Spin and Charge Excitations in

Stripe-Ordered Cuprates

Takami Tohyama

Tokyo University of Science, Japan

Shigetoshi Sota RIKEN

Michiyasu Mori JAEA

Kenji Tsutsui QST

Outline

- Charge stripes in hole-doped cuprates

- Dynamical density-matrix renormalization group

(DDMRG) calculations of dynamical spin and charge

structure factors for a 24x4 t-t’-J ladder

- Comparison with ＲＩＸＳ experiments and some implication

for electron-doped cuprates

T.T., M. Mori, and S. Sota, Phys. Rev. B 97, 235137 (2018)

- Effect of dimensionality on dynamical spin and charge

structure factors

- RIXS in one-dimensional extended Hubbard model

Schematic diagram of stripe order in

1/8-doped La1.875La0.125CuO4

Review: M. Fujita et al., J. Phys. Soc. Jpn. 81, 011007 (2012)

Spin dynamics in stripe-ordered hole-doped cuprates: experiments

M. Fujita et al., J. Phys. Soc. Jpn. 81, 011007 (2012)

Inelastic neutron scattering

“hourglass” around (p,p)

(p,p)(0.6p,p) (1.4p,p)

Resonant inelastic x-ray scattering

“plateau” around (0.5p,0)

stripe-ordered La1.875Ba0.125CuO4

H. Miao et al.,

PNAS 114, 12430

(2017)

(0.5p,0)

Ground-state properties in the 1/8-hole-doped 12x4 t-t'-J ladder

S. White and D. J. Scalapino, PRB 60, R753 (1999)

J/t = 0.35

charge densityd-wave paring

correlation

Hole-doped four-leg t-t'-J ladder gives a charge-stripe orderS. White and D. J. Scalapino, PRB 60, R753 (1999)

We calculate dynamical spin and charge structure factors

0

1 1, Im 0 0z z

tJ

S S SH E i

p

q qq

0

1 1, Im 0 0

tJ

N N NH E i

p

q qq

t-t’-J model

1 2 1

, , , ,

, , , , ,

1' h.c.

4tJ i j i j i j i j

i j i j i j

H t c c t c c J n n

S S

24x4 t-t’-J ladder

x (leg) direction open boundary

y (rung) direction periodic boundary

Dynamical density-matrix renormalization group (DDMRG)

J/t = 0.4

t’/t = -0.25

T.T., M. Mori, and S. Sota, Phys. Rev. B 97, 235137 (2018)

Dynamical charge structure factor in the 24x4 t-t'-J ladder

hole doping x=1/12 x=1/8 x=1/6

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xh=1/12

N(q,)

qy=0

24x4 t-t'-J hole x=8/96=0.083, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xh=1/8

N(q,)

qy=0

24x4 t-t'-J hole x=12/96=0.125, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xh=1/6

N(q,)

qy=0

24x4 t-t'-J hole x=16/96=0.167, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

Minimum-energy position is close to qx= p(1- 4x) as expected from stripe order.

(p,0)(0,0)

0.0 0.4 0.80.0

0.2

0.4

0.6

0.8

1.0

S(q,)

qy=p

24x4 t-t'-J hole x=16/96=0.167, m=3000, /t=0.08

qx/p

/t

0.0

0.5

1.0

1.5

2.0

1.2 1.6 2.0

xh=1/6

0.0 0.4 0.80.0

0.2

0.4

0.6

0.8

1.0

S(q,)

qy=p

24x4 t-t'-J hole x=12/96=0.125, m=3000, /t=0.08

qx/p

/t

0.0

0.5

1.0

1.5

2.0

1.2 1.6 2.0

xh=1/8

Dynamical spin structure factor in the 24x4 t-t'-J ladder

0.0 0.4 0.80.0

0.2

0.4

0.6

0.8

1.0

S(q,)

qy=p

24x4 t-t'-J hole x=8/96=0.083, m=3000, /t=0.08

qx/p

/t

0.0

0.5

1.0

1.5

2.0

1.2 1.6 2.0

xh=1/12

hole dopingx=1/12 x=1/8 x=1/6

(p,p)(0.6p,p) (1.4p,p)

- Calculated q=(p,p) excitations

are higher in energy than the

experimental data.

- Incommensurate qx is p(1-2x).

