Orbital Physics

Andrzej M. Oleś

Marian Smoluchowski Institute of Physics, Jagiellonian University, Prof. S. Łojasiewicza 11, Kraków, Poland Jülich, 29 September 2017

Correl17 This research is supported by Narodowe Centrum Nauki (NCN) 1 (National Science Center) under Project MAESTRO 2012/A/ST3/00331 Outline

• Orbital models: intrinsic quantum frustration • -orbital superexchange: entanglement

• Construction of Kugel-Khomskii model: KCuF3

• Spin-orbital model for LaMnO3

• Spin-orbital models with t2g orbitals; orbital fluctuations • of ; double exchange; perspective Mosaic from 12th century, Veneto frustration Thanks to: Cattedrale di San Giusto, Trieste

• Wojciech Brzezicki Jagiellonian & Unita degli Studi di Salerno • Piotr Czarnik Jagiellonian University, Kraków, Poland • Jacek Dziarmaga Jagiellonian University, Kraków, Poland • Krzysztof Wohlfeld University of Warsaw, Warsaw, Poland • Lou-Fe’ Feiner Technical University, Eindhoven, Netherlands • Peter Horsch Max-Planck-Institut FKF, Stuttgart, Germany • Giniyat Khaliullin Max-Planck-Institut FKF, Stuttgart, Germany • Jiri Chaloupka Masaryk University, Brno, Czech Republic • Wen-Long You Soochow University, Suzhow, China Correl17 • Adolfo Avella Universita degli Studi di Salerno, Italy 2 • Mario Cuoco Universita degli Studi di Salerno, Italy 3d orbitals

3z2-r2 x2-y2

correlated insulators zx yz xy orbitals are quantum

Correl17 3 Intrinsic frustration of orbital interactions •= SSJH SU(2) symmetry: spin ∑ i j ij>< 7 9 eg systems (dd , ) z-like active 1 2 4 t2g systems ( d , d , d ) two active orbitals along each axis, eg orbitals t2g orbitals e.g. zx and xy along a axis Orbital interactions are directional !

classical quantum

frustrated frustrated order out of square lattice: kagome lattice triangular lattice disorder no frustration Cubic symmetry of directional orbital interactions: γγ)()( orb = ∑ i TTJH j ij>< Interaction depends on the bond direction => frustration on a square lattice Correl17 4 At large U: from Hubbard to the t-J model

Canonical transformation in the regime of t << U Perturbation: processes P1HP2 and P2HP1 charge excitations => superexchange

with

(Erik Koch) kinetic exchange

Correl17 5 [K.A. Chao, J. Spałek, A.M. Oleś, J. Phys. C 10, L271 (1977)] Toward superexchange model: hybridization and d-d hopping t

Effective hopping t via oxygen 3d-2p-3d => bandwidth W

strong weak e orbitals t orbitals g JT coupling 2g Correl17 6 [J. Zaanen and AMO, PRB 48, 7197 (1993)] Hopping and orbital superexchange for t2g

1 2 In t2g systems ( d ,d , …) two states no hopping ||c are active along each cubic axis, xy orbital called c e.g. yz & zx for the axis c –

[A.B. Harris et al., PRL 91, 087206 (03)] We introduce convenient notation Orbital interactions have cubic symmetry

they are described by quantum operators:

Scalar product but for JH>0 also other terms breaking the „SU(2)” symmetry

Correl17 7 Hopping for eg orbitals

Real basis:

( FM manganites )

0 has cubic symmetry

Correl17 with interaction => 8 Intraatomic Coulomb interactions for 3d orbitals

rotational invariance Intraorbital Coulomb

Hund’s exchange Correl17 9 Orbital operators: eg define:

Local projection operators:

c axis:

Bond projection operators:

Correl17 + + 10 Orbital superexchange for eg orbitals

Consider high spin ( S=1 ) excitations:

effective U

Correl17 11 Orbital waves and gap for eg orbitals

Two sublattices – orbital waves as for a quantum antiferromagnet 3D 2D

Quantum corrections are small Correl17The gap vanishes at Ez = zJ 12 and in 3D for Ez = 0 [J. van den Brink et al., PRB 59, 6795 (99)] 2D model with increasing frustration: from Ising to compass From Ising to compass:

Ising e g orbital compass A QPT from 2D Ising order to 1D compass nematic order at θ ~ 85 degrees 2D compass

Correl17 13 Frustration in the 2D quantum compass model Orbital (pseudospin) model with competing Ising-like interactions:

Jz=0 Jx=0

θ Artist’s view of the LxL cluster Low-energy states for the 4x4

for the 2D compass model (L=4) cluster for Jx= Jcosθ, Jz= Jsinθ

L High 2x2 degeneracy at the isotropic Jx=Jz point Model for quantum qubits => quantum computation Correl17 14 [J. Dornier, F. Becca, F. Mila, PRB 72, 024448 (05)] Order at finite T in 2D compass model

compass model

nematic order

Ordered nematic phase in 2D FM classical compass model Interpolation in the 2D compass model from QMC CMCCorrel17 for 12x12 lattice at T =0.1J 15 [S. Wenzel and W. Janke, PRB 78, 064402 (08)] A -J J Classical vs quantum for 2D frustrated models

B -J J => C -J J

interactions

Onsager C B A T>0 tensor Correl17 network 16 Spin-orbital physics

AF phases with some FM bonds Frustration can be removed C-AF A-AF Simplest case: Mott insulators with spin-orbital order manganites, nickelates, vanadates, titanates, … Goodenough-Kanamori rules: AO order supports FM spin order FO order supports AF spin order

Are these rules sufficient? LaVO LaMnO 3 3 Qualitative changes due to t2g orbitals e g orbitals spin-orbital entanglement Correl17 17 Inventors of spin-orbital physics at Blois (06)

Correl17 18 Degenerate Hubbard model & charge excitations ( t<

Two parameters: U – intraorbital Coulomb interaction, J – Hund’s exchange H r r = −+ 5 )( − 2 ⋅ SSJnnJUnnUH int ∑ ii αα↓↑ 2 ∑ iiH βα ∑ iiH βα iα i, <βα i, <βα + ( + + + + + ddddddddJ ) H ∑ iα↑ iα↓ ii ββ↑↓ ↑ ii ββ↓ ii αα↑↓ i, <βα

[A.M. Oleś, PRB 28, 327 (1983)] eg systems t2g systems

In a Mott insulator (t<

single parameter: HS η=JH /U

Correl17 19 Low energy Hamiltonian: Spin-orbital superexchange (t<

Two parameters: U – intraorbital Coulomb interaction, JH – Hund’s exchange A t large U >> t ( J=4t2/U ): η=JH /U charge excitation spin interactions: SU(2) symmetry

=> superexchange

spin-orbital model contains orbital operators of cubic symmetry ( γ =a,b,c ) Averaging over orbital (dis)ordered state => anisotropic spin model:

+⋅= ⋅ SSJSSJH jiabjics γ )( ∑ ∑ γ ≡ JJ ij ij ij c ab Correl17 20 Here spin and orbital operators are disentangled Kugel-Khomskii model We follow the general scheme:

KK model with

+ +

Correl17 21 Kugel-Khomskii model

8 Equidistant multiplet structure for d ions η=JH /U LS Charge excitations fully characterized by:

HS

KK model Experimental observations:

Correl17 22 In this regime behaves as the 1D quantum antiferromagnet Which kind of spin-orbital order ? spin exchange Strong anisotropy for G -AF

FM AF

A-AF C-AF

Correl17 FM G-23AF Mean field analysis of spin-orbital order

Two-sublattice ground state:

anisotropic exchange constants

Correl17 24 Phase diagram: 2D Kugel-Khomskii model

η=JH /U

a

a

Much weaker AF interactions between holes in orbitals than in Correl17 25 Spin-orbital entanglement near the QCP Example: Kugel-Khomskii (KK) model (d9)

A-AF

Spin disordered states

A-AF phase in KCuF3 A-AF eg orbitals T=1/2 spins S=1/2

Parameters: (1) EJ z/ – eg orbital splitting (2) JUH/ – Hund´s exchange

Quantum critical point: (Ez,JH) = (0,0)

Entanglement near the QCP ?

