Dynamical properties of strongly correlated electron systems - PowerPoint PPT Presentation

Dynamical properties of strongly correlated electron systems studied by the density-matrix renormalization group (DMRG) Takami Tohyama Tokyo University of Science Shigetoshi Sota AICS, RIKEN

Outline Density-matrix renormalization group （ DMRG) ► DMRG ► Dynamical DMRG ► Extension to two dimensional systems Recent results obtained by DDMRG ► Spin excitations in 1D quantum spin systems ► Optical excitations in 1D Mott insulator coupled to phonon ・ linear absorption ・ Third-harmonic generation (THG) - H. Matsuzaki, H. Nishioka, H. Uemura, A. Sawa, S. Sota, T. Tohyama, and H. Okamoto, Phys. Rev. B 91 , 081114(R) (2015) - S. Sota, T. Tohyama, and S. Yunoki, J. Phys. Soc. Jpn. 84 , 054403 (2015) ► Spin and charge excitations in square t-t’-U Hubbard model - T. Tohyama, K. Tsutsui, S. Sota, and S. Yunoki, Phys. Rev. B 92 , 014515 (2015)

Dynamical properties in strongly correlated electron systems (SCES) External field: Quantum beam: SPring-8, J-PARC photon, neutron Pump-probe spectroscopy high-temperature superconductors, spin charge quantum spins, Mott insulators, … SCES orbital lattice ・ constructing lattice model with correlation ・ numerical techniques to calculate dynamics response to external field ： excitation dynamics equibrium/ nonequibrium

Setting lattice models e.g. Hubbard model i site ∑ ∑ + = − + σ H t c c U n n spin σ + δ σ ↑ ↓ i , i , i , i , δ σ i , , i Parameters: from first-principles calculations, experiments, etc Dynamical correlation functions e.g. current-current correlation: optical absorption 1 1 ∑ χ ω = − = − − † ( ) Im 0 j j 0 j it H c . .) ( c c + σ σ π i 1, i , ω + − − γ H i σ E i , 0

Density-matrix renormalization group （ DMRG) [S. R. White, PRL 69 , 2863 (1992)] System | i > Environment | j > Renormalize the states of the Environment into those of the System for each step, by using the density-matrix given by the ground-state wave function. ground-state wave function ∑ ψ = ψ i j ij i , j ・ ・ ・ density matrix of system ・ ・ ・ = ∑ ・ ・ ・ ρ ψ ψ ′ ′ ii ij i j j m ∑ ∑ = = α α ≈ α α ρ A ω ω A Tr( ) A A u u u u α α α α = 1 ω ≈ 0 discard unimportant states: α α ρ : eigenstate of u m : truncation number ≥ ω ρ ( 0): eigenvalue of α m − ∑ ω 1 : truncation error α α = 1

E. Jeckelmann, Phys. Rev. B Dynamical DMRG 66 , 045114 (2002) = ψ system environment ij j i ∑ ∑ ∑ ∑ , ρ = ψ ψ ρ = ψ ψ = p α p 1 α α α i’j ii’ i j ii’ , ij , i’j α α j j Multi targets: α Single target 1 1 χ ω = − ˆ ˆ ( ) Im 0 O O 0 π ω + − + γ H i E 0   0   ψ α ω = ⇒ ρ ω ˆ  ( ) O 0 ( )  1  ˆ 0 O Correction vector ω + − + γ   H i E 0 The reduced density matrix depends on ω  perform DMRG for a given energy ω .

Process of ω A given energy Dynamical DMRG Ground state ： 0 Lanczos method Target states ： 1 ˆ ˆ O 0 , O 0 ω + − + γ H i E 0 ・ ・ ・ Generation and diagonalization ・ ・ ・ ( ) ρ ω ρ ・ ・ ・ † of U U † Transformation of operators: UAU 1 1 Calculation of χ ω = − ˆ ˆ † ( ) Im 0 O O 0 π ω + − + γ physical quantities H i E 0

How to calculate the correction vector 1 ( ) φ ω = ˆ 0 O Lorentzian broadening ω − + γ H i 1. Modified conjugate gradient method ( ) ( ) E. Jeckelmann, ω − + γ φ ω = ˆ 0 H i O Phys. Rev. B 66 , 045114 (2002) Solve this equation iteratively for a given ω . 2. Lanczos method M ∑ 1 ( ) φ ω ˆ   ~ n n O 0  ω − + γ E i =   n 1 n  ˆ 0 n O Lanczos vector starting from Independent of ω

How to calculate the correction vector 1 ( ) φ ω = ˆ 0 O ω − + γ H i 3. Polynomial expansion using Legendre functions [S.Sota, T.T., PRB 82 , 195130 (2010)] L ∑ [ ] ( ) φ ω ω − π ω ˆ ~ 2 Q ( ) i P ( ) P H O ( ) 0 l l l = l 0 Legendre polynomial of the first kind P l Q Legendre polynomial of the second kind l Recursive relation + ω = + ω ω − ω ( l 1) P ( ) (2 l 1) P ( ) lP ( ) + − l 1 l l 1 ω and H : separated polynomials.

