Unitarity potentials and Unitarity potentials and neutron matter at unitary neutron matter at unitary limit limit

T.T.S. Kuo (Stony Brook)

H. Dong (Stony Brook), R. Machleidt (Idaho)

Collaborators:

Atom-Atom interaction V for trapped cold fermionic gases can be

experimentally tuned by external magnetic field, giving many-body problems with tunable interactions:

By tuning V to Feshbach resonance, scattering length . At this limit (unitary limit), interesting physics observed.

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(H0 + VAA )Ψ0(A) = E0Ψ0(A),

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A → ∞

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as → ∞

AA

AA

Near the Feshbach resonance ( )

At , the equation of state (EOS)

Above often known as ‘Bertsch challenge problems’

€

1

as

= 0

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−1

as

< 0

€

−1

as

> 0

gas in BEC

gas in BCS

€

1

as

= 0

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E0 = ξE0free

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E0free

A=

3h2kF2

10m

⎛

⎝ ⎜

⎞

⎠ ⎟has an ‘universal’ form:

with ξ=0.44 for ‘all’ gases.

depends only on

€

E0

€

kF

this is BCS-BEC cross-over

Experimental values for ξof atomic gases

ξ Authors

0.39(15) Bourdel et al., PRL (2004)

0.51(4) Kinast et al., Science (2005)

0.46(5) Partridge et al., Science (2006)

0.46 Stewart et al., PRL (2006)+0.05-0.12

Neutron matter is a two-species fermionic system,

should have same unitary-limit properties as cold fermi gas, and neutron-neutron fm, it is rather long.

We study the EOS of neutron matter at and near the unitary limit, using different unitarity potentials

If is universal, then results should be independent of the potentials as long as their

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as ≈ −19

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E0 = ξE0free

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as → ∞

How to obtain unitarity potentials with ?

tuned CDBonn meson-exchange potential

tuned square-well ‘box’ potentials

How to calculate ground state energy ?

ring-diagram and model-space HF methods

Results and discussions

€

as → ∞

€

E0

Atom-Atom interactions V can be experimentally varied by tuning

external magnetic field.

How to vary the NN interaction ? Can we tune experimentally? May use Brown-Rho scaling to tune

, namely slightly changing its meson

masses.

Ask Machleidt to help!

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VNN

€

VNN

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VNN

AA

CD-Bonn ( S ) of different

We tuned only m , as attraction in S mainly from σ-exchange. depends sensitively on m .

m [MeV] a [fm]

original 452.0 -18.98

tunned 475.0 -5.0

447.0 -42.0

442.850 -∞(-12070)

442.800 +∞

434.0 +21

10

€

VNN

€

as

sσ

σ1

0

€

as

σ

We have also used hard-core square-well (HCSW) potentials

Their scattering length ( ) and effective range ( ) can be obtained analytically.

We can have many HSCW unitarity potentials

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V (r) = Vc;

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r < rc

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rc < r < rb

€

as

€

re €

r > rb

€

Vb;

€

0;

Phase shiftsδfor HCSW potentials:

with

where E is the scattering energy.

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tan(δ + K3rb ) =K3

K2

tan(K2rb + α )

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tan(α + K2rc ) =K2

K1

tanh(K1rc )

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K1 = (Vc − E)m

h2,

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K2 = (E −Vb )m

h2,

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K3 = Em

h2

From phase shift δ, the scattering length is

where

The effective range also analytically given.

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as = −B

A

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A = K10K20 − K202 tanh(K10rc )tan[K20(rb − rc )],

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B = K20 tanh(K10rc ) + K10 tan[K20(rb − rc )]

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K10 = Vc

m

h2,

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K20 = −Vb

m

h2.

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−rbK10K20 + rbK202 tanh(K10rc )tan[K20(rb − rc )],

with

Condition for unitarity potential is

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rb − rc =1

K20

tan−1[K10

K20 tanh(K10rc )]

Potentials

Vc /MeV

rc /fm

Vb /MeV

rb /fm

as /x10 fm

reff /fm

HCSW01 3000 0.15 -20 2.31

15.2 2.36

HCSW02 3000 0.30 -30 2.03

3.38 2.21

HCSW03 3000 0.50 -50 1.81

-4.58 2.20

c c b c s e

Three different HCSW unitarity potentials

6

Ground state energy shift

Above is quasi-boson RPA

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ΔE0 = E0 − E0free

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ΔE0 = dλ Ym (ij,λ )Ym* (kl,λ ) ijVlow−k kl

ijkl<Λ

∑m

∑0

1

∫

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Ym*(kl,λ ) = Ψm (λ , A − 2) alak Ψ0(λ ,A)

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AX + BY = ωX

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A*Y + B*X = −ωY

By summing the pphh ring diagrams to all orders,the transition amplitudes Y are given by the RPA equations:

Model-space approach:Space (k > Λ) integrated out: renormalized to has strong short range repulsion is smooth and energy independentSpace (k ≤ Λ) use to calculate all-order sum of

ring diagrams

Note we need of specific scattering length

including

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Vbare

€

Vbare

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Vlow−k

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Vlow−k

€

Vlow−k

€

Vlow−k

€

as → ±∞

of specific scattering length Starting from a bare CD-Bonn potential

of scattering length a, given by

obtained from solving the above T-matrix equivalence equations using the iteration method

of Lee-Suzuki-Andreozzi

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T(k ',k,k 2) = V a (k ',k) + q2dqV a (k ',q)T(q,k,k 2)

k 2 − q2 + i0+0

∞

∫

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Tlow−k (p', p, p2) = Vlow−ka (p', p)

);,,(),,( 2'2' pppTpppT klow−= Λ≤),( ' pp

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+ q2dqVlow−k

a ( p',q)Tlow−k (q, p, p2)

p2 − q2 + i0+0

Λ

∫

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Vlow−ka

€

Vlow−ka

€

Vlow−ka

Ring diagram unitary ratio Ring diagram unitary ratio given by different unitarity given by different unitarity potentials potentials

Diagonal matrix elements Diagonal matrix elements of of VV NN

The ring-diagram unitary ratio The ring-diagram unitary ratio near the unitary limitnear the unitary limit

When choosing Λ= k , ring-diagram methodbecomes a Model-Space HF method,

and E /A given by simple integral

Here means Λ= k .

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ΔE0 =1

2

r k 1

r k 2 Vlow−k

kFr k 1

r k 2

k1 ,k2 ≤kF

∑

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E0

A=

3

5εF +

8

πk 2dk[1−

3k

2kF

+k 3

2kF3

]0

kF∫

€

× (2Jα +1) α ,k Vlow−kkF α ,k

α

∑

F

0

F

€

Vlow−kkF

MSHF has simple relation between ξand :

is highly accurately simulated by momentum

expansions:

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3εF

5(ξ −1) =

8

πk 2dk[1−

3k

2kF

+k 3

2kF3

]0

kF∫

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× 1S0,k Vlow−kkF 1S0,k

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k Vlow−kkF k = V0 + V2(

k

kF

)2 + V4 (k

kF

)4

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3π

10(ξ −1) = kF (

V0

3+

V2

10+

3V4

70)

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Vlow−kkF

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Vlow−kkF

0 2 4where V , V and V are constants. Then

Above is a strong sum-rule and scaling constraint

for at the unitary limit.

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Vlow−kkF

Checking for four unitarity potentials

V k /fm

V /fm

V /fm

V /fm

Sum ξ

CDBonn 1.2 -2.053 3.169 -1.801 -0.445 0.434

HCSW01

-2.001 2.865 -1.402 -0.441 0.439

HCSW02

-1.904 2.373 -0.999 -0.440 0.440

HCSW03

-1.893 2.261 -0.825 -0.440 0.439

HCSW01

1.0 -2.102 2.202 -1.070 -0.526 0.442

HCSW01

1.4 -1.945 3.584 -1.983 -0.375 0.443

NN F 0 2 4-1

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3π

10(ξ −1) = kF (

V0

3+

V2

10+

3V4

70)

Comparison of V from four differentunitarity potentials (Λ= k =1.2 fm)

F

low-kkF

ξ Method Ref.

0.326,0.568 Padé approximation Baker et al., PRC(1999)

0.326 Galitskii resummation Heiselberg et al., PRA(2001)

0.7 Ladder approximation Bruun et al., PRA(2004)

0.455 Diagrammatic theory Perali et al, PRL(2004)

0.42 Density functional theory

Papenbrock et al., PRA(2005)

0.401 NSR extension with pairing fluctuations

Hu et al., EPL(2006)

0.475 εexpansion Nishidaet al., PRL(2006)

0.360 Variational formalism Haussmann et al., PRA(2007)

0.475 εexpansion Chen et al., PRA(2007)

0.44(1) Quantum Monte Carlo Carlson et al., PRL(2003)

0.42(1) Quantum Monte Carlo Astrakharchik et al., PRL (2004)

0.44 Ring-diagram and MSHF

This work (Dong, et al., PRC 2010)

Comparison of recent calculated values on ξ

MSHF single-particle (s.p.) potential is

with k =(k - k )/2, k =(k + k )/2.

The MSHF s.p. spectrum is which can be well approximated by

m* is effective mass andΔis effective ‘well-depth’ .

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ε(k1) =h2k1

2

2m+ U(k1)

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ε(k1) =h2k1

2

2m*+ Δ

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+2

πk1

kdk[kF2 − k1

2 + 4k(k1 − k)] α ,k Vlow−kkF α ,k

k_

k+∫ }

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U(k1) = (2Jα +1){16

πk 2dk

0

k−∫ α ,k Vlow−kkF α ,k

α

∑

− F 1 + F 1

At the unitary limit, m* and Δof MSHF should obey the linear constraint

We have checked this constraint.

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ξ =1

2+

m

2m *+

5Δ

6εF

Check at unitary limit

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ξ =1

2+

m

2m *+

5Δ

6εF

Summary and Summary and outlook:outlook:Our results have provided strong

‘numerical’

evidences that the ratio ξ= E / E is a universal constant, independent of the interacting potentials as long as they have .

However, it will be still challenging to prove

this universality analytically !

0 0free

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as → ∞

Thanks to organizers

R. Marotta and N. Itaco