Magnetism and doping effects in spin-orbit coupled Mott insulators
Max Planck Institute for Solid State Research, Stuttgart
Giniyat Khaliullin
Witczak-Krempa, Chen, Y-B.Kim, Balents (Annu. Rev. 2014)
Multiorbital Hubbard: U vs λ phase diagram
spin-orbit coupling λ
corr
elat
ion
U
spin-orbit coupling
S p i n – o r b i t a l a l g e b r a
Fe r m i o n i c s t a t i s t i c s
next 40 min: stay in this area
corr
elat
ion
weak
strong
Spin-orbit coupled Mott insulators: „unconventional“ magnetism and doping effects
spin-orbit coupling
corr
elat
ion
Spin-orbit coupled Mott insulators: from 3d-Co to 5d-Ir
Ti Co Ru Rh Os Ir
Sr2IrO4
CoO2
Ca2RuO4
KOs2O6
3d 4d 5d
La2CuO4
spin-orbit quenched
spin-orbit coupling
corr
elat
ion Sr2IrO4
CoO2
Ca2RuO4
KOs2O6
D o
p i
n g
D o
p i
n g
D o
p i
n g
Fe r m i o n i c s t a t i s t i c s
„unconventional“ magnetism „unconventional“ metal & SC?
D O P I N G:
La2CuO4
Jahn-Teller Spin-orbit
Lifting the orbital degeneracy
moment J=S+L
trigonal field
real orbital L=0
complex L=1
|xy + yz + zx
λ
SL
quadrupole spin
Spin ½ algebra pseudospin spin
t2g t2g
g= 2 g= -2
reality is often „in-between“
coplanar, single-Q
H = (SS)
Magnetism Jahn-Teller
S=1/2 Spin-orbit
J=1/2
GKh (PTPS, 2005)
large magnetic unit cell non-coplanar, multi-Q
hosts spin vortex
H= (SS) + Ising(x,y,z)
yy
xx zz
coplanar, single-Q
H = (SS)
Magnetism Jahn-Teller
S=1/2 Spin-orbit
J=1/2
GKh (PTPS, 2005)
large magnetic unit cell non-coplanar, multi-Q
hosts spin vortex
H= (SS) + Ising(x,y,z)
yy
xx zz
JT vs SO incommensurate orderings
SC pairing Jahn-Teller Spin-orbit
GKh, Koshibae, Maekawa (PRL 2004) GKh (PTPS, 2005)
Spin-singlet d-wave SC Baskaran (2003) Kumar, Shastry (2003) Ogata (2003) P. Lee et al.(2004)
Pseudospin-triplet p-wave SC
J=1/2 S=1/2
nondegenerate
-
Knight-shift finite in all directions
SC pairing Jahn-Teller Spin-orbit
Nax(Co,Rh,Ir)O2
GKh, Koshibae, Maekawa (PRL 2004) GKh (PTPS, 2005)
Spin-singlet d-wave SC Baskaran (2003) Kumar, Shastry (2003) Ogata (2003) P. Lee et al.(2004)
Pseudospin-triplet p-wave SC
J=1/2 S=1/2
Orbital moment L has a „shape“:
The origin of „unconventionality“: ORBITAL magnetism
Lz=1 z(x+iy)
c
a
Lx=1 x(y+iz)
b
Ly=1 y(z+ix)
Hopping amplitude:
nontrivial band topology Shitade et al. (2009)
A B complex
(no inversion)
A B real
(inversion)
Orbital moment L interactions:
ORBITAL MAGNETISM
c
a b
1) non-Heisenberg 2) bond-dependent
unconventional models: bond-dependent Ising, biquadratic…
L-moment shape links magnetism and chemical bonding
Orbital moment L interactions:
ORBITAL MAGNETISM
c
a b
1) non-Heisenberg 2) bond-dependent
„orbital frustration“
H
GKh & Okamoto (PRL 2002)
Simple cubic lattice: Qx
Qy Qz
non-coplanar multi-Q no LRO at finite T
?
