CCMSハンズオン: HΦ講習会@柏の葉キャンパス

HΦの概要-プログラムとアルゴリズムOverview of HΦ: Program & Algorithm
山地 洋平
東京大学大学院工学系物理工学専攻
Youhei Yamaji
Department of Applied Physics, The University of Tokyo

1. Introduction to HΦ
2. Algorithm: Lanczos & LOBCG
3. Algorithm: TPQ
Appendix:
Formulation of many-body quantum systems in a nutshell

1	The slides used in the hands-on
uploaded in HΦ home page:
First, Google “Materiapps HPhi,” you will ﬁnd,
http://issp-center-dev.github.io/HPhi/index.html

2	HΦFor direct comparison between experiments and theory
and promoting development of other numerical solvers
Numerical diagonalization package for lattice hamiltonian-For wide range of quantum lattice hamiltonians
Ab initio eﬀective hamiltonians-Lanczos method [1] and LOB(P)CG [2]:Ground state and low-lying excited states
Excitation spectra of ground state-Thermal pure quantum (TPQ) state [2]: Finite temperatures-Real-time evolution-Parallelization with MPI and OpenMP[1] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994) .
[2] A. V. Knyazev, SIAM J. Sci. Cumput. 23, 517 (2001).
[3] S. Sugiura, A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012).

Open source program package (latest release: ver.3.1.2)
License: GNU GPL version3
Project for advancement of software usability in materials science” by ISSP

Target Hamiltonian
・Standard Hamiltonian 1
Itinerant electrons: Hubbard-type model

Fermion Hubbard: Particle # & total Sz conserved
HubbardNConserved: Particle # conserved & total Sz not
Fermion HubbardGC: Particle # & total Sz not conservedTarget Hamiltonian
・Standard Hamiltonian 2
Localized spin: Heisenberg-type modelprohibited

Spin: total Sz conserved
SpinGC: total Sz not conserved

S > 1/2 can be simulated
if your memory is enough large

Target Hamiltonian
・Standard Hamiltonian 3
Mixture: Kondo-lattice-type modelprohibited
Kondo Lattice: Particle # & total Sz conserved
Kondo LatticeGC: Particle # & total Sz not conserved

7	

Primitive Standard Input File
W = 4
L = 4
model = "Hubbard"//method = "Lanczos”
method = "TPQ"//method = "FullDiag"lattice = "Square"t = 1.0t' = 0.5
U = 8.0
nelec = 16
2Sz = 0	8	Output
Ground-state/ﬁnite-temperature/time-evolution of-Energy-Square of energy-One-body equal time Greenʼs function-Two-body equal time Greenʼs/correlation function-Dynamical Greenʼs function is also available

9	An Example of Expert Input:
Ab Initio Spin Hamiltonian
Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, & M. Imada, Phys. Rev. Lett. 113, 107201 (2014).

14010	

Overview of Software HΦ
-Language: C-Compiler: C & Fortran compiler-Library: BLAS, LAPACK, Kω (distributed with HΦ)(optional: MPI, Scalapack, MAGMA)-Parallelization: OpenMP & MPI
For installation, cmake is required11	Flow of Simulation

Standard input
W = 4
L = 4
model = "Hubbard”
method = "TPQ”
lattice = "Square"t = 1.0t' = 0.5
U = 8.0
nelec = 16
2Sz = 0	

Making input ﬁlesfrom scratch

Standard interface

Expert input
Def. files for Hamiltonian	Def. files for controlling simulation	

Expert interface
Subroutines:-Lanczos-CG-TPQ-TimeEvolution-Full diag.
(LAPACK, Scalapack, MAGMA)

Standard output
Output ﬁles12	

Algorithm Implemented in HΦ:Lanczos & LOBCGKrylov Subspace Method
for Sparse and Huge Matrices
Alexey Krylov
Aleksey Nikolaevich Krylov
1863-1945
Russian naval engineer and applied mathematician

