Transcript Diffusion Monte Carlo Study of Semiconductor Quantum Rings

Superfluidity of Neutron and Nuclear
Matter
F. Pederiva
Dipartimento di Fisica
Università di Trento I-38050 Povo, Trento, Italy
CNR/INFM-DEMOCRITOS
National Simulation Center, Trieste, Italy
Coworkers
S. Gandolfi (SISSA)
A. Illarionov (SISSA)
S. Fantoni (SISSA)
K.E. Schmidt (Arizona S.U.)
Why is it of interest?
• “Superfluidity” of nuclei as been long known. Attractive
components of the NN force induce a pairing among
nucleons. A few outcomes of it are the even-odd staggering
of binding energies or anomalies on the momentum of
inertia.
• More recently superfluidity of bulk nuclear matter has
been recognized to play a role in the cooling process of
neutron stars.
Superfulidity and cooling of neutron
stars
Superfluidity of nucleons has essentially three
effects on neutrino emission in neutron stars (see
e.g. Yakovlev, 2002):
1. Suppresses neutrino processes involving nucleons
(e.g. direct URCA process)
2. Initiates a specific mechanism of neutrino emission
associated with Cooper pairing of nucleons
3. Changes the nucleon heat capacity
Superfulidity and cooling of neutron
stars
The equations of thermal evolution of a NS are due to Thorne
(assuming the internal structure independent on the temperature):
cv T
1
2Gm  2 
1 2
e Lr  Q  
2 2
4r e
c r r
e t


Lr
2Gm  



1

e
Te
4r 2
c2r
r
 
Changes occur when T=Tc in a given pairing channel.
It is therefore necessary to know the critical temperature Tc as a function of
the density of the nuclear matter.
Superfluid gap
The easier way to estimate Tc is through the evaluation of
the pairing gap. For instance, in the BCS model we have

Tc 
1.76
The pairing gap  has been estimated by various theories
and in different channels (mainly 1S0 and 3P2-3F2).
WE USE AFDMC to estimate the pairing gap as a function
of density.
Nuclear Hamiltonian
The interaction between N nucleons can be written
in terms of an Hamiltonian of the form:
M
pi2
H 
  v p ( rij )O ( p ) (i, j )  V3
i 1 2mi
i  j p 1
N
where i and j label the nucleons, rij is the distance
between the nucleons and the O(p) are operators
including spin, isospin, and spin-orbit operators.
M is the maximum number of operators (M=18
for the Argonne v18 potential).
Nuclear Hamiltonian
The interaction used in this study is AV8’ cut to
the first six operators.
O
p 1... 6
 (1, σi  σ j , Sij )  (τ i  τ j )
where
Sij  3(rij  σi )(rij  σ j )  σi  σ j
EVEN AT LOW DENSITIES THE DETAIL OF THE
INTERACTION STILL HAS IMPORTANT EFFECTS
(see Gezerlis, Carlson 2008)
DMC for central potentials
The formal solution
( R, τ)  e
 ( H  ET ) τ
e
( E0  ET ) τ
( R,0)
c00 ( R,0)   e
n 0
( En  ET ) τ
cn n ( R,0)
converges to the lowest energy eigenstate not
orthogonal to (R,0)
Auxiliary Fields DMC
The use of auxiliary fields and constrained paths is originally
due to S. Zhang for condensed matter problems (S.Zhang, J.
Carlson, and J.Gubernatis, PRL74, 3653 (1995), Phys. Rev. B55. 7464
(1997))
Application to the Nuclear Hamiltonian is due to S.Fantoni
and K.E. Schmidt (K.E. Schmidt and S. Fantoni, Phys. Lett. 445, 99
(1999))
The method consists of using the HubbardStratonovich transformation in order to reduce
the spin operators appearing in the Green’s
function from quadratic to linear.
Auxiliary Fields
For N nucleons the NN interaction can be re-written as
V  Vsi  Vsd  Vsi 
sA

 
i
s
i α; j β j β
i ;j
where the 3Nx3N matrix A is a combination of the various
v(p) appearing in the interaction. The s include both spin
and isospin operators, and act on 4-component spinors:
 i  ai n  bi p   ci n  di p 
THE INCLUSION OF TENSOR-ISOSPIN TERMS HAS BEEN THE
MOST RELEVANT DIFFICULTY IN THE APPLICATION OF AFDMC
SO FAR
Auxiliary Fields
We can apply the Hubbard-Stratonovich transformation to
the Green’s function for the spin-dependent part of the
potential:
e
Vsd Δτ
3N
 e
1
 λ n On2 Δτ
2
Commutators
neglected
n 1
e
1
 λ n On2 Δτ
2
1

2π

 xn2

dxn exp  2   λ n Δτ xnOn 
The xn are auxiliary variables to be sampled. The effect of
the On is a rotation of the spinors of each particle.
Nuclear matter
Wave Function
The many-nucleon wave function is written as the product
of a Jastrow factor and an antisymmetric mean field
wave function:
 (r1...rN ; σ1...σ N ; τ1...τ N )  fJ (rij ) A(r1...rN ; σ1...σ N ; τ1...τ N )
i j
The functions fJ in the Jastrow factor are taken as the
scalar components of the FHNC/SOC correlation operator
which minimizes the energy per particle of SNM at
saturation density r0=0.16 fm-1. The antisymmetric
product A is a Slater determinant of plane waves.
1S
0
gap in neutron matter
AFDMC should allow for an accurate estimate of the gap in superfluid
neutron matter.
INGREDIENT NEEDED: A “SUPERFLUID” WAVEFUNCTION.
Nodes and phase in the superfluid are better described by a
Jastrow-BCS wavefunction


T ( R)   f J (rij )fBCS ( R, S )
 i j

where the BCS part is a Pfaffian of orbitals of the form
f (rij , si , s j )  
a
vka
uk a
e
ik rij
 ( si , s j )
Coefficients from CBF calculations! (Illarionov)
Gap in neutron matter
The gap is estimated by the even-odd energy difference at
fixed density:
1
( N )  E ( N )  E ( N  1)  E ( N  1)
2
•For our calculations we used N=12-18 and N=62-68. The gap
slightly decreases by increasing the number of particles.
•The parameters in the pair wavefunctions have been taken by CBF
calculatons.
Gap in Neutron Matter

3
1
0  2.5 10  2.110
Gandolfi S., Illarionov A., Fantoni S., P.F., Schmidt K., PRL 101, 132501 (2008)
Pair correlation functions
Gap in asymmetric matter
Conclusions
• AFDMC can be successfully applied to the study of
superfluid gaps in asymmetric nuclear matter and pure
neutron matter. Results depend only on the choice of the
nn interaction.
• Calculations show a maximum of the gap of about 2MeV
at about kF=0.6 fm-1
•Large asymmetries seem to increase the value of the gap
at the peak
•A more systematic analysis is in progress.