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Nuclear Effective Interaction

From Correlated Basis Functions

Omar Benhar

INFN and Department of Physics, “Sapienza” University

I-00185 Roma, Italy

Bridging nuclear ab-initio and energy-density-functional theories

IPN, Orsay, October 2-6, 2017

O UTLINE

? Motivation

? The paradigm of Nuclear Many-Body Theory

? Derivation of the CBF effective interaction

? Applications: the nuclear matter equation of state and beyond

? Perspectives & Open Questions

1 / 18

M OTIVATION

? Astrophysical applications require theoretical approaches

capable to provide a consistent description of properties of

nuclear matter other than the equation of state at zero

temperature, including the neutrino emission and absorption

rates, the transport coefficients as well as the superfluid and

superconducting gaps

? Effective interactions obtained from a microscopic nuclear

dynamics—strongly constrained by phenomenology—combine

the flexibility of the effective interaction approach with the

ability to provide a realistic description of a variety of

equilibrium and non equilibrium properties.

2 / 18

T HE PARADIGM OF ab initio N UCLEAR M ANY-B ODY T HEORY

? Nuclear systems can be described as a collection of pointlike

protons and neutrons interacting through the hamiltonian

H=

X p2

X

X

i

+

vij +

Vijk = H0 + HI

2m j>i

i

k>j>i

? The Hamiltonian is determined from the properties of exacltly

(A = 2, 3)—or nearly exactly (A → ∞)—systems. In principle,

describing the properties of more complex systems should not

require additional adjustable parameters.

? The matrix elements of HI between eigenstates of H0 , are

generally large, and cannot be used to do standard perturbation

theory

? Effective interactions are designed to obtain accurate estimates of

nuclear properties at lowest order perturbation theory

? Ideally, effective interactions should be derived from the bare

Hamiltonian

3 / 18

I NTRODUCING THE E FFECTIVE I NTERACTION

? Consider nuclear matter. The eigenstates of H0 are Fermi gas

states {|nF G i}

? Taming the matrix element of the Hamiltonian

hmF G |H|nF G i ⇒

hmF G |Heff |nF G i (H ⇒ Heff )

hm|H|ni

({|nF G i} ⇒ {|ni})

. Use the effective Hamiltonian Heff in standard perturbation theory

with Fermi gas basis states, as in the G-matrix approach

. Use the bare Hamiltonian to do perturbative calculations in the new

basis, as in the approach based on Correlated Basis Functions (CBF)

? The effective interaction must be designed in such a way as to

provide accurate estimates of nuclear matter properties at lowest

order of standard perturbation theory

4 / 18

? In principle, the two approaches may be merged defining the

new basis states through the transformation

|ni = F |nF G i

leading to

Heff = F † HF

? Implementing this simple prescription requires qualifications,

associated with the definition and determination of F

? In the generalised Jastrow ansatz

Y

X

(n)

f (n) (rij )Oij

F =

Fij , Fij =

j>i

n

with the operator structure of Fij reflecting the one of the

nucleon-nucleon (NN) potential. As rij → ∞, Fij → 11

5 / 18

D ETERMINATION OF THE CORRELATION FUNCTION

? The shape of the f (n) (rij ) is determined variationally,

minimising the expectation value of the Hamiltonian in the

correlated ground state, evaluated using the cluster expansion

techinque

h0|H|0i

≥ E0

hHi =

h0|0i

? Accurate nuclear calculations can be carried out exploiting the

cluster expansion formalism

X

∂

hHi = EF G +

lnh0|eβ(H−EF G) |0i

= EF G +

∆E` [F ]

∂β

β=0

`≥2

where ∆E` [F ] is the contribution arising from clusters of `

correlated particles, and summing up all relevant contributions

solving the FHNC/SOC integral equations

? hHi is minimized with respect to a set of variational parameters

determining the range of the correlation functions

6 / 18

F ROM hHi TO hHeff i

? The effective interaction is defined through the equation

X

hHiFHNC/SOC = EF G +

∆` [F˜ ]

`≤`max

= EF G + h0F G |Veff |0F G i

with

Veff =

X

j>i

(n≤6)

Oij

ij

ij

veff

, veff

=

X

(n)

(n)

veff (rij )Oij

n

= [1, (σi · σj ), Sij ] ⊗ [1, (τi · τj )]

3

Sij = 2 (σi · rij )(σj · rij ) − (σi · σj )

rij

and the range of F˜ is adjusted in such a way as to obtain the

FHNC/SOC results at low order of the cluster expansion

7 / 18

? Pedagogical example: neglecting three-nucleon forces, one may

set `max = 2, and obtain

1 ˜ 2 ˜

ij

veff

=

∇Fij + Fij vij F˜ij

m

? Note that the correlation function F˜ij depends on density, and so

does the effective interaction

? Adding three-body cluster terms allows to take into account the

leading contributions arising from the three-nucleon potential,

the inclusion of which is essential to obtain saturation in

isospin-symmetric nuclear matter

? The resulting effective interaction reproduces the FHNC/SOC

ground-state energies of both isospin-symmetric nuclear matter

(SNM) and pure neutron matter (PNM). It can be used to

describe matter at fixed baryon density and arbitrary proton

fraction

8 / 18

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