spin and charge excitations in stripe-ordered cuprates ... · schematic diagram of stripe order in...
TRANSCRIPT
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