Unitary Fermi gas in the expansion Yusuke Nishida18 January 2007 Contents of this talk 1. Fermi gas...
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Transcript of Unitary Fermi gas in the expansion Yusuke Nishida18 January 2007 Contents of this talk 1. Fermi gas...
Unitary Fermi gasin the expansion
Yusuke Nishida 18 January 2007
Contents of this talk1. Fermi gas at infinite scattering length2. Formulation of expansions
in terms of 4-d and d-2
3. LO & NLO results at zero T & 4. Summary and outlook
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
Contents of thesis
3/23
Introduction :Fermi gas at infinite scattering length
4/23Interacting Fermion systems
Attraction Superconductivity / Superfluidity
Metallic superconductivity (electrons)Kamerlingh Onnes (1911), Tc = ~9.2 K
Liquid 3HeLee, Osheroff, Richardson (1972), Tc = 1~2.6 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
BCS theory
(1957)
5/23Feshbach resonance
Attraction is arbitrarily tunable by magnetic field
C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90, (2003)
S-wave scattering length : [0, ]
Strong coupling|a|
a<0 No bound state40K
a (rBohr)
Weak coupling|a|0
a>0
Bound stateformation
Feshbach resonance
6/23BCS-BEC crossover
0
BCS state of atomsweak attraction: akF-0
BEC of moleculesweak repulsion: akF+0
Eagles (1969), Leggett (1980)Nozières and Schmitt-Rink (1985)
Strong coupling limit : |a kF|• Maximal S-wave cross section Unitarity limit• Threshold: Ebound = 1/(2ma2) 0
-B
Superfluidphase
?
Strong interaction
Fermi gas in the strong coupling limit a kF= : Unitary Fermi gas
7/23Unitary Fermi gas George Bertsch (1999), “Many-Body X Challenge”
r0
V0(a)
kF-1
kF is the only scale !
Atomic gas : r0 =10Å << kF-1=100Å << |a|=1000Å
Energy per particle
0 r0 << kF-1 << a
cf. dilute neutron matter |aNN|~18.5 fm >> r0 ~1.4 fm
is independent of systems
What are the ground state properties ofthe many-body system composed of
spin-1/2 fermions interacting via a zero-range,
infinite scattering length contact interaction?
8/23
• Mean field approx., Engelbrecht et al. (1996): <0.59• Linked cluster expansion, Baker (1999): =0.3~0.6• Galitskii approx., Heiselberg (2001): =0.33• LOCV approx., Heiselberg (2004): =0.46• Large d limit, Steel (’00)Schäfer et al. (’05): =0.440.5
Universal parameter • Simplicity of system is universal parameter
• Difficulty for theory No expansion parameter
Models
Simulations
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).
Systematic expansion for and other observables (,Tc,…) in terms of (=4-d)
• Carlson et al., Phys.Rev.Lett. (2003): =0.44(1)• Astrakharchik et al., Phys.Rev.Lett. (2004): =0.42(1)• Carlson and Reddy, Phys.Rev.Lett. (2005): =0.42(1)
This talk
9/23
Formulation of expansion
=4-d <<1 : d=spatial dimensions
10/23
T-matrix at arbitrary spatial dimension d
Specialty of d=4 and d=22-component fermionslocal 4-Fermi interaction :
iT =
(p0,p)
2-body scattering in vacuum (=0)
1 n
“a”
Scattering amplitude has zeros at d=2,4,…Non-interacting limits
11/23
T-matrix at d=4- (<<1)
T-matrix around d=4 and 2
iT =ig ig
iD(p0,p)
Small coupling b/w fermion-boson
g = (82 )1/2/m
T-matrix at d=2+ (<<1)
iT =ig2 Small coupling
b/w fermion-fermiong = (2 /m)1/2
12/23
Boson’s kinetic term is added,
and subtracted here.
=0 in dimensional regularization
Expand with
Ground state at finite density is superfluid :
Lagrangian for expansion
• Hubbard-Stratonovish trans. & Nambu-Gor’kov field :
• Rewrite Lagrangian as a sum : L = L0+ L1+ L2
13/23Feynman rules 1
• L0 :
Free fermion quasiparticle and boson
• L1 :
Small coupling “g” between and
(g ~ 1/2)
Chemical potential insertions ( ~ )
14/23
+ = O()
Feynman rules 2
• L2 :
“Counter vertices” to cancel 1/ singularitiesin boson self-energies
p p
p+k
k+ = O()
p p
p+k
k
1.
2.
