Spin-Orbit Mott Insulators: Magnetism, Order, and Excitations
George Jackeli Max-Planck Institute & University of Stuttgart, Germany Andronikashvili Institute of Physics, Tbilisi, Georgia
Winter School “Computational Magnetism’ Vienna, 20-23 Feb , 2017
1 Thanks to
Judit Romhanyi Leon Balents (MPI -> OIST) (KITP)
H. Takagi (MPI Stuttgart & Uni Tokyo)
BJ. Kim (MPI Stuttgart), V. Kataev (IFW Dresden)
2 ‘There is no learning without having to pose a question’
Richard Feynman (Nobel prize’65)
3 Transition metal oxides: many challenges
d- electrons are partially filled and not very extended Interactions dominate over kinetic energy Many degrees of freedom are at work
Orbital
Spin Charge
Lattice
Metal to Magnetic Insulator High-Tc Superconductivity Colossal Magnetoresistance
Unexpected variety of phases and transitions between them 4 Most of the elements are Transition metals
5 Relativistic in origin, Coulomb force Spin-orbit coupling
Orbital
Spin Charge
Lattice
Enhance interplay by going to heavy TM elements: A Route to Novel Collective States & Phenomena
6 Symmetry breaking by various multipoles
Monopole
charge order broken sym. translational
7 Symmetry breaking by various multipoles
Monopole Dipole
charge order ferroelectricity broken sym. magnetism translational inversion time reversal rotations
8 Symmetry breaking by various multipoles
Monopole Dipole Quadrupole
charge order ferroelectricity broken sym. magnetism orbital order translational inversion spin nematics time reversal rotations rotations
9 Symmetry breaking by various multipoles
Monopole Dipole Quadrupole Octupole
charge order ferroelectricity broken sym. magnetism orbital order translational inversion spin nematics hidden order time reversal rotations inversion rotations time reversal
10 No Symmetry breaking: Quantum Liquid
plain
11 No Symmetry breaking: Quantum Liquid
plain full bodied
12 Outline
Introduction Mott insulator, Orbital degeneracy, Spin-orbital exchange interactions Spin-orbit coupled Mott insulators Local electronic structure, Exchange in total angular momentum j-basis
Experimental implications: j=1/2 systems: Iridates and Ruthenates, j=3/2 systems: Vanadates, Molybdates,and Osmates
13 Mott Insulator one electron per site
14 Mott Insulator one electron per site half-filled band: metal
DOS
E
15 Mott Insulator one electron per site half-filled band: metal
DOS
U t E
16 Mott Insulator
one electron per site half-filled band: insulator
DOS
gap U t E
Coulomb repulsion U>>t Kinetic energy, Charges become localized and gapped Sir Nevill Mott (Nobel prize’77)
17 Mott Insulator
one electron per site half-filled band: insulator
DOS
gap U t E
Coulomb repulsion U>>t Kinetic energy, Charges become localized and gapped Sir Nevill Mott (Nobel prize’77) Low energy degrees are magnetic: Spins interacting via Heisenberg exchange
Werner Heisenberg (Nobel prize’32) 18 Ferro and Antiferromagnetism
19 Ferro and Antiferromagnetism
known to ancient Greek tribe Ferromagnet near town Magneisa
20 Ferro and Antiferromagnetism
known to ancient Greek tribe Ferromagnet near town Magneisa
Antiferromagnet proposed by
Louis Neel (Nobel prize’70)
21 Ferro and Antiferromagnetism
known to ancient Greek tribe Ferromagnet near town Magneisa
Antiferromagnet proposed by
Louis Neel (Nobel prize’70)
22 Ferro and Antiferromagnetism
known to ancient Greek tribe Ferromagnet near town Magneisa
Antiferromagnet proposed by
Louis Neel (Nobel prize’70) criticized by
Lev Landau (Nobel prize’62)
AFM order parameter was hidden 23 Ferro and Antiferromagnetism
known to ancient Greek tribe Ferromagnet near town Magneisa
Antiferromagnet proposed by
Louis Neel (Nobel prize’70) detected by neutron scattering criticized by
Lev Landau (Nobel prize’62)
AFM order parameter was hidden Bertram Brockhouse Clifford Shull (Nobel prize’94) (Nobel prize’94) 24 Orbital Degeneracy
eg Z2 X2-Y2
d-level t2g XY XZ Transition metal ion YZ
25 Three different couplings & regimes in spin-orbital systems
Jahn-Teller coupling-EJT
EJT
orbitals are rigidly ordered: spin-only Heisenberg model
26 Three different couplings & regimes in spin-orbital systems
Jahn-Teller coupling-EJT exchange interactions- J
EJT spin exchange depends on orbital occupancy: directional character of orbitals are rigidly ordered: orbitals induces frustration spin-only Heisenberg model
?
