3D ASLDA solver3D ASLDA solver- status report- status report

Piotr Magierski (Warsaw), Aurel Bulgac (Seattle), Kenneth Roche (Oak Ridge),

Ionel Stetcu (Seattle)Ph.D. Student: Yuan Lung (Alan) Luo

General outlookGeneral outlookGoal: Goal: Provide a 3D DFT solver.Provide a 3D DFT solver.

Main features:Main features:- Defined on a rectangular spatial lattice Defined on a rectangular spatial lattice (by means of DVR),(by means of DVR),- FFT is used for calculations of derivatives,- FFT is used for calculations of derivatives,- No symmetry constraints,No symmetry constraints,- Able to describe superfluid neutron-proton Able to describe superfluid neutron-proton system with arbitrary spin asymmetrysystem with arbitrary spin asymmetry (ASLDA – Asymmetric Superfluid Local (ASLDA – Asymmetric Superfluid Local Density Approach)Density Approach)- Solves BdG (HFB) equations.Solves BdG (HFB) equations.- Parallelizable.Parallelizable.- Possible extensions: 1D, 2D, other large Possible extensions: 1D, 2D, other large Fermi systems (ultracold atoms)Fermi systems (ultracold atoms)

Summary (2008)Summary (2008)

1. Is your Year-2 plan well on track? Absolutely.2. What are the aspects of your science that require high-performance computing? Data and computational complexity.3. Plan for Year-2 and Year-3: - profile current code. - parallelization of the code. - self-consistent version. - interface between TDSLDA code.

At the moment ‘the non-selfconsistent DFT solver’ exist.It can handle arbitrary shape of potentials in p-h and p-p channels.The accuracy of the results is stable with respect to variations of the potentials,chemical potentials, energy cutoff, effective masses, etc.

Accomplishments since the last meeting:1) self-consistent version of the code (Broyden method applied)2) nuclear density functional coded3) interface between TDSLDA code4) 2D version of ASLDA (parallelized) for ultracold atomic gas

Procedure

Diagonalizationof the norm matrix

Imaginary time step(using p-h potential)

Time step isvaried (decreasing)

G-S method

Construct and solveBdG equations with achosen energy cutoff.

p-h and p-p potentialsare updated

(N - large) (N - smaller)

N – number of evolved wave functions

Particle-hole mean field:Particle-hole mean field:

20 0 0 0

20

( )

( ) ( ( ))

s sskyrme

sT J

H C C s s C C s s C j j

C s T J C J s j

���

( ) ( ) ( )

( ) ( ) ( ) ( )

1( ) ( ) ( ) ( ) ( )

2 ext

h r B r C r

h r B r C r iA r

V r U r S r i A r V r

ˆ( 1) ( )

( 1) ( 1)

, 1,...,

- imaginary time step

n h nj j

orthogonalizationn nj jnormalization

e j N

1ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) ( ( ) ( ) )2

ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( )

ˆ ˆ( ) ( )

h r B r U r iW r S r C r A r A ri

h r h rh r h r iW r V r

h r h r

Skyrme density functional:

Imaginary time step evolution:Imaginary time step evolution:

ˆ 1

1

( ) ˆ1 ( )!

inh i n

i

e h Oi

Usually used for n=1Drawbacks: - slow convergence. - divergent for too large imaginary time step. - numerically costly for n>1.

2ˆ ˆ ˆ ˆˆ 2 32 2 2 2ˆ ˆ(1 ) ( ), - second order method2

where

ˆ - is local in momentum space

ˆ- is local in space

ˆ -contains mixed terms (e.g. spin-orbit)

T V V The e e M M e e O

T

V

M

ˆ( ) ( ) ( ) ( ) ( ) ( )h r h r V r h r iW r

REAL SPACEREAL SPACE MOMENTUM SPACEMOMENTUM SPACEFFTFFT

2

ˆ 2

ˆ ( )

( ) ( )

( ) ( )

pT m

V V r

e p e p

e r e r

The above expansions are particularly useful if the Hamiltonian can be expressed as a sum of terms either local in the coordinate space (potential),

or in the momentum space (kinetic energy). In such a case one is adviced to use Fast Fourier Transform algorithm

during the evolution of the wave function.

