Eiji Nakano,

1
Epsilon Expansion Approach for BEC-BCS Crossover
J-W Chen EN (cond-mat/0610011)
Eiji Nakano, Dept. of Physics, National Taiwan
University

• Outline
• Experimental and theoretical background
• Epsilon expansion method at finite scattering
  length
• Application to energy per particle
• Summary and outlook

2
Cold Trapped Atoms
1) Experimental and theoretical background

Source C. Regal
3
Superfluidity of
2004
Closed channel
Open channel
Feshbach resonance
4
Review Scattering Length
Binding energy
5
BEC-BCS Crossover

Changing a at will Technique of Feshbach
Resonance
Source C. Regal
6
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).

e.g.,
Usual diagrammatic method is not reliable. (There
is no expansion parameters. )
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QMC calculations
Chang. et al. (2004)
Astrakharchik. et al. (2004)
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Approach from different spatial dimensions, dgt4
(1) Study at arbitrary dimension by Nussinov and
Nussinov (cond-mat/0410597)
N-body wave function and variational method
Its normalization diverges at
Twod-body bound state.
Free Bose gas at
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(2) Epsilon expansion at unitary point
by Nishida and Son (cond-mat/0604500)
(3) Pionless EFT for dilute nuclear matter,
specific ladder diagram at dgNinfinity,
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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In Unitary limit and at Region of akFgt0
Free Bose Gas (approximately) Mean field gives
exact solution.
Fluctuation develops as one goes to lower
dimension
Non-trivial vacuum the unitary Fermi gas
If expansion coefficients of epsilon are
convergent, extrapolation to d3 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
1)
which is determined uniquely so as to make boson
wave function be unit.
Here we impose the scaling to boson chemical
pot.
2)
so that
reflecting free Bose gas.
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Reorganization of Lagrangian
e.g.,
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Effective Field Theory
Pole
15
Around the unitary limit Expansion in B (binding
energy)
16
For instance, Chemical potential,
Energy/particle, to next-to-leading order in
epsilon and up to O(B)
Steps to
1,
2,
3,
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In the Unitary limit
In BEC limit from large B expansion up to
B2, we find
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In BCS limit
Since we can not expect that physics at d4 is
trivial as free Bose gas anymore, counting rules
should be changed
And B serves as an effective Boson mass at region
of akFlt0.
Mean-field is exponentially small Two-loop gives
a slope.
Comparable to result by K. Huang and C.N. Yang
(1956)
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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

• Summary

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.
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Why is 4d special?
has a singularity at
for
ground state a free Bose gas