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Eiji Nakano, Dept. of Physics, National Taiwan University
Outline:
1) Experimental and theoretical background2) Epsilon expansion method at finite scattering length3) Application to energy per particle 4) Summary and outlook
Epsilon Expansion Approach for BEC-BCS Crossover
J-W Chen+ EN (cond-mat/0610011)
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Cold Trapped Atoms
Source: C. Regal
1) Experimental and theoretical background1) Experimental and theoretical background
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Feshbach resonance:
Superfluidity of 2004
Open channel
Closed channel
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Review: Scattering Length
Source: C. Regal
maB
2
1
Binding energy:
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BEC-BCS Crossover
Source: C. Regal
Changing a at will: Technique of Feshbach Resonance
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Studies on Unitary Fermi gas:
• Zero-rang interaction,
• Infinite scattering length,
•The only parameter akF goes to infinity (no expansion parameter ) Physical quantities become universal (scaled by Fermion density).
Fkr /10
01
Fak
Usual diagrammatic method is not reliable.(There is no expansion parameters. )
5
3/)/( FAEe.g.,
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QMC calculations:
Chang. et al. (2004)Astrakharchik. et al. (2004)
44.042.0
210 )()(/ FF akak
A
E
A
E
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(1) Study at arbitrary dimension by Nussinov and Nussinov (cond-mat/0410597)
Approach from different spatial dimensions, d>4
N-body wave function and variational method
Its normalization diverges at 4dTwod-body bound state.
Free Bose gas at 4d
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(2) Epsilon expansion at unitary point by Nishida and Son (cond-mat/0604500)
475.0
(3) Pionless EFT for dilute nuclear matter, specific ladder diagram at d=gN=infinity, by T. Schaefer, C-W Kao, S. R. Cotanch, (cond-mat/0604500)
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Epsilon Expansion
• Computing in dim.
• Expanding in
• Setting
(Nishida and Son)
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4.dat
In Unitary limit and at Region of akF>0
Free Bose Gas (approximately)Mean field gives exact solution.
3.dat
Fluctuation developsas one goes to lower dimension
Non-trivial vacuum: the unitary Fermi gas
If expansion coefficients of epsilon are convergent, extrapolation to d=3 might give reliable results, a la, Wilsonian epsilon approach.
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2) Epsilon expansion method at finite scattering length
After Hubbard-Stratonovich transformation,
Condensation and Bosonic fluctuation:
which is determined uniquely so as to make boson wave function be unit.
Here we impose the scaling to boson chemical pot.:
so that reflecting free Bose gas. 4.dat0 B
1)
2)
0
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Reorganization of Lagrangian:
parts Free:0L
ions)(perturbat nsInteractio:1L
ies.singularit1 tormscounter te as serves:2 L
e.g.,
22BB
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Pole:
Effective Field Theory: ac 0
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Around the unitary limit: Expansion in B (binding energy)
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For instance, Chemical potential, Energy/particle, to next-to-leading order in epsilon and up to O(B)
1,
2,
3,
Steps to
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In the Unitary limit:
In BEC limit:
from large B expansion up to B^2, we find
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In BCS limit:
Comparable to result by K. Huang and C.N. Yang (1956)
Mean-field is exponentially small Two-loop gives a slope.
Since we can not expect that physics at d=4 is trivial as free Bose gas anymore, counting rules should be changed:
)1(~ O
And B serves as an effective Boson mass at region of akF<0.
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Energy per particle relative to that of free gas:
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Blow-up of around unitary limit:
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4) Summary and outlook
We have extended the epsilon expansion method to finite scattering region.Result, Slope and curvature of E/A and Chemical pot., is in overall good agreement with QMC and other low energy theorems.
•Summary
•Outlook
1, Application to Nuclear matter (Neutron star)
2, Investigation of finite range correction.
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Why is 4d special?
has a singularity at
for
ground state a free Bose gas