情報基盤研究開発センターハンズオン：HΦ講習会­ITOを用いたハンズオン­

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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 (→並列化性能の紹介)[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

Primitive Standard Input File
W = 4
L = 4
model = "Hubbard"//method = "Lanczos”
method = "TPQ"//method = "FullDiag"lattice = "Square"t = 1.0t' = 0.5
U = .0nelec = 16
2Sz = 0

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

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).

An example: Frustrated magnet Na2IrO3Step (sec)

140
120
100
80

Conventional

3rd n.

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).

ec)

140RESPACK
cf.)
120

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 required

Flow of Simulation

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

Expert input

Making input ﬁlesfrom scratch

Def. files for HamiltonianDef. files for controlling simulation

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

Standard output
Output ﬁles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 Method

Lanczos Method

Lanczos Method

Orthogonalization

Lanczos Method

Hamiltonian projected onto m D Krylov subsace

Eigenvalues of projected Hamiltonian→ Approximate eigenvalues of original HamiltonianLanczos Method: # of Vectors Required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 eigenvalue

24 site cluster of Kitaev-Γ model (frustrated S =1/2 spins)

Dimension of Fock space: 224=16777216
6

4

Eigenvalues

Sparse intermediate eigenvalues: Not correct2
Smallest
and largest eigenvalues: Well converged
0

-2

-4

-6

-8
0

10

20

30

40

50

60

70

Dimension of Krylov subspace80How Lanczos Method Works-7

Eigenvalues

-7.2

-7.4

-7.6

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

-7.8

-8

0

10

20

30

40

50

60

70

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

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?

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 increases

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 essential

Example of TPQ: Eﬀective Hamiltonianof α-RuCl3, K-Γ-J3 Model

Nearest neighbor
3rd neighbor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

Heat Capacity of K-Γ-J3 Model876

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

5
4
3
2
1
0
0.01

0.1

1

10

Standard Deviation
Standard deviation in TPQ
100

18 site24 site

10-1
10-2
10-3
10-4
0.01

0.1

1

10

Algorithm Implemented in HΦ:Real time evolution
Univ. of Tokyo, ISSP
Kota IdoQuantum'Dynamics  

896

Bloch, Dalibard, and Zwerger: Many-body phy

resultithree stancespotent
neous

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ex1)

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fs~ psR

K-TCNQ

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pump

(a)

dR/dE

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(b)
ν

Photocurrent (a.u.)

Basov et al.: Electrodynamics of correlated electron materials

496

(c)

Bloch'et'al.','Rev.'Mod.'Phys.80,885431(2008).'

FIG. 4. "Color online# Optical lattices. "a# Two- and "b# threedimensional
optical lattice potentials formed by superimposing
FIG. 19Basov'et'al.','Rev.'Mod.'Phys.83,431(2011).'(color online). Depiction of a photoinduced phase tranPhoton energy (eV)sition showing the potential energy as a function of a generalized
two or three orthogonal standing waves. For a twostructural coordinate. The basic idea is the so-called ‘‘domino
FIG. 20 (color online). (a) Reflectivity and photocurrent measured
dimensional optical lattice, the atoms are confined to an array
effect’’: Photoexcitation initiates a structural change which, inon K-TCNQ as a function of frequency taken at room temperatureofbelow
tightly
confining
turn, drives an electronic phase transition with an order parameter
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TSP ¼ 395
K. The peakone-dimensional
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A(t) = A0 exp<latexit sha1_base64="+cSPuw09pGK7cmjMjoKqJDbaat4=">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</latexit>!
tshiU: t0
freq:<latexit sha1_base64="27QTRJ5B0Msc8prmAph2oMpNiGM=">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</latexit><latexit<latexit sha1_base64="D01cAYEImkvvcT99Ej8t9d3OXGY=">AAACuHichVE7SwNBEP483/GRqI1gEwwRK5mIoKQSbSx9xQca4t25xiX34m4T0OAfsLWwUAsFC/Fn2NhYWvgTxFLBxsK5y4GoGOe43Zlvv292dsbwLBkooqcWrbWtvaOzqzvR09vXn0wNDK4FbtU3RcF0LdffMPRAWNIRBSWVJTY8X+i2YYl1ozIfnq/XhB9I11lVB54o2nrZkXvS1BVDm6pU3/btNB2VUhmaoMjSv51c7GQQ26KbesA2duHCRBU2BBwo9i3oCPjbQg4Ej7Ei6oz57MnoXOAICdZWmSWYoTNa4bXM0VaMOhyHOYNIbfItFv8+K9PI0iPd0Cvd0y0908efuepRjrCWA96NhlZ4peTx8Mr7vyqbd4X9L1UThcGV6RwLVoT8Zky7KSN8v8IeZqJ3S+6DFyFhR8xGLbXD09eV/HK2PkZX9MK9uKQnuuNuOLU383pJLJ8hwcPM/Rzdb2dtciLH/tJUZnYuHmsXRjCKcZ7dNGaxgEUU+F4bJzjHhZbXdrSyJhtUrSXWDOGbaf4nLyicPw==</latexit>

t0 )]A(t) = A0 t
<latexit sha1_base64="kLf4eWDCL+iKmWgydVBra32yCWA=">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</latexit>⇥(tt0 ) /(2t2dump )tdump:

2

tdump<latexit sha1_base64="9wN49jYQhENkVkz/n8T4HbVdFos=">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</latexit>

A0
VecPotW,'L:'''''''<latexit sha1_base64="PtBNGlJpijTMN16BmGBAhNazPb0=">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</latexit>⇤cos [!(tt0 )]Output
Norm.dat'

SS.dat'##_cisajs_step%%.dat''
OneBodyG''	poten_al.dat''

:'Gaussian
A(t) = A0 exp<latexit sha1_base64="+cSPuw09pGK7cmjMjoKqJDbaat4=">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</latexit>

⇥

(t

2

t0 ) /(2t2dump )⇤cos [!(tt0 )]A0 = (0.1, 0.1),!/t = 84<latexit sha1_base64="6mQvaVVJmQ3QKsZEis7j7uoeu5E=">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</latexit>

hHi<latexit sha1_base64="WaTmssL0AOL9XoLCfGbET9rgkBQ=">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</latexit>

A0 = (2, 2),

2 !/t = 25<latexit sha1_base64="cN2NvcpxcHyG+zt0RbtXqqeQpJ0=">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</latexit>

0
-2

8

5
1

-4
1
1

0

2

4

6

Time

8

10

•

TwoBodyTE

OneBodyTE•
hbp://isspdcenterddev.github.io/HPhi/manual/userguide_HPhi_ja.pdf

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 of
Quantum 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