eiji nakano, dept. of physics, national taiwan university outline: 1)experimental and theoretical...
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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)
Cold Trapped Atoms
Source: C. Regal
1) Experimental and theoretical background1) Experimental and theoretical background
Feshbach resonance:
Superfluidity of 2004
Open channel
Closed channel
Review: Scattering Length
Source: C. Regal
maB
2
1
Binding energy:
BEC-BCS Crossover
Source: C. Regal
Changing a at will: Technique of Feshbach Resonance
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.,
QMC calculations:
Chang. et al. (2004)Astrakharchik. et al. (2004)
44.042.0
210 )()(/ FF akak
A
E
A
E
(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
(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)
Epsilon Expansion
• Computing in dim.
• Expanding in
• Setting
(Nishida and Son)
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.
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
Reorganization of Lagrangian:
parts Free:0L
ions)(perturbat nsInteractio:1L
ies.singularit1 tormscounter te as serves:2 L
e.g.,
22BB
Pole:
Effective Field Theory: ac 0
Around the unitary limit: Expansion in B (binding energy)
For instance, Chemical potential, Energy/particle, to next-to-leading order in epsilon and up to O(B)
1,
2,
3,
Steps to
In the Unitary limit:
In BEC limit:
from large B expansion up to B^2, we find
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.
Energy per particle relative to that of free gas:
Blow-up of around unitary limit:
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.
Why is 4d special?
has a singularity at
for
ground state a free Bose gas