Pseudospin J=S+L inherits bond-dependent and frustrated
nature of orbitals
„unconventional“ magnetism
L S J
spin-nematics, multipolar order, spin-liquids…
Spin-orbit multiplels of TM-ions
GS-degeneracy:
Co,Rh,Ir
Nb,Mo,Re Ru,Re,Os
Mo,Re,Os
„pseudospin“
Kramers: dipole, octupole,… non-Kramers: quadrupole,…
large J multipoles: spin nematic, „hidden“ order quantum spin „nonmagnetic“
Abragam & Bleaney (1970)
„The old“ pseudospin J=1/2
Two branches split by spin-orbit: magnon & exciton NaxCoO2 ?
GKh et al. (2004,2005)
„Heisenberg“ magnon
exciton
3d Co-fluoride, neutron scattering
M a g n o n
E x c i t o n 600 meV
40 meV 3d
5d
× 10
K2CoF3
from 3d Co to 5d Ir B.J.Kim, Takagi,…
Holden et al. (1971) neutron scatteting
Kim et al. (2012) RIXS data
Sr2IrO4
spin one-half 2D Mott systems similar to cuprates ?
d1 d5 d7 d9
Ti ...
t2g electron eg electron
Ni …
eg hole
Cu
t2g hole
Co …
JAF=130 meV La2CuO4
d1 d5 d7 d9
t2g electron eg electron eg hole t2g hole
NaxCoO2 triangular lattice small J, low Tc
JAF=130 meV La2CuO4
spin one-half 2D Mott systems similar to cuprates ?
superlattices …/NiO2/…
?
Cu Ni Co
spin one-half 2D Mott systems similar to cuprates
d1 d5 d7 d9
t2g electron eg electron eg hole
Cu
t2g hole
Co Ir
Sr2IrO4
Cuprate-like magnetism and SC?
TN~240 K
(Cava, Cao,…) TN~320 K
JAF=130 meV La2CuO4
Layered “213” iridate Na2IrO3
honeycomb lattice planes
edge-shared octahedra 90° Ir-O-Ir bonds
…similar to NaxCoO2
spin one-half 2D Mott systems similar to cuprates
d1 d5 d7 d9
t2g electron eg electron eg hole
Cu
t2g hole
Ir
La2CuO4 Sr2IrO4 Na2IrO3
t2g eg „orbital chemistry“ spin-orbit coupling
Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013)
J=1/2 magnetism in iridates: theory
Two-parameter Hamiltonian = Ising(x,y,z) + Heisenberg:
dominant in 180-bonding (Sr2IrO4 perovskite)
dominant in 90-bonding (honeycomb Na2IrO3)
= „cuprate“ model high-Tc SC ?
= Kitaev model (2006) „Majorana world“ ?
Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013)
J=1/2 magnetism in iridates: theory
Two-parameter Hamiltonian = Ising(x,y,z) + Heisenberg:
dominant in 180-bonding (Sr2IrO4 perovskite)
dominant in 90-bonding (honeycomb Na2IrO3)
= „cuprate“ model high-Tc SC ?
= Kitaev model (2006) „Majorana world“ ?
Kim et al.(2012)
Sr2IrO4 TN~240 K
Coldea et al.(2001)
La2CuO4 TN~320 K
?
?
-theory predicts „nearly“ Heisenberg AF
Sr2IrO4
Pseudospin ½ in perovskites:
Fermiology of electron doped Sr2IrO4
B.J. Kim et al. (Science 2014)
„Fermi-arcs“ at low doping
Potassium K overlayer
„normal“ FS
B.J. Kim et al. (Science 2014)
„Fermi-arcs“ at low doping
Pseudogap opens at low T
…and closes at 110 K
„normal“ FS
T-dependent „pseudogap“ in Sr2IrO4
many-body effect !
Sr2IrO4 magnetism, fermiology & lattice: same as in
superconductivity?
SUPER! GREAT!
