algorithm for two-dimensional strongly correlated systems and its - - PowerPoint PPT Presentation

Massively parallel density matrix renormalization group method algorithm for two-dimensional strongly correlated systems and its applications

1st R-CCS Symposium (Kobe) Shigetoshi Sota1, Takami Tohyama1,2, Seiji Yunoki1

1RIKEN R-CCS, 2Tokyo Univ. of Sci.

2 茨城県東海村 兵庫県佐用町

SPring-8 SACLA J-PARC

Large scale quantum beam facilities

Quantum beams

Strongly correlated matter

Responses for the external fields： Quantum dynamics Spin Charge Orbit Phonon

K/Post-K computer

Collaboration between theoretical and experimental researchers

Quantum fluctuation and excitation dynamics

• f quantum many-body system

☑ Constructions of theory and computation to

accurately understand complex experiment results

☑ Predicting characteristics from the numerical

calculations and proposing experiments for large quantum beam facilities

Collaboration with large scale quantum beam experiments

Introduction

3

• Quantum many-body systems
• Physical quantities

E.g. N=100  2100 ≈ 1031 N spin ½ system: degree of freedom is not N but 2N !! 2N×2N matrix 2N vector 2N×2N matrix

4

• The spirit of density matrix renormalization (DMRG)
• Optimize the basis set to describe the state to be calculated
• Use only m2 bases instead of 2N bases (m2 << 2N)

Introduction

2

2 1 1

N

m n n n n

c n n  

  

   

 

• S. R. White, PRL 69, 2863 (1992).

5

• How to choose the optimized bases
• m eigenstates with largest

eigenvalues of the reduced density matrix of the superblock ground state

system environment superblock

• Ground state of the superblock
• Reduced density matrix
• m eigenstates with largest

eigenvalues:

Introduction

SB SB , i i ij i j j

  

 



2

2 1 1 , 1 1

N

m n n n n m m i j i j

c n n a i j  

        

      

  

(sys) i



Dynamical DMRG method

6

• Dynamical Correlation function

Multi target procedure ˆ: arbitrary operator A

ˆ   H

 

†

1 1 ˆ ˆ Im 0 ˆ 2

A

A A N H i          

ˆ ˆ 1 , ˆ , A i H A      

• Target state

: ground state Basis set is optimized to describe these states Kernel polynomial method

• E. Jeckelmann, Phys. Rev. B 66, 045114 (2002)

1

1 ˆ 0 ˆ ˆ ˆ {2 ( ) ( )} ( )

l l l l l

A H i w Q iP P H A    

  

   



SS, M. Ito, J. Phys. Soc. Jpn. 76, 054004 (2007). SS , T. Tohyama, PRB 82, 195130 (2010).

Massively parallel Dynamical DMRG

7

Density Matrix Renormalization Group (DMRG) Kernel Polynomial method (KPM) Massively Parallelization Quantum Dynamics of strongly correlated quantum systems Dynamical DMRG (https://www.r-ccs.riken.jp/labs/cms/DMRG/Dynamical_DMRG_en.html)

Efficiency

8

0% 20% 40% 60% 80% 100% 500 1,000 2,000 4,000 8,000 16,000 32,000 48 192 768 3072 12288 49152 FLOPS[%] Time[s] Process Strong Elapse Weak Elapse Strong FLOPS% Weak FLOPS%

SS, S. Yunoki, T. Tohyama, A. Kuroda,Y. Kitazawa, K. Minami, and F. Shoji, in preparation

7.8 PFLOPS on K computer

9

Spin excitation dynamics on spin frustrated system S=1/2 triangular lattice Heisenberg antiferromagnet

• Typical spin frustrated system.
• Ground state properties have been already well known.
• The magnetic excitations are less well understood.
• i.e. uniform triangular lattice: three-sublattice 120° Néel ordered state.

Hamiltonian:

, , i j i j i j

H J

 

 



S S

Ba3CoSb2O9

10

https://www.titech.ac.jp/news/2012/025500.html

layer interlayer , ,

( )

z z i j i j l m i j l m

H J S S J

   

      

 

S S S S

• Spin-1/2 XXZ model with small easy-plane anisotropy

Magnetic Co2+ ions forms a uniform triangular lattice. The effective magnetic moment of Co2+ ions with an octahedral environment can be described by the pseudospin-1/2. Co2+ ion is located at the center of octahedra.

J=1.67meV, Δ=0.046, and J’=0.12meV

• T. Suzuki, et al. Phys. Rev. Lett 110, 267201 (2013).

Magnetic Excitations (1)

11

• Inelastic neutron scattering spectra of Ba3CoSb2O9
• J. Ma, et. al., Phys. Rev. Lett. 116, 087201 (2016).
• S. Ito, et. al, Nat. Communi. 8, 235 (2017).

Magnetic excitations cannot be understood by linear spin wave theory.

Magnetic excitation (2)

12

• S. Ito, et. al, Nat. Communi. 8, 235 (2017)

At present, theory cannot explain the high energy excitations continua

• bserved in Ba3CoSb2O9.

We investigate the magnetic excitations by the Dynamical DMRG.

Model and computational conditions

13

, i j i j

H J

 

 



S S

• Hamiltonian:
• lattice: 12×6 triangular lattice (cylindrical boundary condition)

Open boundary Periodic boundary

(We assume )

1.67meV. J 

• S. Ito, et. al, Nat. Communi. 8, 235 (2017)
• DMRG truncation number m=6000.
• Half width at half maximum is 0.1J. (Kernel polynomial method)

Dynamical spin structure factor S(q,) along Γ→M

14

Experiment Ito, et al, Nat. Commun. 8, 235 (’17) Γ q ω(meV) M ω(meV) S(q,) DMRG (J=1.67meV) DMRG result

15

• 6
• 4
• 2

• 6
• 4
• 2

1.8-2.0meV

qy qx

S(q,)

• 6
• 4
• 2

• 6
• 4
• 2

2.6-2.8meV

qy qx

S(q,)

• 6
• 4
• 2

• 6
• 4
• 2

3.4-3.6meV

qy qx

S(q,)

DMRG (J=1.67meV) Experiment Ito, et al, Nat. Commun. 8, 235 (’17)

S(q,): constant energy map

Summary

16

 Our developed massively parallel dynamical DMRG

shows high performance on K computer

 Spin dynamics of S=1/2 AFMHM on triangular lattice

 In good qualitative agreement with experiments  What is the nature of high energy excitations??

Experiments Dynamical DMRG

https://www.r-ccs.riken.jp/labs/cms/DMRG/Dynamical_DMRG_en.html