O()
O()
15/23
1. Assume justified later
and consider 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 will be Ng/2 + N
5. The only exception is = O(1) O()
Power counting rule of
or
or
Number of insertions
Number of couplings “g ~ 1/2”
16/23Expansion over = d-2
1. Assume justified later
and consider 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 will be Ng/2
Lagrangian
Power counting rule of
17/23
Results at zero temperature
Leading and next-to-leading orders
18/23
Assumption is OK !
Thermodynamic functions at T=0
k k
p
q
p-q
O(1) O()
+ +Veff (0,) =
• Effective potential : Veff = vacuum diagrams
+ O(2)
• Gap equation of 0 C=0.14424…
• Pressure : with the solution 0()
19/23Universal parameter
• Universal parameter around d=4 and 2
Systematic expansion of in terms of !
Arnold, Drut, Son (’06)
• Universal equation of state
20/23Quasiparticle spectrum
+- i (p) =p pk
p-k
p pk
k-p
• Fermion dispersion relation : (p)
Energy gap :
Location of min. :
Self-energydiagrams
0
Expansion over 4-d
Expansion over d-2
O()
21/23Extrapolation to d=3 from d=4-• Keep LO & NLO results and extrapolate to =1
J.Carlson and S.Reddy,
Phys.Rev.Lett.95, (2005)
Good agreement with recent Monte Carlo data
NLOcorrectionsare small5 ~ 35 %
NLO are 100 %cf. extrapolations from d=2+
22/23Matching of two expansions in • Borel transformation + Padé approximants
• Interpolated results to 3d
2d boundary condition
d
♦=0.42
4d
2d
Expansion around 4d
23/23
1. Systematic expansions over =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
2. LO+NLO results on , , 0
• NLO corrections around d=4 are small
• Extrapolations to d=3 agree with recent MC data
3. Future problems• Large order behavior + NN…LO correctionsMore understanding Precise determination
Summary
Picture of weakly-interacting fermionic &bosonic quasiparticles for unitary Fermi gas
may be a good starting point even at d=3
24/23
Back up slides
25/23Specialty of d=4 and 2
2-body wave function
Z.Nussinov and S.Nussinov, cond-mat/0410597
Pair wave function is concentrated near its origin
Unitary Fermi gas for d4 is free “Bose” gas
Normalization at unitarity a
diverges at r0 for d4
At d2, any attractive potential leads to bound states
“a” corresponds to zero interaction
Unitary Fermi gas for d2 is free Fermi gas
26/23Unitary Fermi gas at d≠3
BECBCS Strong coupling
Unitary regime
d=4
d=2
g
g
• d4 : Weakly-interacting system of fermions & bosons, their coupling is g~(4-d)1/2
• d2 : Weakly-interacting system of fermions, their coupling is g~(d-2)
Systematic expansions for and other observables (, Tc, …) in terms of “4-d” or “d-2”
27/23NNLO correction for • O(7/2) correction for
Arnold, Drut, and Son, cond-mat/0608477
• Borel transformation + Padé approximants
d
NLO 4dNLO 2d
NNLO 4d
Interpolation to 3d
• NNLO 4d + NLO 2d
cf. NLO 4d + NLO 2d
28/23Critical temperature
Veff = + + + insertions
• Gap equation at finite T
• Critical temperature from d=4 and 2
NLO correctionis small ~4 %
Simulations : • Lee and Schäfer (’05): Tc/F < 0.14• Burovski et al. (’06): Tc/F = 0.152(7)• Akkineni et al. (’06): Tc/F 0.25
• Bulgac et al. (’05): Tc/F = 0.23(2)
29/23
d
Tc / F
4d
2d
Matching of two expansions (Tc)
• Borel + Padé approx.
• Interpolated results to 3d
Tc / F P / FN E / FN / F S / N
NLO 1 0.249 0.135 0. 212 0.180 0.698
2d + 4d 0.183 0.172 0.270 0.294 0.642
Bulgac et al. 0.23(2) 0.27 0.41 0.45 0.99
Burovski et al. 0.152(7) 0.207 0.31(1) 0.493(14) 0.16(2)
30/23
• Borel summation with conformal mapping=1.23550.0050 & =0.03600.0050
• Boundary condition (exact value at d=2)=1.23800.0050 & =0.03650.0050
expansion in critical phenomena
O(1) 2 3 4 5 Lattice Exper.
1 1.167 1.244 1.195 1.338 0.892 1.239(3)
1.240(7) 1.22(3) 1.24(2)
0 0 0.0185 0.0372 0.0289 0.0545 0.027(5) 0.016(7) 0.04(2)
Critical exponents of O(n=1) 4 theory (=4-d 1)
expansion isasymptotic seriesbut works well !
How about our case???