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