/- $(WM]U_VKRl%43-=#1I\YRPAdkh9$
:LaFeAsO, -(ET)2Cu(SCN)2, EtMe3Sb[Pd(dmit)2]2

中村 和磨 (東大工, A03-9)

Post constrained RPA Project:Reduction of spatial dimension KN-Yoshimoto-Nohara-Imada, J. Phys. Soc. Jpn. 79, 123708 (2010)

z1
z2

z2
z3
z4

z1

z3
z4

KAZUMA NAKAMURA (A03-9)
YOSHIHIDE YOSHIMOTO (A02-5)
MASATOSHI IMADA (A03-9)
Acknowledge:
YOSHIRO NOHARA (Max Plank Institute)Aim and Background
Strong correlation and quantum fluctuationfrom first principles and prediction of newphases and functions of correlated materials
Ab initio construction of effective model
describing Low-energy property
Model analysis of derived model consideringstrong correlation and quantum fluctuationin high accuracy
LDA+Dynamical-Mean-Field Theory,
V. I. Anisimov, et al. J. Phys. Cond. Mat., 9, 767 (1997)
LDA+path-integral-renormalization-group;
Y. Imai, I. V. Solovyev, M. Imada, PRL 95, 176405 (2005)

Feasibility Studies (2006-prenent)(1) Iron-bansed superconductors:- KN-Arita-Imada, JPSJ 77, 093711 (2008)- Miyake-KN-Arita-Imada, JPSJ 79, 044705 (2009)- Misawa-KN-Imada, JPSJ, 80, 023704 (2011)- KN-Yoshimoto-Nohara-Imada, JPSJ 79, 123708 (2010)(2) Organic compounds:- KN-Yoshimoto-Kosugi-Arita-Imada, JPSJ 78, 083710 (2009)- Shinaoka-Misawa-KN-Imada, in preparation(3) Alkali-cluster-in-zeolite systems:- KN-Koretsune-Arita, PRB 80, 043941 (2009)(4) Transition metal and its oxides:- KN-Arita-Yoshimoto-Tsuneyuki, PRB 74, 235113 (2006)- Miyake-Aryasetiawan-Imada, PRB 80, 155134 (2009)(5) Excited states of semiconductors:- KN-Yoshimoto-Arita-Tsuneyuki-Imada, PRB 77,195126(2008)(6) Review:- Imada-Miyake, JPSJ 79, 112001 (2010)

Low-energy Hamiltonian

1) Basis function
2) Transfer integral
3) Screened Coulomb,
Screened exchange1)  Maximally localized Wannier function (MarzariEEEEVanderbilt 1997, Souza-Marzari-Vanderbilt 2002) 
2)  Matrix elements for DFT Kohn-Sham Hamiltonian3) constrained RPA; 
Original idea Aryasetiawan et.al. 2004, Solovyev-Imada 2005
Practical detail KN-Arita-Imada, JPSJ 77, 093711, 2008

RPA polarizability:

Virtual (V)(1)

Ef (2)
Target (T)Occupied (O)

(3)
(4)

LaFeAsO: constrained RPA
Interaction (eV)

KN-Arita-Imada, JPSJ 77, 093711 (2008)

1/r
bare
constrained RPAfull RPA

1/(6.7r)

cRPA is 3D interaction
with long-range taildecaying with powerr (Angstrom)

LaFeAsOWhat‘s the problem ? We derive ab initio parameters for
3D model, while we solve 2D modelin the analysis stage
LaO layer

FeAs layerWe have serious problem on“dimensional inconsistency”
LaFeAsO is quasi-2D system
Derived model = 3D model,Analyzed model = 2D model
Considering strong quantum
fluctuation effects with highaccuracy is considerably
difficult for the 3D modelReducing 3D to 2D
KEY IDEA: renormalize spatial dimension“Dimensional Downfolding”
We extend cRPA idea to the degreeof freedom of “spatial space”

3D
Interlayerinteraction
Intralayerinteraction

2D
deleteRenormalizedinteractiondeleteInterlayer screening
"a.@S
8,O_LTV\^'WV :_(d) WiekjX
cfg; LW "X$: &AS
5M`_[WPJ]Computational details:

1.
2.
3.
with

4.
5.
6.
with

LaFeAsO: Band & Wannier
Typical quasi-2D system,
good target of present study

xyyz

LaO
FeAs
LaO
FeAs

z2zx

x2-y2t 300 meVtD 10 meV

Interaction (eV)

LaFeAsO: 2D downfoldedbare
3D-cRPAfull RPAr (Angstrom)

2D-cRPA
2D-cRPASummary Nakamura-Yoshimoto-Nohara-Imada,J. Phys. Soc. Jpn. 79, 123708 (2010)

Interaction (eV)

We developed a new ab initio downfolding
scheme for deriving effective low-energymodels in low dimensions
It justify 2D short-ranged Hubbard models
as effective models from first principles

r (Angstrom)

r (Angstrom)

Performance Report for
Massively-Parallel Project
For constrained-RPA code 
KAZUMA NAKAMURA (A03-9)
YOSHIHIDE YOSHIMOTO (A02-5)
Acknowledge:
YOSHIRO NOHARA (Max Plank Institute)
YUICHIRO MATSUSHITA (OSHIYAMA Lab)HIROAKI ISHIZUKA (MOTOME Lab)

Computational cost
Nk

Nk Nb Nb

NPWcost  (Nk )2(Nb)2 NPW(Nk )2(Nb)2 NPW NkNb
=NkNbNPWO(10) O(10)
= 10,000 (if Nk =100, Nb =1,000)

Need: distributed-memory codeEtMe3Sb[Pd(dmit)2]2

Memory size of  ~ 400
Gbyte with Nband=2000,Nk=125, NPW = 100000
The  data cannot be
stored by single nodealone

Need of development for“distributed-memory RPA code”

For massively parallelization I 
Proposed byYOSHIHIDEYOSHIMOTODivision of  data; Step1

Step2

Step3

Step4

occ
unocc
occ

Data split

unocc
12345678910

1E23456789unocc 12345678910 Data send
to MPI
10

12345678910

1E2345678910

unocc 12345678910

occ

occ

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

Calc 

For massively parallelization II 
Step6

Step7

12345678910

12345678910

10123456789 

12345678910

12345678910

10123456789 Data Rotation
MPI_SENDRECV

Calc 

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10

Step8

Step9

12345678910

12345678910

910

12345678

12345678910

12345678910

910

123456781 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +10

Data Rotation
MPI_SENDRECV

Calc 

MPI_COMM_SPLITTNR
core=128
W2a5H_MPI_COMM_SPLIT(q1)

(q2)

(q3)

(q4)
- 4comm(q ; 7?) - 8MPI*4OMP/comm(;0; B7 ?) Performance of ourCode: Benchmark for small
System: SrVO3

SrVO3@kashiwa 2q**;0B7WZC(n=MPIOMP)

ntime(sec)1341.4
4(1x4)
89.9
8(2x4)
49.5
12(3x4)33.8
16(4x4)27.3

+CCCE

98.2
97.7
98.3
98.1

+CCCCC

94.9
86.2
84.1
78.2>+
3.8
6.9
10.1
12.5

SrVO3@kashiwa 20q**;0+7C(n=COMMMPIOMP)
+CCCCC

91.389.3
87.685.8>+
7.3
28.6
70.1
137.3





+
db

+CCCE

98.6
99.6
99.8
99.9+

ntime(sec)135908( 1x2x4)
49232( 4x2x4)
12680(10x2x4)
51160(20x2x4)
26>+

B7

db

db

Test Run at 2011/1/14:
2048-cores calculation
Performance of ourcode: Benchmark for large
system: C60

C60@kashiwa 1q**;0B7WZC(n=MPIOMP)

ntime(sec) +CCCC
+CCCCCC >+115639.14( 1x4) 4077.0
98.6
95.9
3.88( 2x4) 2108.3
98.9
92.7
7.4
16( 4x4) 1071.3
99.4
91.2
14.6
32( 8x4)
542.6
99.6
90.1
28.864(16x4)
297.9
99.7
82.0
52.5