Advantages:- Much faster convergence (order of magnitude difference between the first order and the second order method).- The methods do not diverge even for large time steps.- The low cost of FFT instead of matrix multiplication.

ˆ( ) ( ) ( ) ( ) ( ) ( )h r h r V r h r iW r

How to decrease magnitude of mixed terms:

( ) ( )ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )r rn nh r r e h r V r h r iW r e r

2

( ) ( )

( ) ( ) 2 ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

where

h r h r

h r h r h r r

V r V r h r r r h r r

ˆ ˆ( ) ' ( )( ) ' ( ), ' ( ) ( )h r h rn n n n

then

e r e e r r e r

*

In particular transformation:

( )' ( ) ( )

allows to get rid of the effective mass term.

n nm r

r rm

† *

' ' '

' 3'

'

* *' '*

*''

0ˆ0

* ( )0 ( )* ( ) * ( )

* ( )( ) 0

( ) ( ) ( ) ( )( )

( ) (

nn n n nn

nnn n n

n

k k k kT

kkk k

h U UE

V Vh

h h

rrr r d r

rr

r r r rr V V

r

*, ' '

'*

'', '

;) ( ) ( )

0 ( ) ( ')( )

( ) ( ') 0

k k k k

k kT

kkk kk k

r r r

r rr V U

r r

In the basisIn the basis( )

( )( )

nn

n

rr

r

one solves BdG(HFB) eq.

0 02

020

0 2

0

2

0

1( ) ( )

/ 21 log

42

( )( )

2

( )( )

2

cut cut

cut

cutcut

r g rg k k k k

k kC

m

V rk r

Cm

E V rk r

Cm

kkyy

kkxx

2π/L

x

x

1

1 2

2 2 2 2 21

1 2

External potential in p-h channel

( )1 cosh 1 cosh 1 cosh

0.4 ( ) ; 0.2

120 , 250 , 3.5 , 2.0 , 5

xyext r r z

a b b

xy

V VV r

r x y z r x y

V MeV V MeV a fm b fm fm

External potential in p-p channel: ( ) 2ext r MeV

Preliminary results (neutrons – SkM*):

Number of wave functions: 40Number of particles: 20 Box size: 24x24x24 fm^3

= 0 - spin symmetric systemN N

N N

= 0.8 - spin asymmetric system

N N

N N

( , 0, )x y z

( , 0, )s x y z

( , 0, )j x y z

( , 0, )J x y z

( , 0, )x y z ( , 0, )x y z ( , 0, )x y z

( , 0, )J x y z

z(fm)

z(fm)

z(fm)

z(fm)

x(fm

)

x(fm

)x(

fm)

Relations between the codes (Map of the codes)

Nuclear problem Ultracold gas problem

Stationary

Time-dependent

3D ASLDA(neutrons coded)

3D TDSLDA(p-n, parallelized

version exists)

2D ASLDA(parallelized version

exists)

2D, 3D TDSLDA(parallel)

Generation of initial conditions

See talks of Aurel Bulgac (today), Kenny Roche and Ionel Stetcu (tomorrow)

1. Is your Year-3 plan well on track? Absolutely.2. What are the aspects of your science that require high-performance computing? Data and computational complexity.3. Plan for Year-3 - profile current code. - parallelization of the code + proton-neutron extension - applications and generation of initial conditions for TDSLDA4. Plan for Year-4 - further applications: calculations of odd nuclei, neutron droplets...

Accomplishments since the last meeting:1) self-consistent version of the code (Broyden method applied)2) nuclear density functional coded3) interface between TDSLDA code4) 2D version of ASLDA (parallelized) for ultracold Fermi gas

Summary 2009