27 Coupled spin-orbital exchange (Kugel-Khomskii type models)
Cambridge University Press 2014 28 Coupled spin-orbital exchange (Kugel-Khomskii model)
weaker FM
29 Coupled spin-orbital exchange (Kugel-Khomskii model)
weaker FM
stronger AFM
30 Coupled spin-orbital exchange (Kugel-Khomskii model)
weaker FM
stronger AFM
Exchange couplings:
Orbital degrees are static Pott’s-like! 31 Directional orbitals on a family of lattices
XY XZ
YZ
32 Ground state manifold: Hard core dimers
GJ & Ivanov PRB (2007) GJ & Khomskii PRL (2008)
Ground state manifold is spanned by hard-core dimer coverings
Spins are bound into spin-singlet on dimer bonds: Spin gaped phase = Extensive orientational degeneracy infinitely many ways of covering Different coverings are orthogonal due to orbitals
33 Three different couplings & regimes in spin-orbital systems
Jahn-Teller coupling-EJT exchange interactions- J spin-orbit coupling-λ
EJT spin exchange depends on orbital occupancy: directional character of spins & orbitals locally orbitals induces frustration orbitals are rigidly ordered: entangled: spin-only Heisenberg model orbital frustration & directional character are converted to isospins
?
34 Orbital Degeneracy
eg Z2 X2-Y2 d-level
t2g L=2 XY YZ XZ l=1
35 Orbital Degeneracy
eg Z2 X2-Y2 d-level
t2g L=2 XY YZ XZ l=1
L=-l=1
36 spin-orbit 37 77 Ir= Ir= Ising Ising couplings - (lucky (lucky number) Heisenberg vs
bond correlations spin-orbit 38 77 Ir= Ir= Ising Ising couplings - (lucky (lucky number) Heisenberg vs bond V= 23 (V+Ir=100)
octupolar octupolar order correlations spin-orbit 39 77 Ir= Ir= Ising Ising couplings - (lucky (lucky number) Heisenberg vs bond Mo=42 valence valence glass bond (answer to ultimate question…) (answer ultimate to V= 23 (V+Ir=100)
octupolar octupolar order correlations spin-orbit 40 77 Ir= Ir= Ising Ising couplings - (lucky (lucky number) Heisenberg vs bond Mo=42 valence valence glass bond 76 (???) (answer to ultimate question…) (answer ultimate to Os= unusual ‘spin’ unusual textures‘spin’ V= 23 (V+Ir=100)
octupolar octupolar order correlations Spin-orbit coupling in t2g shell
Cambridge University Press 1970 41 Spin-orbit coupling in t2g shell A. Abragam and B. Bleaney (1970)
42 Spin-orbit coupling in t2g shell A. Abragam and B. Bleaney (1970)
3λ/2
43 j=1/2 systems: Iridates and Ruthenates
44 Spin-orbit coupling in t2g shell A. Abragam and B. Bleaney (1970)
d5 (Ir4+, Ru3+…) g=2
3λ/2∼600µες
45 ’Zoo’ of Iridate compounds
A2IrO4 A3Ir2O7 A2BIrO6
, , - A IrO α β γ 2 3 A4Ir3O8
46 ’Zoo’ of Iridate compounds
A2IrO4 A3Ir2O7 A2BIrO6
, , - A IrO α β γ 2 3 A4Ir3O8
47 ’Zoo’ of Iridate compounds
A2IrO4 A3Ir2O7 A2BIrO6
, , - A IrO α β γ 2 3 A4Ir3O8
48 ’Zoo’ of Iridate compounds
A2IrO4 A3Ir2O7 A2BIrO6
, , - A IrO α β γ 2 3 A4Ir3O8
All of them are Mott insulators! 49 Spin-Orbit assisted Mott transition 50 Kramers doublet of Ir4+
Isospin exchange inherit the bond symmetry
51 Kramers doublet of Ir4+ GJ & Khaliullin PRL (2009)
Isospin exchange inherit the bond symmetry
52 Kramers doublet of Ir4+ GJ & Khaliullin PRL (2009)
Isospin exchange inherit the bond symmetry
53 Kramers doublet of Ir4+ GJ & Khaliullin PRL (2009)
Isospin exchange inherit the bond symmetry
+
_
Destructive quantum interference Heisenberg term is suppressed 54 Kramers doublet of Ir4+ GJ & Khaliullin PRL (2009)
Isospin exchange inherit the bond symmetry
X
Destructive quantum interference Heisenberg term is suppressed 55 Kramers doublet of Ir4+ GJ & Khaliullin PRL (2009)
