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

Density-matrix renormalization group （DMRG) ► DMRG ► Dynamical DMRG ► Extension to two dimensional systems Recent results obtained by DDMRG ► Optical excitations in 1D Mott insulator coupled to phonon ・ linear absorption ・ Third-harmonic generation (THG) ► Spin and charge excitations in square t-t’-U Hubbard model

• 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)
• T. Tohyama, K. Tsutsui, S. Sota, and S. Yunoki, Phys. Rev. B 92, 014515 (2015)

Outline

► Spin excitations in 1D quantum spin systems

Dynamical properties in strongly correlated electron systems (SCES)

high-temperature superconductors, quantum spins, Mott insulators, …

External field: photon, neutron

SCES

response to external field：

excitation dynamics equibrium/ nonequibrium

spin charge

• rbital

lattice Quantum beam: SPring-8, J-PARC Pump-probe spectroscopy

・constructing lattice model with correlation ・numerical techniques to calculate dynamics

Setting lattice models

e.g. Hubbard model

Dynamical correlation functions

e.g. current-current correlation: optical absorption

1 1 ( ) Im 0 j j H i E χ ω π ω γ = − + − −

† 1, , ,

. .) (

i i i

j it H c c c

σ σ σ +

= − −

∑

Parameters: from first-principles calculations, experiments, etc

∑ ∑

↓ ↑ + +

+ − =

i i i i i i

n n U c c t H

, , , , , , σ δ σ δ σ

i

σ

site spin

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.

∑

=

j i ij

j i

,

ψ ψ

ii ij i j j

ρ ψ ψ

′ ′

=∑

ground-state wave function density matrix of system

1

Tr( ) : eigenstate of ( 0): eigenvalue of

m

A ρA A A u u u u ρ u ρ

α α α α α α α α α α

ω ω ω

=

= = ≈ ≥

∑ ∑

discard unimportant states:

α

ω ≈

1

1

m α α

ω

=

−∑

m: truncation number : truncation error

Dynamical DMRG

system environment

ij

ψ =

i j

∑ ∑ ∑

= =

α α α α α α

ψ ψ ρ 1

,

, ,

p p

j i’j ij ii’ Multi targets: α

∑

= ψ ψ ρ

j i’j j i ii’ Single target

1 1 ˆ ˆ ( ) Im 0 O O H i E χ ω π ω γ = − + − +

ˆ ( ) ( ) 1 ˆ 0 O O H i E

α ω

ρ ω ψ ω γ     = ⇒    + − +  

The reduced density matrix depends on ω  perform DMRG for a given energy ω.

Correction vector

• E. Jeckelmann, Phys. Rev. B

66, 045114 (2002)

Generation and diagonalization

• f

Process of Dynamical DMRG

Ground state： Lanczos method

・ ・ ・ ・ ・ ・ ・ ・ ・

Target states：

( )

ρ ω

1 ˆ ˆ 0 , O O H i E ω γ + − + ω

A given energy

†

U U ρ

Transformation of operators:

†

UAU

Calculation of physical quantities

†

1 1 ˆ ˆ ( ) Im 0 O O H i E χ ω π ω γ = − + − +

How to calculate the correction vector

( )

1 ˆ 0 O H i φ ω ω γ = − +

• 2. Lanczos method

( )

1

1 ˆ ~

M n n

n n O E i φ ω ω γ

=

− +

∑

 

   n  Lanczos vector starting from Independent of ω ˆ 0 O Lorentzian broadening

• 1. Modified conjugate gradient method

( ) ( )

ˆ 0 H i O ω γ φ ω − + = Solve this equation iteratively for a given ω.

• E. Jeckelmann,
• Phys. Rev. B 66,

045114 (2002)

How to calculate the correction vector

( )

1 ˆ 0 O H i φ ω ω γ = − +

• 3. Polynomial expansion using Legendre functions

[S.Sota, T.T., PRB 82, 195130 (2010)]

( ) [ ]

ˆ ~ 2 ( ) ( ) ( )

L l l l l

Q i P P H O φ ω ω π ω

=

−

∑

l

Q

l

P

Legendre polynomial of the first kind Legendre polynomial of the second kind

ω and H : separated polynomials.