Krylov subspace

Numerical cost to construct Kn:
Numerical cost to orthogonalize Kn:
Cornelius Lanczos 1950
Walter Edwin Arnoldi 1951

*nnz: Number of non-zero
entries/elements

Krylov Subspace Method15	Lanczos Method16	Lanczos Method

Orthogonalization17	

Lanczos Method

Hamiltonian projected onto m D Krylov subsace

Eigenvalues of projected Hamiltonian→ Approximate eigenvalues of original Hamiltonian18	Lanczos Method: # of Vectors Required19	

Convergence of Lanczos Method
Yousef Saad,
Numerical Methods for Large Eigenvalue Problems (2nd ed)
The Society for Industrial and Applied Mathematics 2011

Assumption:
Convergence theorem for the largest eigenvalue20	24 site cluster of Kitaev-Γ model (frustrated S =1/2 spins)
Dimension of Fock space: 224=16777216
64Eigenvalues

Sparse intermediate eigenvalues: Not correct
2Smallest
and largest eigenvalues: Well converged
0-2-4-6-80

1020

30

40506070

Dimension of Krylov subspace80

How Lanczos Method Works-7Eigenvalues-7.2-7.4-7.6

E0 = -7.8870906929 (converged)
E0 = -7.8870906928 (75 steps)-7.8-80102030

40506070

Dimension of Krylov subspace80LOB(P)CG
Algorithm of LOBCG: m lowest eigenstates
Initial condition:←approximation of i th smallest eigenvalue←m smallest eigenstates are chosen from 3m states

Algorithm Implemented in HΦ:TPQ(Finite-Temperature Simulation)

Finite-Temperature Physical Quantity:
Heat Capacity
Spread of energy distribution
32 site cluster of S=1/2 spin

-Average with Boltzmann distribution

Complexity
MemoryHamiltonian〜3x108TB!

〜69GB
25	

Typical Pure State Approach
Imada-Takahashi (1986)
Lloyd (1988)
Jacklic-Prelovsek (1994)
MemoryHams-De Raedt (2000)
Sugiura-Shimizu (2012, 2013)
M. Imada & M. Takahashi, J. Phys. Soc. Jpn. 55, 3354 (1986).

Complexity

Typical state: Random vector
N. Ullah, Nucl. Phys. 58, 65 (1964).

-Uniform distribution onunit sphere in

At ﬁnite temperature
Average over the distribution
How large is the standard deviation?

26	

Typical Pure State Approach
Seth Lloyd, Ph.D. Thesis, Rockefeller University (1988); arXiv:1307.0378.
A. Hams & H. De Raedt, Phys. Rev. E 62, 4365 (2000).
A. Sugita, RIMS Kokyuroku (Kyoto) 1507, 147 (2006).
P. Reimann, Phys. Rev. Lett. 99, 160404 (2007).
S. Sugiura & A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012).
S. Sugiura & A. Shimizu, Phys. Rev. Lett. 111, 010401 (2013).

Exponetially small when system size increases27	

有限温度計算手法: TPQ

Construction of Typical Pure State
Thermal Pure Quantum (TPQ) States
Sugiura & Shimizu, Phys. Rev. Lett. 108, 240401 (2012)

Initial state ( at T = + ):do k=1,Nstep

If possible, taking random average

enddo

Hamiltonian-wave function product is essential28	Example of TPQ: K-Γ-J3 Model

Nearest neighbor
3rd neighbor
29	How to Simulate K-Γ-J3 Model
model = "SpinGC”
method = "TPQ"
lattice = "Honeycomb”
a0w = 2a0l = 2a1w = 4a1l = -2J0x = -0.80901699437
J0yz = 0.58778525229
J0zy = 0.58778525229
J1zx = 0.58778525229
J1y = -0.80901699437
J1xz = 0.58778525229
J2xy = 0.58778525229
J2yx = 0.58778525229
J2z = -0.80901699437
h = 0.07071067811
Gamma = -0.07071067811
2S=1	Add Exchange	  and Ising byExpert mode

