Dynamical properties of strongly correlated electron systems studied by the density-matrix renormalization group (DMRG) Takami Tohyama

Transcription

1 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

2 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, (R) (2015) - S. Sota, T. Tohyama, and S. Yunoki, J. Phys. Soc. Jpn. 84, (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, (2015)

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

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

5 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 ij i ρ i, j density matrix of system α= 1 = ψψ A = Tr( ρa) = ωα u A u ωα u A u u α : eigenstate of ωα( 0): eigenvalue of ρ α ρ ii ij i j j m α α α α 1 discard unimportant states: ωα 0 m: truncation number ωα : truncation error m α = 1 j

6 ρ ψ ij = = ψ ψ ii ij j Single target i j Dynamical DMRG system environment i j ρ = ψ ψ, pα = 1 ii α p α Multi targets: α j α, ij α, i j 1 1 χω ( ) = Im 0 Oˆ Oˆ 0 π ω+ E 0 H + iγ 0 ψ ( ω ) ˆ α = O 0 ρω ( ) 1 Oˆ 0 ω + E 0 H + iγ The reduced density matrix depends on ω perform DMRG for a given energy ω. E. Jeckelmann, Phys. Rev. B 66, (2002) α Correction vector

7 Process of Dynamical DMRG A given energy Ground state: Lanczos method 0 ω Target states: Oˆ 1 0, Oˆ 0 ω+ E 0 H + iγ Generation and diagonalization of ρ ω ( ) UρU Transformation of operators: UAU Calculation of physical quantities χω 1 ˆ 1 ˆ π ω+ E 0 H + iγ ( ) = Im 0 O O 0

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

9 How to calculate the correction vector ( ) φ ω = 1 O ˆ 0 ω H + iγ 3. Polynomial expansion using Legendre functions [S.Sota, T.T., PRB 82, (2010)] L ( ) [ ] φ ω ~ 2 Q( ω) iπp( ω) PHO ( ) ˆ 0 l= 0 Pl Q l Recursive relation l l l Legendre polynomial of the first kind Legendre polynomial of the second kind ( l + 1) P ( ω) = (2l + 1) ωp ( ω) lp ( ω) l+ 1 l l 1 ω and H : separated polynomials.

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

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

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

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

14 Performance in K computer 3 8 triangular Hubbard model elaplsed time (sec.) FLOPS/PEAK (%) region number region number All most perfect road balance

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

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

17 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:

18 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, (R) (2015) - S. Sota, T. Tohyama, and S. Yunoki, J. Phys. Soc. Jpn. 84, (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, (2015)

19 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 Hδα = J (1 ( 1) δ ) S S + α S S δ =0.022 at T=0 i 1 i i+ 1 i i+ 2 i i

20 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, (2002)] In-chain soft phonon in organic SP material (TTF)CuS 4 C 4 (CF 3 ) 4 ω ph = 1.4 mev, = 1.8 mev > 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.

21 H Spin-Peierls model = J S S + α S S SP i i+ 1 0 i i+ 2 i i + ω 0 i i i J ( + λ b + b b ) 1 + b 1 S S 2 i bb i i i+ i+ i i+ 1 λ : unknown

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

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

24 J 3 J 2 J 5 J 1 J 4 J m J d A J 1 J 2 J 3, J 4 J 5 J m, J d, J d g K Rb Cs

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

26 Optical excitation of one-dimensional Mott insulator coupled to phonons Origin of in-gap state generated by just after photo irradiation for Ca 2 CuO 3 Extended Hubbard-Holstein model

27 Optical absorption calculated by DDMRG Low-energy excitation consistent with experiment H. Matsuzaki, H. Nishioka, H. Uemura, A. Sawa, S. Sota, T. T., H. Okamoto, Phys. Rev. B 91, (R) (2015) Spin excitations formed by polarons

28 Comparison between two broadenings N=12 Conjugate gradient Lorentzian N=24 Polynomial expansion Gaussian T. T., H. Matsueda, Prog. Theor. Phys. Suppl. 175, 165 (2008) S. Sota, T. T., Phys. Rev. B 82, (2010)