1 1

( 1) ( ) (2 1) ( ) ( )

l l l

l P l P lP ω ω ω ω

+ −

+ = + −

Recursive relation

A problem of polynomial expansion

Gibbs oscillation

2 ( ') ~ ( ') ( ) ( ') 2 1

L L l l l

P P l δ ε ε δ ε ε ε ε

=

− − = +

∑

L

Recursion relations: ( )

2 ' 1 1 ' ' 1 1

2 1 2 1 ( ) ( ) ( ) ( ) 1 1 1 ( ) 2 1 ( ) ( )

l l l l l l l

l l l P H H P H P H P H l l l P H l P H P H

σ σ σ σ σ σ σ

σ

+ − + −

+ + = + + + + + = + +        

( )

0.2

L

δ ε −

2 / L σ π =

without Gaussian averaging L L

( )

0.2

L σ

δ ε −

with

( )

2 2

1 2 2 1

1 ( ) ( ) ( ) 2

H l l l

P H P H d e P H

ε σ σ

ε πσ

− − −

→ =

∫



   2 / L σ π =

Introduce Gaussian-type broadening to remove Gibbs oscillation

[S. Sota and M. Itoh,

• J. Phys. Soc. Jpn. 76, 054004 (2007)]

Gaussian broadening

Other applications of polynomial expansion

( ) ( ) ( ) ( ) ( )

~ 1 2 1 ( ) ( )

iH t L l l l l

t t e t l j t P H t

δ

φ δ φ δ φ

− =

+ = − +

∑

Time-evolved wave function

spherical Bessel function

l

j

Thermodynamic properties

modified spherical Bessel function

l

i

Partition function

( )

2

2 1 ~ ( 2) ( ) 2

H L l l l

e l C i P H

β

ξ β ξ β ξ

=

= + −

∑



( ) ( )

Z ξ β ξ β =  

• S. Sota, T. T., Phys. Rev. B 78, 113101 (2008)

c.f. E. M. Stoudenmire, S. R. White,

• Phys. Rev. B 87, 155137 (2013)

Added sites for the sweep of a fraction of system

added sites sweeping direction of sweeping update update the information of operators by MPI communications.

Extension to two dimensions (2D-DMRG) real-space parallelization method

Performance in K computer 3×8 triangular Hubbard model

5 10 15 20 500 1000 1500

FLOPS/PEAK (%) region number

5 10 15 20 10 20 30 40 50

elaplsed time (sec.) region number

All most perfect road balance

elapsed time efficiency

• Ex. One-dimensional extended Hubbard model

Performance in K computer

15

Triangular Hubbard model

• J. Kokalj, R. H. McKenzie, Phys.
• Rev. Lett. 110, 206402 (2013)

Uc1 U/t Uc1 Uc2

metal 120°AF Spin liquid?

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

Density-matrix renormalization group （DMRG) ► DMRG ► Dynamical DMRG ► Extension to two dimensional systems Recent results obtained by DDMRG ► Optical excitations in 1D Mott insulator coupled to phonon ・ linear absorption ・ Third-harmonic generation (THG) ► Spin and charge excitations in square t-t’-U Hubbard model

• 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)
• T. Tohyama, K. Tsutsui, S. Sota, and S. Yunoki, Phys. Rev. B 92, 014515 (2015)

Outline

► Spin excitations in 1D quantum spin systems

Spin-Peierls (SP) compound CuGeO3

• TSP=14K
• the first inorganic SP system
• edge-shared Cu-O chain with S=1/2 on Cu2+
• deviation from Heisenberg model

(Bonner & Fisher curve)

• M. Hase, I. Terasaki, K. Uchinokura, PRL 70, 3651 (1993)

1 1 2

(1 ( 1) )

i i i i i i i

H J

δα

δ α

+ +

  = − − ⋅ + ⋅    

∑ ∑

S S S S

SP

T T <

δ =0.022 at T=0

Phonons in CuGeO3

• 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 ωph= 1.4 meV, ∆= 1.8 meV (TTF)CuS4C4(CF3)4 ∆ > ωph In-chain phonon in CuGeO3 ωph= 26 meV, 13 meV ∆= 2 meV ∆ << ωph antiadiabatic limit

[G. Uhrig, PRB 57, R14004 (1998)]

In-chain phonon may couple to spin even below the SP transition.

Spin-Peierls model

( )

SP 1 2 † † † 1 1 1

2

i i i i i i i i i i i i i i i i

H J b b J b b b b α ω λ

+ + + + +

  = ⋅ + ⋅     + + + − + ⋅

∑ ∑ ∑ ∑

S S S S S S

λ : unknown

S(q,ω) of spin-Peierls model by DDMRG

α0= 0.36 ω0 = 1.5J λ= 0.5ω0 /J λ= 0 ω0 Phonon-assisted spin excitation is expected above the upper edge of spin continuum for CuGeO3. L=16, T=0

• T. Sugimoto, S. Sota, and T.T.,

JPSJ 81, 034706 (2012)

New diamond quantum spin lattice

K3Cu3AlO2(SO4)4

• M. Fujihala et al., J. Phys. Soc. Jpn. 84, 073702 (2015)

J1 J3 J2 Jm Jd J5 J4

A J1 J2 J3, J4 J5 Jm, Jd, Jd’ 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

240-site ring (80 unit cells) m=360 S(q,ω) of K3Cu3AlO2(SO4)4 by DDMRG

Optical excitation of one-dimensional Mott insulator coupled to phonons

Origin of in-gap state generated by just after photo irradiation for Ca2CuO3 Extended Hubbard-Holstein model

Optical absorption calculated by DDMRG Low-energy excitation consistent with experiment Spin excitations formed by polarons

• H. Matsuzaki, H. Nishioka, H.