„…find 10 differences…“ Sr2IrO4 La2CuO4
La2CuO4
Current status: no definite answer
Jackeli, GKh (2009) Chaloupka, Jackeli, GKh (2010, 2013)
J=1/2 magnetism in iridates: theory
Two-parameter Hamiltonian = Ising(x,y,z) + Heisenberg:
dominant in 180-bonding (Sr2IrO4 perovskite)
dominant in 90-bonding (honeycomb Na2IrO3)
= „cuprate“ model high-Tc SC ?
= Kitaev model (2006) „Majorana world“ ?
The Kitaev model
Exactly solvable
Low-energy excitations: free Majorana fermions
Short-range RVB, large spin gap
Dirac cones EF
SxSx SySy
SzSz
Sα = c fα
Itinerant free band (uncharged)
local & high-energy
Honeycomb lattice theory: K should be dominant but J is finite
Na2IrO3
What is in between ? -If „some liquid“ is still left ?
Kitaev model Heisenberg model
order liquid
Chaloupka, Jackeli, GKh (2010)
Spins, magnons Majorana land
Quantum phase transition: spin fractionalization
spin-orbit coupling
J K
YES!
pseudospin spin
Na2IrO3
AM order TN~15K
(1) Mag. bandwidth: 40 meV~30 TN (Gretarrson et al.)
(3) SW gap is small ~1 meV (Coldea et al.)
(2) Intense q=0 scattering (Gretarrson et al.)
Exp.data
(2) non-Heisenberg (3) C3 symmetric
(1) strongly frustrated
Interactions:
Kitaev-Heisenberg K-J model with large K > J makes all three points „for free“
- Kitaev term seems to be dominant - Other terms have yet to be sorted out
(current status) Data collected so far suggests that
Na2IrO3
2014 (H.-Y. Kee et al. van den Brink et al. Imada et al.)
measure and fit, measure and fit…
most wanted: single-crystal S(q,w) data
-Did you lose your Majoranas here?
Na2IrO3
The streetlight effect
-No, but the light is much better over here !
New candidate: RuCl3 Plumb et al, 2014
honeycomb plane
„dreamland“
J=1/2 d5 Co, Rh, Ir
„nonmagnetic“ Mott insulators
…farther away from the streetlight
L S
L + S = 0
d4 Ru, Os,..
competing
singlets magnetic LRO
QCP
spin-orbit driven magnetic QCP J=0 physics:
singlet
triplet Jex λ λ
d4 d4
(iii) Condensation of spin-orbit exciton
magnetic LRO
d4 ion: Van-Vleck magnetism
(i) There are no „pre-existing“ moments
L S
M=2S-L AFM exci
ton
gap
exchange interaction
J=0
J=1 λ (ii) J=0 to J=1 transition: off-diagonal magnetic moment M=2S-L
(„spin-orbit exciton“) 0
Singlet-triplet examples
QCP
para magnetic
J
(C) eg orbital FeSc2S4 (Chen, Balents, Schnyder, 2009)
(D) Spin-state-crossover in Fe-based SC (Chaloupka & GKh, 2013)
(A) Weakly coupled dimers
(B) 4f Pr compounds (broad literature since 1970‘s)
Bilayer AF
(Nat.Phys. 2009)
birthplace for „unconventional“ physics
d4 : Intra-ionic „dimer“ made of S and L
λ S L
1) energetic (~100 meV) 2) generic (any lattice)
QUANTUM CRITICALITY
GKh (2013)
Sachdev, Keimer, Phys.Today, 2011
Inter-ionic dimers
-small energies -special geometry
LRO moment:
cond.density
distance from QCP
M
AFM (triplon gas) PM
J QCP
spin-orbit exchange
J
S=1 boson
d4 Mott insulator: singlet-triplet model (180°) (perovskites)
GKh (2013)
Excitations: 1. The amplitude mode changing the angle between S & L
2. The phase modes in-phase rotation of S & L
S L Goldstone
„Higgs“ S L
1 2
k
ω
Bond-dependent „xy-model“:
c-bond exchange:
hopping pair-generation
(triangular, honeycomb, kagome…)
x and y type bosons only involved
GKh (2013)
„Heisenberg“ „Kitaev“
NB: T is hard-core boson, not spin!