Isospin exchange inherit the bond symmetry
JH
X
56 Crystal structure of Sr2IrO4
Staggered rotation of octahedra around c-axis by α∼11ο
57 Magnetic structure of Sr2IrO4
Cao et al., PRB ‘98
FM moment MFM~0.1µB : too small for a saturated FM
too large for a weak FM -2 [e.g. in La2CuO4: 0.2x10 µB]
Canted AFM in xy-plane ? Spin canting angle φ∼α
58 Microscopic Hamiltonian of Sr2IrO4 GJ & Khaliullin PRL (2009) Dominant interactions (no Hund’s coupling)
φ
59 Microscopic Hamiltonian of Sr2IrO4 GJ & Khaliullin PRL (2009) Dominant interactions (no Hund’s coupling)
φ No true anisotropy (Heisenberg model in rotated basis):
-if spins are confined in xy-plane they are canted -if not they form collinear Neel order along z-axis
60 Microscopic Hamiltonian of Sr2IrO4 GJ & Khaliullin PRL (2009) Dominant interactions (no Hund’s coupling)
φ No true anisotropy (Heisenberg model in rotated basis):
-if spins are confined in xy-plane they are canted -if not they form collinear Neel order along z-axis Hund’s coupling selects in-plane canted structure 61 62 Heisenberg-like Magnetism of Sr2IrO4
Kim et al, PRL’12
63 Heisenberg-like Magnetism of Sr2IrO4
Kim et al, PRL’12 Fujiyama et al, PRL’12
64 Heisenberg-like Magnetism of Sr2IrO4
Kim et al, PRL’12 Fujiyama et al, PRL’12 Takayama, GJ, et al PRB’16
65 Hexagonal Iridates A2IrO3 (A=Na, Li)
66 Pseudospins on a Honeycomb layer of A2IrO3 GJ & Khaliullin PRL (2009)
O z
x y Ir
67 Pseudospins on a Honeycomb layer of A2IrO3 GJ & Khaliullin PRL (2009)
Kitaev Model O z Ann. Phys’06
x y Ir
Fundamental Physics Prize’14
68 Kitaev Honeycomb Model
A. Kitaev, Ann. Phys’06
exactly soluble 2D quantum model spin liquid ground-state zz Fractional excitations: xx yy Majorana Fermions, Dirac spectrum only NN two-spin correlations (Baskaran et al PRL’07) Exact dynamic spin structure factor (Knolle et al PRL’14) Raman spectra (Knolle et al PRL’15)
69 Kitaev Honeycomb Model
A. Kitaev, Ann. Phys’06
exactly soluble 2D quantum model spin liquid ground-state zz Fractional excitations: xx yy Majorana Fermions, Dirac spectrum only NN two-spin correlations zz (Baskaran et al PRL’07) Exact dynamic spin structure factor (Knolle et al PRL’14) Raman spectra (Knolle et al PRL’15)
70 Kitaev Honeycomb Model
A. Kitaev, Ann. Phys’06
exactly soluble 2D quantum model spin liquid ground-state zz Fractional excitations: yy xx yy Majorana Fermions, Dirac spectrum only NN two-spin correlations zz (Baskaran et al PRL’07) Exact dynamic spin structure factor xx yy (Knolle et al PRL’14) Raman spectra (Knolle et al PRL’15)
71 Experimental Findings (more in later slides)
Susceptibility & Thermodynamics: Singh and Gegenwart, PRB’10; Singh et al PRL’12
Ordering ~ 15K in both Na2IrO3 (θ=-125K) & Li2IrO3 (θ =-33K)
Zig-Zag order in Na2IrO3 XRS : Liu et al, PRB’11 INS: Choi, Coldea et al, PRL’12 N&XR Difraction: Ye et al PRB’12
72 Phase Diagram of Kitaev-Heisenberg Model
J.Chaloupka, GJ & G.Khaliullin PRL 2010 & 2013 73 Spin-wave dispersions of Zigzag phase
Finite intensity Bragg Peak
Zero intensity Soft mode
74 α, β, γ- A2IrO3 α-RuCl3 Sr2IrO4
Isotropic: Bond directional Ising: Heisenberg-like ‘Kitaev-like’ High-Tc SC? Quantum Spin Liquid? 75 ‘The great tragedy of Science is the slaying of a beautiful hypothesis by an ugly fact.’ theoretical
Tomas Henry Huxley 19th century British biologist
76 77 Sr2IrO4
Fermi arcs
78 Sr2IrO4
Fermi arcs
79 Sr2IrO4
Fermi arcs
80 81 A2IrO3 α-RuCl3
QSL?