Uemura, A. Sawa, S. Sota, T. T., H. Okamoto, Phys. Rev. B 91, 081114(R) (2015)

Comparison between two broadenings

• T. T., H. Matsueda, Prog. Theor.
• Phys. Suppl. 175, 165 (2008)
• S. Sota, T. T., Phys. Rev. B

82, 195130 (2010)

N=12 Conjugate gradient Lorentzian N=24 Polynomial expansion Gaussian

Third-harmonic generation (THG)：Sr2CuO3

☑ shift of peak position ☑ emergence of low- energy spin-related excitation without electron- phonon interaction, in contrast to linear absorption

• S. Sota, T. T., S. Yunoki, J. Phys. Soc. Jpn. 84, 054403 (2015)

Pseudo gap Anomalous metal Antiferro. d-wave SC Fermi liquid d-wave SC Fermi liquid

Mott Insulator

Phase diagram of high-Tc cuprates

“stripe”

（charge

• rder）

Nd2-xCexCuO4 La2-xSrxCuO4

doping x

Static spin structure factor S(q) by DMRG

electron doping x=0.06 x=0.17 x=0.11 x=0.22 6x6 sites Parameters: t=0.3eV U/t=8, t’/t=-0.3 x: carrier concentration

Static spin structure factor S(q) by DMRG

hole doping x=0.06 x=0.17 x=0.11 x=0.22 6x6 sites Parameters: t=0.3eV U/t=8, t’/t=-0.3 x: carrier concentration

Research collaborating with quantum-beam facilities ESRF (soft x-ray resonant inelastic scattering) SPring-8 (hard x-ray resonant inelastic scattering) J-PARC (inelastic neutron scattering)

Spin and charge dynamics of electron-doped cuprate superconductors

• K. Ishii, M. Fujita, T.T. et al., Nat. Commun. 5, 3714 (2014)

+ numerical techniques

Electron-doped cuprate: Nd2-xCexCuO4

[K. Ishii, M. Fujita, T.T. et al., Nat. Commun. 5, 3714 (2014)]

Observation of the enhancement of magnetic-excitation energy with increasing carrier density

2 4

q=(π/7,π/3)

x=0 x=0.06 x=0.11 x=0.22

S(q,ω) (arb. units)

2 4

q=(π/7,2π/3)

0.0 0.2 0.4 0.6 2 4

q=(π/7,π) ω (eV)

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

[C. J. Jia et al., Nat.

• Commun. 5, 3314 (2014)]

☑ Consistent with quantum Monte Carlo calculations ☑ 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)]

0.0 0.5 1.0 1.5

q=(π/7,π/3)

x=0.06 x=0.11 x=0.16 x=0.22

N(q,ω) (arb. units)

0.0 0.5 1.0 1.5 q=(π/7,2π/3) 0.0 0.2 0.4 0.6 0.0 0.5 1.0 1.5

q=(π/7,π) ω (eV)

0.0 0.5 1.0 1.5

x=0.06 x=0.11 x=0.22

q=(π/7,π/3) N(q,ω) (arb. units)

0.0 0.5 1.0 1.5 q=(π/7,2π/3) 0.0 0.2 0.4 0.6 0.0 0.5 1.0 1.5

q=(π/7,π) ω (eV)

Hole-doping ☑ Strong intensity at low-q, low-energy Electron-doping Prediction for experiments

Dynamical charge structure factor N(q,ω) by DDMRG

• T. T., K. Tsutsui, M. Mori, S. Sota, S. Yunoki, Phys. Rev. B 92, 014515 (2015)

Doping dependence of N(q,ω) in electron-doped t-t’-U Hubbard model ☑ Lower in energy than spin excitations

Peak in S(q,ω)

Charge motion in doped Mott insulator Incoherent motion

energy scale: hopping t already observed by RIXS (Ishii et al.)

Coherent motion

energy scale: magnetic J Prediction to RIXS (T. T. et al.)

[G. Khaliullin, P. Horsch, PRB 54, R9600 (1996)]

N(q,ω) of t-J model

Spin and charge velocities in the Hubbard-type model 1 0.5 n

1D: spin-charge separation Velocity

spin velocity vs charge velocity vc 1 0.5 n

2D: approximate spin-charge separation Velocity

vs vc

[T. T. and S. Maekawa, JPSJ 65, 1902 (1996)]

Density-matrix renormalization group （DMRG) ► DMRG ► Dynamical DMRG polynomial expansion ► Extension to two dimensional systems Recent results obtained by DDMRG ► Optical excitations in 1D Mott insulator coupled to phonon ・ linear absorption ・ Third-harmonic generation (THG) ► Spin and charge excitations in square t-t’-U Hubbard model

Summary

► Spin excitations in 1D quantum spin systems Usefulness of Gaussian broadening