30	Heat Capacity of K-Γ-J3 Model87
6

B=0.1, 28 samples
B=0.1, 56 samples
B=0.1, 112 samples
B=0.0, 28 samples
B=0.0, 56 samples
B=0.0, 112 samples

Error bars:
B=0.1, 112samples
Error bars:
B=0.0, 112samples

5432100.010.111031	

Standard Deviation
Standard deviation in TPQ
1001018 site
24 site-110-210-3
10-4
0.01

0.1

110

Parallelization of HΦ有限温度計算手法: TPQ

Parallelization

・Hybrid parallel-Shared memory (OpenMP) Thread-Distributed memory (MPI) Process
Parallel computer
node

node

T1	
T0	 core2	
core1	

core1	 core2	

T3	
core3	
T2	 core4	

core3	 core4	

core5	 core6	

core5	 core6	

T0	
T1	
T2	
T3	

有限温度計算手法: TPQ

Parallelization

・Hubbard/Kondo Lattice・Hybrid parallel/HubabrdGC→4n process-Shared memory (OpenMP) Thread ・Spin/SpinGC-Distributed memory (MPI) Process →(2S+1)n processParallel computer

Example of MPI: 2 process

rank 0rank 1

nodenode

P1	
P0	 core2	
core1	

P3	P2	 core2	
core1	

core3	 core4	

core3	 core4	

core5	 core6	

core5	 core6	

P0	

P2	

P1	

P3	Speedup

Lanczos method: Up to 6.87x1010 dimension@K computer & ISSP supercomputer
From 4096 32768 cores: Parallelization eﬃciency 80%

HΦ Developers
Dr. Takahiro Misawa
The Institute for Solid State Physics,
The University of Tokyo

Prof. Synge Todo
Department of Physics,
The University of Tokyo

Dr. Mitsuaki Kawamura
The Institute for Solid State Physics,
The University of Tokyo

Prof. Naoki Kawashima
The Institute for Solid State Physics,
The University of Tokyo

Dr. Yoshikazu Yoshimi
The Institute for Solid State Physics,
The University of Tokyo

Kota Ido
The Institute for Solid State Physics,
The University of Tokyo

Acknowledgement:
Dr. Yoshinori Nakanishi-Ohno (Univ. of Tokyo)
Prof. Takeo Hoshi (Tottori Univ.)
Prof. Tomohiro Sogabe (Nagoya Univ.)“Project for advancement of software
usability in materials science” by ISSP

Youhei Yamaji
Department of Applied Physics,
The University of Tokyo

Appendix: Formulation ofQuantum Many-Body Problem

Quantum Many-Body Problems
An example: 3 Quantum dots
F. R. Braakman, et al., Nat. Nano. 8, 432 (2013)

Quantum dot:
量子ドット:-電子１つを閉じ込めることができる“箱”-Aquantum box can conﬁne a single-1電子トランジスタや量子コンピュータをelectron-Utilized作るための量子ビットとして使われる
for single electron transistor,
quantum computers

Three-body problem:→ Number of states = 23 (factor２ from spin)

States represented bysuperposition

Quantum Many-Body Problems
Mutual Interactions
Operators acting ona single qubit
A two dimensional representationof Lie algebra SU(2)Vectors in Fock Space
Correspondence between spin and bit
2N-dimensional Fock space:

Decimal representation of orthonormalized basis

Wave function as a vector

Vectors and Matrices in Fock Space
Inner product of vectors

Hamiltonian matrix
Orthonomalized basis:

Example: Two Spins
Decimal representation of orthonormalized basis

Problem: Find 4 by 4 Hamiltonian matrix that describes

Useful transformation:
Ladder operators

Answer of the Problem

Matrix element

4 by 4 Hamiltonian matrix

Answer of the Problem 2:
Energy Spectrum of the Two Spins