Phase diagram of the d9 model Correl17 26 [L.F. Feiner, AMO, J. Zaanen, PRL 78, 2799 (97)] Phase diagram: 3D Kugel-Khomskii model

Exotic magnetic order arises at boarder lines Correl17 between phases AF due to spin-orbital entanglement 27 Spin-orbital model for LaMnO3

Weak AF superexchange due to t2g

Bond projection operators: Mn3+

+ +

Correl17 The spin-orbital part is similar to KCuF3 ( S28=2 ) [L.F. Feiner and AMO, PRB 59, 3295 (99)] Spin-orbital physics: optical spectral weights

Structure of the spin-orbital model: Terms originate from charge excitations to multiplet states n

Spectral weight for an excitation at energy ωn

These weights are found from the superexchange terms:

for excitation at ωn

total weight

Correl17 29 Optical spectral weights for LaMnO3 spin-orbital model tot

spectral weight for excitation at energy ωn

FM

Theory reproduces the spectral weights tot for low energy ω1 high-spin excitations Spin and orbital correlations AF are here disentangled

TN Note: S = 2

Correl17 30 Phase diagrams La A MnO 1-x x 3 Colossal magnetoresistance in manganites

Possible magnetic phases in doped manganites [M. Stier and W. Nolting, PRB 78, 144425 (08)]

3+ [Sang-Wook Cheong (98)] Mn

Orbital and charge degree of freedom

Theory has to include strong

Correl17 correlations of eg electrons 31 [Y. Tokura, RPP 67, 797 (06)] Avoiding correlations: electron doped Sr1-xLaxMnO3

Kondo-like model for (few) noninteracting eg electrons :

G-AF C-AF

Correl17 C-AF phase stabilized by double exchange for x >0.02 32 Degenerate Hubbard model & charge excitations ( t<

Two parameters: U – intraorbital Coulomb interaction, J – Hund’s exchange H r r = −+ 5 )( − 2 ⋅ SSJnnJUnnUH int ∑ ii αα↓↑ 2 ∑ iiH βα ∑ iiH βα iα i, <βα i, <βα + ( + + + + + ddddddddJ ) H ∑ iα↑ iα↓ ii ββ↑↓ ↑ ii ββ↓ ii αα↑↓ i, <βα

eg systems t2g systems

In a Mott insulator (t<

single parameter: HS η=JH /U

Correl17 33 Spin-orbital superexchange in RTiO3

Ti constraint: Multiplet structure =>

superexchange

orbital fluctuations

Correl17 Nonconservation of orbital flavor 34 => spin-orbital liquid ? Spin-Orbital Model for RVO3 (R=La,Y, …)

t 2 configurations of V3+ ions with S =1 spins t2g hopping 2g each orbital is inactive along one axis

A.B. Harris et al., For T

orbital fluctuations

spin fluctuations

spin fluct. Energies of t2g orbitals in YVO3

orbital fluct. Superexchange for t<

Correl17 35 [G. Khaliullin, P. Horsch, AMO, PRL 86, 3879 (01)] Spin-orbital superexchange in RVO3

1 2 In t2g systems ( d ,d ) two states are Local constraints: active, e.g. yz i zx for the axis c – they are described by quantum operators: c orbitals occupied {a,b} – orbital degree of freedom

Orbital interactions Scalar product but for η>0 also: have cubic symmetry Orbital SU(2) symmetry is broken !

2 Spin-orbit superexchange in RVO3 d ( S = 1 ):

Ti V

Multiplet structure => Correl17 Entanglement expected ! 36 Strong orbital fluctuations in C-AF phase ( LaVO3 ) {a,b}={yz,zx} fluctuations on bonds || c axis

⇒ finite value of FM -Jc>0 for η =0 ( no Hund’s mechanism! ) ⇒ similar values of exchange AF Jab and FM - Jc for η =0.13

shadow Exchange constants in C-AF phase C-AF phase for increasing Hund’s exchange η

Correl17 3737 [G. Khaliullin P. Horsch, AMO, PRL 86, 3879 (01)] η=JH /U=0.13 Spin-orbital superexchange in RVO3: spectral weights

cubic symmetry broken by C-AF orbital fluctuations:

more classical:

Correl17 Spin-orbital entanglement along the c axis 38 Spin and orbital phase transitions in RVO3

C-AF phase G-AF phase

TN1 modified due to entanglement !