29 Third-harmonic generation (THG):Sr 2 CuO 3 shift of peak position emergence of lowenergy spin-related excitation without electronphonon interaction, in contrast to linear absorption S. Sota, T. T., S. Yunoki, J. Phys. Soc. Jpn. 84, (2015)

30 Phase diagram of high-tc cuprates Nd 2-x Ce x CuO 4 La Mott Insulator 2-x Sr x CuO 4 Anomalous metal Pseudo gap Fermi liquid d-wave SC Antiferro. stripe (charge order) d-wave SC Fermi liquid doping x

31 Static spin structure factor S(q) by DMRG electron doping 6x6 sites Parameters: t=0.3ev U/t=8, t /t=-0.3 x: carrier concentration x=0.06 x=0.11 x=0.17 x=0.22

32 Static spin structure factor S(q) by DMRG hole doping 6x6 sites Parameters: t=0.3ev U/t=8, t /t=-0.3 x: carrier concentration x=0.06 x=0.11 x=0.17 x=0.22

33 Research collaborating with quantum-beam facilities Spin and charge dynamics of electron-doped cuprate superconductors K. Ishii, M. Fujita, T.T. et al., Nat. Commun. 5, 3714 (2014) ESRF (soft x-ray resonant inelastic scattering) SPring-8 (hard x-ray resonant inelastic scattering) J-PARC (inelastic neutron scattering) + numerical techniques

34 [K. Ishii, M. Fujita, T.T. et al., Nat. Commun. 5, 3714 (2014)] Electron-doped cuprate: Nd 2-x Ce x CuO 4 Observation of the enhancement of magnetic-excitation energy with increasing carrier density

35 Dynamical spin structure factor S(q,ω) by DDMRG Doping dependence of S(q,ω) (two-spin correlation function) in electron-doped 2D Hubbard model 6x6 sites Parameters: t=0.3ev U/t=8, t /t=-0.3 x: carrier concentration S(q,ω) (arb. units) q=(π/7,π/3) q=(π/7,2π/3) x=0 x=0.06 x=0.11 x=0.22 Shift of peak toward high energy with x Consistent with experiment [K. Ishii et al., Nat. Comm. 5, 3714 (2014)] [W. S. Lee et al., Nat. Phys. 10, 883 (2014)] 4 q=(π/7,π) ω (ev) Consistent with quantum Monte Carlo calculations [C. J. Jia et al., Nat. Commun. 5, 3314 (2014)]

36 Dynamical charge structure factor N(q,ω) by DDMRG Doping dependence of N(q,ω) in electron-doped t-t -U Hubbard model T. T., K. Tsutsui, M. Mori, S. Sota, S. Yunoki, Phys. Rev. B 92, (2015) Electron-doping Peak in S(q,ω) q=(π/7,π/3) Hole-doping q=(π/7,π/3) N(q,ω) (arb. units) q=(π/7,2π/3) 1.0 x= x= q=(π/7,π) x=0.11 x=0.22 N(q,ω) (arb. units) q=(π/7,2π/3) 1.0 x= q=(π/7,π) x=0.11 x= ω (ev) Strong intensity at low-q, low-energy Lower in energy than spin excitations ω (ev) Prediction for experiments

37 Charge motion in doped Mott insulator Incoherent motion energy scale: hopping t already observed by RIXS (Ishii et al.) N(q,ω) of t-j model [G. Khaliullin, P. Horsch, PRB 54, R9600 (1996)] Coherent motion energy scale: magnetic J Prediction to RIXS (T. T. et al.)

38 Spin and charge velocities in the Hubbard-type model 1D: spin-charge separation spin velocity v s Velocity charge velocity v c n 2D: approximate spin-charge separation [T. T. and S. Maekawa, JPSJ 65, 1902 (1996)] Velocity n v s v c

39 Summary Density-matrix renormalization group (DMRG) DMRG Dynamical DMRG polynomial expansion 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) Usefulness of Gaussian broadening Spin and charge excitations in square t-t -U Hubbard model