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 Sota 1 , Takami Tohyama 1,2 , Seiji Yunoki 1 1 RIKEN R-CCS, 2 Tokyo Univ. of Sci.

Collaboration with large scale quantum beam experiments Large scale quantum beam facilities Quantum beams SPring-8 J-PARC Spin SACLA Strongly Charge correlated Orbit matter Phonon 茨城県東海村 兵庫県佐用町 Responses for the Collaboration between external fields ： theoretical and experimental Quantum dynamics researchers K/Post-K computer Quantum fluctuation and excitation dynamics of 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 2

Introduction  Quantum many-body systems N spin ½ system : degree of freedom is not N but 2 N !! E.g. N =100  2 100 ≈ 10 31 2 N vector 2 N × 2 N matrix 2 N × 2 N matrix  Physical quantities 3

Introduction  The spirit of density matrix renormalization (DMRG) • Optimize the basis set to describe the state to be calculated • Use only m 2 bases instead of 2 N bases ( m 2 << 2 N ) N 2 2 m         c n n n n    n 1 n 1 S. R. White, PRL 69 , 2863 (1992). 4

Introduction  How to choose the optimized bases superblock • m eigenstates with largest eigenvalues of the reduced density matrix of the superblock ground state system environment • Ground state of the superblock • m eigenstates with largest • Reduced density matrix  eigenvalues: (sys) i      SB SB N 2 2 m           i i , ij i j c n n n n    j n 1 n 1 m m     a i j   i , j   5   i 1 j 1

Dynamical DMRG method E. Jeckelmann, Phys. Rev. B 66 , 045114 (2002) • Dynamical Correlation function 1 1   ˆ ˆ ˆ : arbitrary operator    † Im 0 A A 0 A ˆ        A 2 N H i 0 ˆ   : ground state 0 H 0 0 0 • Target state Kernel polynomial method 1 ˆ ˆ 0 , A 0 , A 0 1 ˆ 0 ˆ       A H i ˆ     H i 0   ˆ ˆ      1 w {2 Q ( ) iP ( )} ( P H A ) 0 l l l l  Basis set is optimized to describe these states l 0 Multi target procedure SS, M. Ito, J. Phys. Soc. Jpn. 76 , 054004 (2007). SS , T. Tohyama, PRB 82 , 195130 (2010). 6

Massively parallel Dynamical DMRG Dynamical DMRG (https://www.r-ccs.riken.jp/labs/cms/DMRG/Dynamical_DMRG_en.html) Density Matrix Renormalization Group (DMRG) Kernel Polynomial method Massively Parallelization (KPM) Quantum Dynamics of strongly correlated quantum systems 7

Efficiency FLOPS[%] Time[s] 32,000 100% 16,000 80% 8,000 60% 4,000 40% 2,000 Strong Elapse 20% 1,000 Weak Elapse Strong FLOPS% Weak FLOPS% 500 0% 48 192 768 3072 12288 49152 Process 7.8 PFLOPS on K computer 8 SS, S. Yunoki, T. Tohyama, A. Kuroda,Y. Kitazawa, K. Minami, and F. Shoji, in preparation

Spin excitation dynamics on spin frustrated system  S=1/2 triangular lattice Heisenberg antiferromagnet Hamiltonian:    H J S S i j , i j   i j , • Typical spin frustrated system. • Ground state properties have been already well known.  i.e. uniform triangular lattice: three-sublattice 120 ° Néel ordered state. • The magnetic excitations are less well understood. 9

Ba 3 CoSb 2 O 9 https://www.titech.ac.jp/news/2012/025500.html Co 2+ ion is located at the center of octahedra. The effective magnetic moment of Co 2+ ions with an octahedral environment can be described by the pseudospin-1/2. Magnetic Co 2+ ions forms a uniform triangular lattice. • Spin-1/2 XXZ model with small easy-plane anisotropy layer interlayer          z z S S S S H J ( S S ) J i j i j l m     i j , l m , J =1.67meV, Δ =0.046 , and J’ =0.12meV T. Suzuki, et al. Phys. Rev. Lett 110 , 267201 (2013). 10

Magnetic Excitations (1) • Inelastic neutron scattering spectra of Ba 3 CoSb 2 O 9 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. 11

Magnetic excitation (2) S. Ito, et. al, Nat. Communi. 8 , 235 (2017) At present, theory cannot explain the high energy excitations continua observed in Ba 3 CoSb 2 O 9 . We investigate the magnetic excitations by the Dynamical DMRG. 12

Model and computational conditions    J  • Hamiltonian: H J S S (We assume ) 1.67meV. i j   i j , S. Ito, et. al, Nat. Communi. 8 , 235 (2017) • lattice: 12 × 6 triangular lattice (cylindrical boundary condition) Periodic boundary Open boundary • DMRG truncation number m =6000 . • Half width at half maximum is 0.1 J . (Kernel polynomial method) 13

Dynamical spin structure factor S( q ,  ) along Γ→ M DMRG ( J =1.67meV ) Experiment Ito, et al, Nat. Commun. 8 , 235 (’17) DMRG result ω(meV) q M Γ S( q ,  ) ω(meV) 14

S( q ,  ): constant energy map DMRG ( J =1.67meV ) Experiment Ito, et al, Nat. Commun. 8 , 235 (’17) 1.8-2.0meV S( q ,  ) 6 36.00 33.75 31.50 4 29.25 27.00 24.75 2 22.50 20.25 q y 0 18.00 15.75 13.50 -2 11.25 9.000 6.750 -4 4.500 2.250 0.000 -6 -6 -4 -2 0 2 4 6 q x 2.6-2.8meV S( q ,  ) 6 36.00 33.75 31.50 4 29.25 27.00 24.75 2 22.50 20.25 q y 0 18.00 15.75 13.50 -2 11.25 9.000 6.750 -4 4.500 2.250 0.000 -6 -6 -4 -2 0 2 4 6 q x 3.4-3.6meV S( q ,  ) 6 36.00 33.75 31.50 4 29.25 27.00 24.75 2 22.50 20.25 q y 0 18.00 15.75 13.50 -2 11.25 9.000 6.750 -4 4.500 2.250 0.000 -6 -6 -4 -2 0 2 4 6 q x 15

Summary  Our developed massively parallel dynamical DMRG shows high performance on K computer https://www.r-ccs.riken.jp/labs/cms/DMRG/Dynamical_DMRG_en.html  Spin dynamics of S=1/2 AFMHM on triangular lattice  In good qualitative agreement with experiments Dynamical DMRG Experiments  What is the nature of high energy excitations?? 16