bose gas
19/20
Critical temperature
• Critical temperature from d=4 and 2
NLO correction
is small ~4 %
• Interpolated results to d=3
Tc / eF
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)
• Akkineni et al. (’06): Tc/eF 0.25
2d
d
Summary
1. Systematic expansions over e=4-d or d-2
• Unitary Fermi gas around d=4 becomes
weakly-interacting system of fermions & bosons
• Weakly-interacting system of fermions around d=2
Very simple and useful starting points to
understand the unitary Fermi gas in d=3 !
2. LO+NLO results on x, D, e0, Tc
• NLO corrections around d=4 are small
• Extrapolations to d=3 agree with recent MC data
3. Future problems
• Large order behavior + NN…LO corrections
More understanding
Precise determination
20/20
NNLO correction for x
22/20
• 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
23/20
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
24/20
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???
Comparison with ideal BEC
25/20
• Ratio to critical temperature in the BEC limit
Boson and fermion contributions to fermion density at d=4
• Unitarity limit at Tc
1 of 9 pairs is dissociated
• BEC limit at Tc
all pairs form molecules
26/20
3 fermions in a harmonic potential
2d
4d
4d
2d
Interacting Fermion systems
Attraction
27/20
Superconductivity / Superfluidity
Metallic superconductivity (electrons)
Kamerlingh Onnes (1911), Tc = ~4.2 K
BCS
Liquid 3He
theory
(1957)
Lee, Osheroff, Richardson (1972), Tc = ~2.5 mK
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
Unitary Fermi gas
George Bertsch (1999),
“Many-Body X Challenge”
28/20
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=
29/20
• 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
Lagrangian for e expansion
30/20
• 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)
31/20
32/20
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
33/20
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
34/20
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
35/20
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)
36/20
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
37/20
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)
38/20
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 !
39/20
Simple application
• Energy of 3 fermions in a harmonic potential
: fermions at unitarity are free fermions
: fermions at unitarity are free bosons + fermion
singlet state
triplet state
in d=3 !?
Cf. exact results in 3d :