Orbital-lattice coupling is crucial

Correl17 39 Goodenough-Kanamori rules

eg orbitals why ? AO order supports FM spin order -- FM order wins FO order supports AF spin order -- only AF terms intraorbital hopping t interorbital hopping t’

complementarity for orbital conserving hopping t

frustration by interorbital hopping t’ ⇒ FM/FO or AF/AO are possible

Exotic spin orders possible in certain situations

Correl17 40 Here A and B are spin and orbital degrees of freedom of the system i.e., the boundary involves the entire system Correl17 41 Correl17 42 Example: triangular lattice

charge excitations in NaTiO2 are possible due to:

⇒ spin-orbital model

interpolates between t orbitals t’ orbital interchanged conserving

Correl17 43 [B. Normand and AMO, PRB 78, 094427 (08)] Example: entangled states in a free hexagon

entangled

Cij is a local measure of entanglement

RVB hexagon

Here spins and orbitals cannot be decoupled

Correl17 44 [J. Chaloupka and AMO, PRB 83, 094406 (11)] Evolution of orbital densities in a free hexagon

entangled

RVB hexagon

Correl17 45 [J. Chaloupka and AMO, PRB 83, 094406 (11)] Fractionalization of orbital excitations: AF/FO

is excited t-J model hole orbiton moves and creates a spinon orbiton propagating with hopping t =J/2 spinon holon

In a 1D spin-orbital model the orbiton fractionalizes into a freely propagating spinon and orbiton, in analogy to spinon and holon in spin t-J model

Correl17 46 [K. Wohlfeld et al., PRL 107, 127201 (11)] t-J-like model for ferromagnetic manganites Double exchange gives FM coupling

between average spins reduces the FM coupling Derivation with Schwinger bosons at doping x magnon dispersion [AMO and L.F. Feiner, PRB 78, 052414 (02)]

[Y. Tokura,Correl17 RPP 67, 797 (06)] 47 Orbital dilution in d4 ( S = 1 ) by d3 impurities ( S = 3/2 )

c = xy doublon a = yz C-AF phase Orbital degree of freedom

Futuristic electronic devices may rely on properties that are highly sensitive to magnetic and orbital order spintronics - orbitronics [W. Brzezicki, AMO, M. Cuoco, PRX 5, 011037 (15)] Correl17 48 Parameters for a hybrid 3d-4d bond

Mismatch potential renormalized by Coulomb U’s and Hund’s JH’s

S=3/2 S1= Parameters to characterize the impurity: Schematic view of a 3d-4d bond with a doublon in the host at orbital c

superexchange Hund’s exchange at 3d

Correl17 49 [W. Brzezicki, AMO, M. Cuoco, PRX 5, 011037 (15)] Enhanced fluctuations for d4-d2 hybrids (charge dilution)

We solve a problem of two sites in the second order perturbation expansion in hopping to get a spin-orbital bond Hamiltonian.

hole doublon

Ground state Fluctuations Here we see the Correl17 action of T+5050T+ ! Enhanced fluctuations for d4-d2 hybrids (charge dilution)

Correl17 51 Frustration & entanglement: orbital and spin-orbital

1. Orbital models: frustration => orbital order at T < Tc 2. Features of spin-orbital superexchange models

3. Different properties of systems with eg and t2g orbitals 4. Spin-orbital entanglement 5. Doped systems: double exchange & novel phases

Quantum Goodenough-Kanamori rules for exchange bonds:

spin triplet spin singlet orbital singlet orbital triplet

Complementary correlations are dynamical !

Challenge: excitations in spin-orbital systems ?

Correl17 52 Thank you for your kind attention !

Correl17 53 http://sces.if.uj.edu.pl/