Transcript Diffusion Monte Carlo Study of Semiconductor Quantum Rings

Recent progress in Auxiliary-Field
Diffusion Monte Carlo computation of
EOS of nuclear and neutron 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.)
Punchlines
 High
quality (=benchmark) Diffusion
Monte Carlo calculations are available
now for pure neutron matter EOS with AV*
and U*-IL* potentials. Can we trust
presently available results?
 We have an accurate estimate of the gap
in superfluid NM
Our general goal
 SOLVE
THE NUCLEAR NONRELATIVISTIC PROBLEM WITH “NO”
APPROXIMATIONS BY DMC (~GFMC).
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
Inclusion of spin-orbit and three body forces is possible
(already done for pure neutron systems).
DMC for central potentials
Important fact:
The Schroedinger equation in imaginary time
is a diffusion equation:
 2 2


 2m   V ( R)  ET  ( R, τ)    τ ( R, τ)


where R represent the coordinates of the
nucleons, and t = it is the imaginary time.
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)
DMC for central potentials
We can write explicitly the propagator only for
short times:
 R|e
 H Δτ
| R'  G ( R, R' , Δτ) 
3A
2
 1 

e
2
 2 πσ 
2

2
σ 4
Δτ
2m
( R  R ') 2

2σ2
e
 V ( R ) V ( R ')


 ET  Δτ
2


DMC and Nuclear Hamiltonians
The standard QMC techniques are easy to apply
whenever the interaction is purely central, or
whenever the wavefunction can be written as a
product of eigenfunctions of Sz.
For realistic potentials the presence of quadratic spin
and isospin operators imposes the summation over all
the possible good Sz and Tz states.
A!
A
4
Z !( A  Z )!
The huge number
of states limits
present calculations
to A14
Auxiliary Fields
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.
Nuclear matter
Simulations
Most simulations were performed in a periodic box
containing 28 nucleons (14 p and 14 n). The density
was changed varying the size of the simulation box.
Particular attention must be paid to finite size effects.
•At the higher densities we performed a summation over
the first shell of periodic replicas of the simulation cell.
• Some checks against simulations with a larger number
of nucleons (N=76,108) were performed at the extrema
of the density interval considered.
Nuclear matter
Finite size effects
r/r0
E/A(28) [MeV]
E/A (76) [MeV]
E/A (108) [MeV]
0.5
-7.64(3)
-7.7(1)
-7.45(2)
3.0
-10.6(1)
-10.7(6)
-10.8(1)
CORRECTIONS ARE LESS THAN 3%!
Nuclear matter
AFDMC EOS differs from all
other previous estimates!
We computed the energy
of 28 nucleons interacting
with Argonne AV8’ cut to
six operators for several
densities*, and we
compare our results with
those given by FHNC/SOC
and BHF calculations**:
Wrong prediction
of PRL
rs 98,(expected)
•S. Gandolfi, F. Pederiva, S. Fantoni, K.E. Schmidt,
102503 (2007)
•**I. Bombaci, A. Fabrocini, A. Polls, I. Vidaña, Phys. Lett. B 609, 232 (2005).
Nuclei
Nuclei can be treated the same way as nuclear matter.
The main technical difference is in the construction of
wavefunctions with the correct symmetry for a given
total angular momentum J. At present we confine
ourselves to closed-shell nuclei (J=0) for which the
many-body wavefunction is expected to have full
spherical symmetry (J=0). In this case it is easy to write
the wavefunction as:
 A c
 ( R)   f ij  Afi (ri  Rcm , si )
 i j 
R: collective coordinate (space, spin, isospin), s: spin, isospin, Rcm: Center of
mass coordinate
Nuclei
We performed calculations for 4He, 8He, 16O, 40Ca with a AV6’ interaction
and without inclusion of the Coulomb potential.
E(4He)
E(8He)
(MeV)
(MeV)
AFDMC
-27.13(10)
-23.6(5)
GFMC1
-26.93(1)
-23.6(1)
EIHH2
-26.85(2)
---
1. R.B. Wiringa, S.C. Pieper, PRL 89, 182501 (2002)
2. G. Orlandini, W. Leidemann, private comm.
OPEN SHELL!! (only
1P3/2 filled, degenerate
with 1P1/2 w/o spinorbit)
Nuclei
E (MeV)
4He
-27.13(10)
E/A (MeV) Eexp/A (MeV)
-6.78
4xE(4He) = -108.52 MeV:
UNSTABLE!!
-7.07
8He
-23.6(5)
-2.95
-3.93
16O
-100.7(4)
-6.29
-7.98
40Ca
-272(2)
Periodic
(A=28)
---
10xE(4He) = -271.3 MeV:
-6.80
-8.55
BARELY
STABLE!!
-12.8(1)
---
Neutron Matter
We revised the computations made on Neutron Matter to
check the effect of the use of the fixed-phase
approximation.
Results are more stable, and some of the issues that were
not cleared in the previous AFDMC work are now under
control.
In particular the energy per nucleon computed with the
AV8’ potential in PNM with A=14 neutrons in a periodic
box is now 17.586(6) MeV, which compares very well
with the GFMC-UC calculations of J. Carlson et al.
which give 17.00(27) MeV. The previous published
AFDMC result was 20.32(6) MeV.
Neutron Matter
+ UIX
Equation of state of
PNM modeled with the
AV8’ potential with and
without the inclusion of
the three-body UIX
potential, compared with
the results of Akmal,
Pandharipande and
Ravenhall1.
1. A. Akmal, V.R. Pandharipande, and D.G. Ravenhall, PRC 58, 1804 (1998)
Neutron Stars
Mass-radius relation in
a neutron star
obtained solving the
Tolman Oppenheimer
Volkov (TOV) equation
using the EOS of pure
neutron matter from
AFDMC and
variational
calculations. Mass in
is units of M., radius
in Km
Neutron Stars
Mass-core density
relation in a neutron
star obtained solving
the Tolman
Oppenheimer Volkov
(TOV) equation using
the EOS of pure
neutron matter from
AFDMC and
variational
calculations. Mass in
is units of M., core
density in fm-3
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 )
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
r
3
1
r0  2.5 10  2.110
Conclusions
• AFDMC can be successfully applied to the study of
symmetric nuclear matter and pure neutron matter. Results
depend only on the choice of the nn interaction.
• The algorithm has been successfully applied to nuclei
• The estimates of the EOS computed with the same
potential and other methods are quite different.
• Pure neutron matter has been revised. The AP EOS
underestimates the hardness when a pure two body
potential is considered
•We have estimates of the gap within range of other DMC
and recent BHF calculations.
What’s next
 Add
three-body forces and spin-orbit in the
nuclear matter propagator (explicit  or
fake nucleons).
 Asymmetric nuclear matter (easy with
some redefinition of the boundary
conditions of the problem)
 Explicit inclusion of pion (and delta) fields:
development of an EFT-DMC (with P.
Faccioli and P. Armani, Trento).