(p,p)(0, p)

- Strong “outward” weight,

inconsistent with experiments

Dynamical charge and spin structure factor in the 24x4 t-t'-J ladder

hole doping x=1/12 x=1/8 x=1/6

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xh=1/12

N(q,)

qy=0

24x4 t-t'-J hole x=8/96=0.083, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xh=1/8

N(q,)

qy=0

24x4 t-t'-J hole x=12/96=0.125, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xh=1/6

N(q,)

qy=0

24x4 t-t'-J hole x=16/96=0.167, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

(p,0)(0,0)

- A step structure at the charge order vector qx for x=1/8 and 1/6

strong coupling between charge and spin; ferromagnetic coupling across the stripes

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

xh=1/12

S(q,)

qy=0

24x4 t-t'-J hole x=8/96=0.083, m=3000, /t=0.08

qx/p

/t

0.00

0.02

0.04

0.06

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

xh=1/8

S(q,)

qy=0

24x4 t-t'-J hole x=12/96=0.125, m=3000, /t=0.08

qx/p

/t

0.00

0.02

0.04

0.06

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

xh=1/6

S(q,)

qy=0

24x4 t-t'-J hole x=16/96=0.167, m=3000, /t=0.08

qx/p

/t

0.00

0.02

0.04

0.06

S(q,)

N(q,)

Dynamical charge and spin structure factor in the 24x4 t-t'-J ladder

electron dopingx=1/12 x=1/8 x=1/6

- No step-like structure in contrast to hole doping because of weak stripe order

- A downward spectral weight toward (p,0) intrinsic for electron doping?

S(q,)

N(q,)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xe=1/12

N(q,)

qy=0

24x4 t-t'-J electron x=8/96=0.083, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xe=1/8

N(q,)

qy=0

24x4 t-t'-J electron x=12/96=0.125, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.0 0.2 0.4 0.6 0.8 1.00.0

0.4

0.8

1.2

1.6

2.0

xe=1/6

N(q,)

qy=0

24x4 t-t'-J electron x=16/96=0.167, m=4000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

xe=1/12

S(q,)

qy=0

24x4 t-t'-J electron x=8/96=0.083, m=3000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.16

0.20

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

xe=1/8

S(q,)

qy=0

24x4 t-t'-J electron x=12/96=0.125, m=3000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.16

0.20

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

xe=1/6

S(q,)

qy=0

24x4 t-t'-J electron x=16/96=0.167, m=3000, /t=0.08

qx/p

/t

0.00

0.04

0.08

0.12

0.16

0.20

(p,0)(0,0)

Doping dependence of magnetic excitation by RIXS

M. Le. Tacon et al., Nat. Phys. 7, 725 (2011);

PRB 88, 020501(R) (2013)

M. P. M. Dean, JMMM 376, 3 (2015)

Hole doping

K. Ishii, M. Fujita, T.T. et al., Nat. Commun.

5, 3714 (2014)

W. S. Lee et al., Nat. Phys. 10, 883 (2014)

Electron doping

Dynamical spin structure factor in the t-t'-J model

J/t=0.4

t’/t=-0.25

4-leg

6-leg

8-leg

The two discrepancies are

resolved with increasing leg

number, i.e., toward square

lattice.

(p,p)(p/2, p)

S(q,)

toward square lattice

Charge stripes from 4-leg to 8-leg ladders

N(q,)

Weakening of stripes from

4-leg to 8-leg systems

toward square lattice

2p

3d

absorption process

(=XAS)

emission process

(=XES)intermediate stateground state

RIXS for L-edge

2

†, 0, i io o

i 0

i o

i o

1, ~ 0 f

f

I f D D E EH E i

kkq

q k k0 : ground state (no Cu2p hole)

f : ‘final’ state with momentum q

: dipole transition operatori(o) i(o) i(o), i(o) , , , ,

, ,

3 ; 2 ;z

z

K z j j

j

D d p jj d P

ε k k k

k

ε r

RIXS for cuprates

†,, i io o

i 0

10 f 0

j j jN

jSf D N

E iSD

H

q qkk

Cu2+: 3d9 1 hole on x2-y2 orbital with in-plane spin orientation

1/2 3/i o

2o i

22

15

j jN N z z

ε ε ε ε 1/2 3/o

2i

2

15

j jS S z

i

ε ε

M. W. Haverkort, PRL 105, 167404 (2010)

J. Igarashi and T. Nagao, PRB 85, 064421 (2012)

S. Kourtis, J. van den Brink, and M. Daghofer, PRB 85, 064423 (2012)

1 3,

2 2j

Non spin-flip process

Spin-flip process

Incident-photon energy i dependence of RIXS intensity

Spin-flip Non-spin-fliphole doping (11%)

YBCO

M. Minola et al., PRL 114, 217003 (2015)

18 sites t’=-0.25t

U=10t Ucore=12t =t

K. Tsutsui, T.T., PRB

94, 085144 (2016)

two-magnon

charge

A change from “two-magnon” to “charge”

fluorescence-like behavior

How do RIXS spectra behave

when attractive interactions are

explicitly taken into account?