82 A2IrO3 α-RuCl3
QSL?
83 A2IrO3 α-RuCl3
QSL?
84 H3LiIr2O6 (Takagi’s Group)
QSL
85 j=3/2 systems: Vanadates, Molybdates,and Osmates
86 Spin-orbit coupling in t2g shell A. Abragam and B. Bleaney, “EPR of Transition Ions”
d5 (Ir4+, Ru3+…) g=2
3λ/2
d1 (V4+,Mo5+, Os7+…) g=0 87 Spin-orbit coupling in t2g shell A. Abragam and B. Bleaney, “EPR of Transition Ions”
d5 (Ir4+, Ru3+…) g=2
Magnetic field mixes j=1/2 with 3/2 3λ/2 and induces van Vleck magnetism
1 4+ 5+ 7+ John Hasbrouck van Vleck d (V ,Mo , Os …) g=0 88 (Nobel prize’77) Octupolar order in Sr2VO4
GJ & G.Khaliullin PRL (2010)
y
x y
x
89 Elementary vs Magnetic Excitations
X Inter-doublet (MY) continuum M
Isospins
„pseudo-magnon“
„octupon“
octupolar Bragg peak
GJ & G.Khaliullin PRL (2010) 90 double perovskites A2LnMoO6
Ln3+ 5+ Mo each sublattice forms FCC
91 92 Rln4
M(H) at 5 (2.3)K: fit Brillouin func, accounting for 7 (2)% of the Mo s=1/2.
, 93 NMR 1/T1 - two component response: paramagnetic & gapped (~ 140 K) 94 broad gapped mode ~ 28 meV (bandwidth ~4 meV) in-gap continuum 9-17 mev 95 96 Overlap of Mo - Mo t2g orbitals
97 Superexchange Spin-Orbital Model
98 Exact ground state manifold: spanned by spin-singlet dimer coverings
Constraint: No neighbouring dimers can lie in the same plane 99 Extensive orientational degeneracy: infinitely many ways of covering
100 Extensive orientational degeneracy: infinitely many ways of covering
101 Low energy excitations & defects,
102 Low energy excitations & defects,
~J
103 Low energy excitations & defects,
104 Low energy excitations & defects,
~J
~J/2
105 ~J
~J/2
106 Low energy excitations & defects,
local singlet to triplet exc: J local SO exc: J/2 non-local SO exc: Pseudogap rather than a Hard Gap 107 Low energy excitations & defects, local singlet to triplet exc: J local SO exc: J/2 non-local SO exc: Pseudogap rather than a Hard Gap 108 J. Romhanyi, L. Balents & GJ arXiv (2016) Ba2NaOsO6 (FM? small net moment ~0.2MB, easy axis [110]) Ba2LiOsO6 Ba2LnMoO6 109 110 Summary Mott insulators with strong spin-orbit coupling: a new class of frustrated systems Orbital frustration directly manifested in magnetic interactions unusual interactions and exotic states 111 Optimistic Outlook Il est bon de savoir que l’utopie n’est jamais rien d’autre que la réalité de demain et que la réalité d’aujourd’hui était l’utopie d’hier. Le Corbusier Swiss-French architect 112 T H A N K Y O U 113