BOND-SELECTIVE boson interactions:
Honeycomb lattice: dimensionality reduction
Each flavor Tx, Ty, Tz has its own zigzag to move along 1D dispersion:
J=Jcrit :
VBS? zigzag order? spin nematic? …
zero-energy lines
unusual singlet-triplet model, yet to be solved…
Spin-orbit driven magnetic QCP
Ruthenates Ca2RuO4? Li2RuO3? or elsewhere?
Van Vleck-type d4 Mott insulators:
d4 d5 „triplon gas“ or AF S=1/2 AF
J=0 singlet
J=1 triplet
J=1/2 doublet
Ru4+ Os4+ Ir 5+
Ru3+ Os3+ Ir 4+
Doping of „nonmagnetic“ d4 Mott insulators
doping
electron doping
λ
d4 d5 PM or AF S=1/2 AF
singlet
triplet
doublet
doping doping
λ
doping
J
AF
doping
J
AF Para
Naive expectation
Doping of „nonmagnetic“ d4 Mott insulators Chaloupka & GKh (unpublished)
Doping of „nonmagnetic“ Mott insulators
d4 S=0
S=1
d5
S=1/2
Quantum mixture of S=0, S=1/2, S=1 objects
Hopping generates S=1 states („vacuum polarization“): t f+ifj (T+-T) - mixes triplons with Stoner continuum Stoner triplon
- activates double-exchange FM
Triplon mixes-up with Stoner continuum, q=0 paramagnon emerges
Paramagnon softens, its intensity increases with hopping t
„bare“ Stoner
paramagnon Van-Vleck exciton
Phase diagram: doping x vs exchange J
AF
Para
FM FM
Para
1. Doping supresses AF (expected)
2. Doping induces FM (new)
3. Larger t wider FM area
AF
„triplon gas“
Phase diagram: doping x vs hopping t
AF
FM
„nonmagnetic“ metal
„nearly-FM“ metal
AF --- nearly AF liquid ---- triplon gas ---- nearly FM --- FM
…emerging out of the „nonmagnetic J=0“ state
„Paramagnon-glue“: triplet or singlet SC-pairing
AF
FM
...undecided…
FM-paramagnons
p-SC
Doping of „nonmagnetic“ d4 Mott insulators
t2g4 ions:
Mott insulator to start with, better 2D Spin-orbit comparable with hopping t
AF
FM
FM-paramagnons Ca2RuO4 ? p-SC
Doping of „nonmagnetic“ d4 Mott insulators
Os
Ir
Re
Ru
Two hopping paths: positive quantum interference
Path A: t/3 Path B: t/3
A + B = (2/3) t Net hopping: large, spin-isotropic
J (SiSj) Heisenberg coupling as in cuprates
Corner-shared octahedra (orbital-diagonal hopping)
teff = A + B = 0 Net hopping amplitude is zero!
Edge-shared octahedra (orbital non-diagonal hopping)
A = iσ t/3 B = - iσ t/3
Usual Heisenberg interaction is suppressed
GKh, Koshibae, Maekawa (2004) GKh (2005) Jackeli & GKh (2009) Shitade et al. (2009)
Two hopping paths: negative interference
…and look for J=1/2 K-J model on other lattices
Pseudospin AND geometrical frustration:
- amplify the chances to find exotic states !
GKh (PTPS, 2005)
spin vortex multi-Q order
(K=-J>0)
Rousochatzakis et al. (condmat, 2012)
Kimchi & Vishwanath (PRB, 2014)
„hidden“ Goldstone
Map H(S,L) onto singlet-triplet model
Spin-orbital exchange Hamiltonian
interchange (x y)j
t2g orbital degeneracy: unquenched L=1
S=1, no orbital degeneracy
J=0
J=1 λ
J=2
2λ
nonmagnetic, J=S+L=0 „Van Vleck-type ion“
Δ
Jahn-Teller Spin-orbit (SL)
d4 : Jahn-Teller vs spin-orbit
xy
xz yz
LDA+U result for Ca2RuO4