One-dimensional extended Hubbard model

†1 1 1

h.c.i i i i i i i ii i i

H t c c U n n V n n n n

F. Iemini, T. O. Maciel, and R. O. Vianna, Phys. Rev. B 92, 075423 (2015).

U/t

V/t

Singlet super

Half-filling

H. Q. Lin and J. E. Hirsch, Phys. Rev. B 33, 8155 (1983).

PS

CDWV/t

RIXS for spin-flip process and S(q,)

CDW singlet super. (SS)

0 2 4 6 8 100

20

40

60

80

S(q

,)

(arb

. unit)

(t)

q=p

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

U=-4, V=+0.25

0 2 4 6 8 100

20

40

60

80

S(q

,)

(arb

. unit)

(t)

q=p

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

U=-4, V=-0.25

S(q,)

0 2 4 6 8 10

0

0.06

0.12

0.18

(t)

IS

=1 (

arb

. unit)

K=p

U=-4, V=+0.25

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

Uc=4,i:edge

0 2 4 6 8 10

0

0.06

0.12

0.18

(t)

IS

=1 (

arb

. unit)

K=p

U=-4, V=-0.25

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

Uc=4,i:edge

RIXS

☑ No difference

between CDW

and SS

☑ Excitations at

high-energy

Hubbard model at

half-filling

U<0

N(q,)

S(q,)

U>0

S(q,)

N(q,)

PS

CDWV/t

RIXS for non-spin-flip process and N(q,)

RIXS

N(q,)CDW singlet super. (SS)

0 2 4 6 8 100

100

200

N(q

,)

(arb

. unit)

(t)

q=p6p/75p/74p/73p/72p/7p/7

U=-4, V=+0.25

0 2 4 6 8 100

20

40

60

80

N(q

,)

(arb

. unit)

(t)

q=p

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

U=-4, V=-0.25

0 2 4 6 8 10

0

0.06

0.12

0.18

(t)I

S=

0 (

arb

. unit)

K=p

U=-4, V=-0.25

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

Uc=4,i:edge

0 2 4 6 8 10

0

0.06

0.12

0.18

0.24

(t)

IS

=0 (

arb

. unit)

K=p

U=-4, V=+0.25

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

Uc=4,i:edge

☑ CDW

RIXS and N(q,)

similar

☑ SS

RIXS detects

high-energy

excitations that

are the same as

spin-flip case.

More incoherent

nature in SS than

CDW

☑ Clear difference

between CDW

and SS

PS

CDWV/t

Incident-photon-energy dependence of RIXS for SS

0 5 10

0

0.05

0.1

0.15

edge (t)

I XA

S (

arb

. unit)

U=-4, V=-0.25 Uc=4

XAS

0 2 4 6 8 10

0

0.06

0.12

0.18

(t)

IS

=0 (

arb

. unit)

K=p

U=-4, V=-0.25

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

Uc=4,i:edge

non-spin-flip

0 2 4 6 8 10

0

0.06

0.12

0.18

(t)

IS

=1 (

arb

. unit)

K=p

U=-4, V=-0.25

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

Uc=4,i:edge

spin-flipabsorption at singly

occupied sites

0 5 10

0

0.05

0.1

0.15

edge (t)

I XA

S (

arb

. unit)

U=-4, V=-0.25 Uc=4

0 2 4 6 8 10

0

0.06

0.12

0.18

(t)

IS

=1 (

arb

. unit)

K=p

U=-4, V=-0.25

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

Uc=4,i:Uc

non-spin-flip spin-flip

XAS

absorption at

empty sites

0 2 4 6 8 10

0

0.06

0.12

0.18

(t)

IS

=0 (

arb

. unit)

K=p

6p/7

5p/7

4p/7

3p/7

2p/7

p/7

0

spin-flip: large intensity at low-energy XAS

non-spin-flip: large intensity at high-energy XAS

Summary

☑ RIXS, S(q,), and N(q,) with charge-stripe order

- “hourglass” in S(q,)

- A step-like behavior with S(q,) along (0,0)-

(p,0)

hole doping: stripe order in the ground state

electron doping: weak stripe order in the ground state

- A downward behavior in S(q,) along (0,0)-(p,0).

← itinerant nature in strong antiferromagnetic correlation

T.T., M. Mori, and S. Sota, Phys. Rev. B 97, 235137 (2018)

☑ Dimensionality effect on S(q,)

T. T., S. Sota, S. Yunoki, in preparation

- Weakening of charge stripe change of “hourglass” behavior

☑ RIXS in 1D extended Hubbard model

K. Tsutsui, T. T., in preparation

- Comparison between CDW and singlet superconducting phases