A problem of polynomial expansion Gibbs oscillation L 2 ∑ δ ε − ε δ ε − ε = ε ε ( ') ~ ( ') P ( ) P ( ') + L l l 2 l 1 = l 0 L

Introduce Gaussian-type broadening to remove Gibbs oscillation ( ) 2  ε − H [S. Sota and M. Itoh, − 1   1  ∫ → = ε σ 2 P H ( ) P H ( ) d e 2 P H ( ) J. Phys. Soc. Jpn. 76 , 054004 (2007)] l l l σ πσ − 2 1 2 σ = π 2 / L + + 2 l 1 l 2 l 1      = + + σ 2 ' P ( H ) H P H ( ) P ( H ) P H ( ) + − + + + l 1 l l 1 l σ σ σ σ l 1 l 1 l 1 Recursion relations: ( )    = + + ' ' P ( H ) 2 l 1 P H ( ) P ( H ) + − l 1 l l 1 σ σ σ ( ) ( ) δ ε − δ ε − 0.2 0.2 L L σ without Gaussian averaging σ = π with 2 / L L L Gaussian broadening

Other applications of polynomial expansion Time-evolved wave function ( ) ( ) − δ φ + δ = φ iH t t t e t L ∑ ( ) ( ) ( ) − l + δ φ ~ 1 2 l 1 j ( t P H ) ( ) t l l = l 0 spherical Bessel function j l Thermodynamic properties ( )  ξ β = β ξ H 2 e + L 2 l 1 ∑ − β ξ ~ C i ( 2) P H ( ) l l 2 = l 0 modified spherical i ( ) ( )   = ξ β ξ β l Z Partition function Bessel function S. Sota, T. T., Phys. Rev. B 78 , 113101 (2008)

Extension to two dimensions (2D-DMRG) real-space parallelization method c.f. E. M. Stoudenmire, S. R. White, Phys. Rev. B 87 , 155137 (2013) sweeping Added sites for the sweep of a fraction of system added sites direction of sweeping update the information of operators update by MPI communications.

Performance in K computer 3 × 8 triangular Hubbard model 50 FLOPS/PEAK (%) 40 30 20 10 0 5 10 15 20 region number elaplsed time (sec.) 1500 1000 500 0 5 10 15 20 region number All most perfect road balance

Performance in K computer Ex. One-dimensional extended Hubbard model efficiency elapsed time 15

Triangular Hubbard model U c1 U / t U c1 U c2 120 ° AF metal Spin liquid? J. Kokalj, R. H. McKenzie, Phys. Rev. Lett. 110 , 206402 (2013)

A recent application of 2D-DMRG for ground state: triangular Hubbard model (6x6 cylinder) T. Shirakawa, T.T., J. Kokalj, S. Sota, S. Yunoki, arXiv:1606.06814

Outline Density-matrix renormalization group （ DMRG) ► DMRG ► Dynamical DMRG ► Extension to two dimensional systems Recent results obtained by DDMRG ► Spin excitations in 1D quantum spin systems ► Optical excitations in 1D Mott insulator coupled to phonon ・ linear absorption ・ Third-harmonic generation (THG) - H. Matsuzaki, H. Nishioka, H. Uemura, A. Sawa, S. Sota, T. Tohyama, and H. Okamoto, Phys. Rev. B 91 , 081114(R) (2015) - S. Sota, T. Tohyama, and S. Yunoki, J. Phys. Soc. Jpn. 84 , 054403 (2015) ► Spin and charge excitations in square t-t’-U Hubbard model - T. Tohyama, K. Tsutsui, S. Sota, and S. Yunoki, Phys. Rev. B 92 , 014515 (2015)

Spin-Peierls (SP) compound CuGeO 3 M. Hase, I. Terasaki, K. Uchinokura, PRL 70 , 3651 (1993) - T SP =14K - the first inorganic SP system - edge-shared Cu-O chain with S=1/2 on Cu 2+ - deviation from Heisenberg model (Bonner & Fisher curve) < T T SP   ∑ ∑ = − − δ ⋅ + α ⋅ i  S S S S  H J (1 ( 1) ) δα + + 1 i i 1 i i 2   i i δ =0.022 at T =0

Phonons in CuGeO 3 - No evidence of soft phonon along the Cu-O chain - 3D character of the structural part of the SP transition q=( π , 0 , π ) [M. Braden et al ., PRB 66 , 214417 (2002)] In-chain soft phonon in organic SP material ω ph = 1.4 meV, ∆ = 1.8 meV (TTF)CuS 4 C 4 (CF 3 ) 4 ∆ > ω ph In-chain phonon in CuGeO 3 ω ph = 26 meV, 13 meV ∆ = 2 meV ∆ << ω ph antiadiabatic limit [G. Uhrig, PRB 57 , R14004 (1998)] In-chain phonon may couple to spin even below the SP transition.

Spin-Peierls model   ∑ ∑ = ⋅ + α ⋅ H J  S S S S  + + SP i i 1 0 i i 2   i i ∑ + ω † b b 0 i i i ( ) J ∑ + λ + − + ⋅ † † b b b b S S + + + i i i 1 i 1 i i 1 2 i λ : unknown

S (q, ω ) of spin-Peierls model by DDMRG T. Sugimoto, S. Sota, and T.T., L =16, T =0 JPSJ 81 , 034706 (2012) α 0 = 0.36 ω 0 = 1.5 J ω 0 λ = 0.5 ω 0 / J λ = 0 Phonon-assisted spin excitation is expected above the upper edge of spin continuum for CuGeO 3 .

New diamond quantum spin lattice K 3 Cu 3 AlO 2 (SO 4 ) 4 M. Fujihala et al., J. Phys. Soc. Jpn. 84 , 073702 (2015)

J 3 J 5 J 2 J m J 4 J 1 J d A J 3 , J 4 J m , J d , J d ’ J 1 J 2 J 5 g K -22.5 -300 -37.5 510 75 2.14 Rb -97.2 -97.2 32.4 445.5 81 2.14 Cs -48 -288 29.4 441.6 96 2.18

S(q, ω ) of K 3 Cu 3 AlO 2 (SO 4 ) 4 by DDMRG 240-site ring (80 unit cells) m =360