C60@kashiwa 32q**;0

B7 +7 C(n=COMMMPIOMP)
7 ?WZVQGRW6<!SF_LTV)
>+
98.81 126.72
97.79 250.24
98.60 504.96
97.62 999.04
94.04 1925.76 

99.98
99.99
100.00100.00100.00+

+CCCCC

+db

+CCCC

>+

ntime(sec)
64( 1x16x4) 9202.83
128( 2x16x4) 4657.06
256( 4x16x4) 2352.64
512( 8x16x4) 1166.69
1024(16x16x4)
589.332048(32x16x4)
305.81db

db

Product Run at 2011/2/11:
4096-cores calculation
Constrained RPA for dmit

Condition of product run:
- Nk=75 (553),- Nband = 1000 (Nocc= 464, Npocc= 4, Nvir= 532),- Ecut() = 36 Ry (100,000 PWs),- Ecut( ) = 4.0 Ry (3,200 PWs)

Architecture and performance:
- SGI Altix ICE 8400EX sytem- X5570(4core)2- Ifort 11.1, SGI-oriented MPI, InfiniBand- 4096 core (4comm128MPI8omp)- Total time = 43h19min

Dielectric function: dmit and -bedt
-bedt|q + G| (a.u)|q + G| (a.u)- 4096 cores- 43h19min- kashiwa- 128 cores- 384h (16days)- SR11000@ITC- 1/6 of dmit M(q+G)

dmit

Convergence:

-bedtdmit
20.0eV M(q(q+G))

Energye gy (eV)

12.5eV||q +G|
G| (a.u)( )|q +G| (a.u)

Interaction (eV)

3D-cRPA Interaction: dmit and -bedt
dmit

-bedtr (Angstrom)

r (Angstrom)

bare
3D-cRPAUnfortunately dmit yet to be converged…

APPENDIXComputational data:
z2

z3

z1

z4

z2

z3

-Cu(SCN)2: Band & Wannier
Geometry

Wanniert 65 meVtD 0.1 meV Interaction (eV)

-Cu(SCN)2: 2D downfolded bare
3D-cRPAfull RPAr (Angstrom)

2D-cRPA
2D-cRPAInteraction (eV)

Screening length
LaFeAsO

zero at 8.4År (Å)

c=8.4Å

-Cu(SCN)2

zero at 16.4År (Å)

c=16.4ÅThus, screening length of interlayer
screening corresponds to the c value

Feynman Diagram for
Screened interaction
z1
z2

Coulomb interaction
between electrons atz1 and z4 are screenedby RPA polarization of (z2,z3)

z3
z4

Interlayer screening
z2

z1

z3

Electrons at z1 and z4
are in target layer, while
screened electrons existz4 in z and z on other layer
23

other types of interlayer screening:
z3
z3
z2
z1
z4
z2
z11
z4
z1
z2

z4
z3

Computational details:
(0) Below is post-cRPA story
(1) Target-band-RPA (2) Fourier transform of 
wave vector in BZ
reciprocal lattice vector in-plane,  out-of-planeLayer 1
CCCLayer 2= targetz2

z1

z3

z3

z4

Layer 3

z2

z3
z3

Layer 1
CCCLayer 2= targetz2
z11

z4

Layer 3

z2

z3
z3

Layer 1
CCCLayer 2= target
Layer 3

z1

z4

z2

z2

z3
Layer 1
CCCLayer 2= targetz2

z1

z4
z3

Layer 3z2We have to cut this polarization to avoid
double counting of it
Layer 1
CCCLayer 2= targetz2

z1

z4
z3

Layer 3z2z3

(3) Polarization cuttingCUT0(4) Inverse FT of cutg, g’: reciprocal lattice vector of super lattice(5) 2D dielectric function2D(6) 2D screened Coulomb(7) 2D screened exchangeInteraction (eV)


LaFeAsO: cRPA (previous slide)bare
3D-cRPA
3D-cRPAfull RPAr (Angstrom)

Program structurePolarization
do q = 1, Nk
do  = 1, Npair
do k = 1, Nkcall FFT module to calculate

enddocall TETRAHEDRON module to calculate
do G=1, NPW
do G’=1, NPW
do  = 1,N
do k=1, Nk
enddo
enddo
enddo
enddo
enndo
enddo