© Simon Trebst

Orbital ordering in eg orbital systemsGround states and thermodynamic of the 120º model

Simon TrebstMicrosoft Station Q

UC Santa Barbara

IFW Dresden June 2010

© Simon Trebst

Orbital ordering in eg orbital systemsGround states and thermodynamic of the 120º model

Andre van RynbachUCSB physics

Synge TodoUniversity of Tokyo

IFW Dresden June 2010

© Simon Trebst

Mott insulators with partially filled d-shells

Mott insulating transition metal oxides with partially filled 3d-shells– such as the manganites – exhibit rich phase diagrams.

Non-trivial interplay of spin, charge, and orbital degrees of freedom.

crystal field splitting (perovskites)

t2g

eg

3d

orbital degree of freedom

|x2 − y2� |3z2 − r2�

LaMnO3

Rb2CrCl4

KCrF3

(d4) KCuF3

LiNiO2(d9)

© Simon Trebst

Orbital degrees of freedomOrbitals are spatially anisotropic degrees of freedom.

Point group symmetry of lattice is picked up by orbitals.

|x2 − y2� |3z2 − r2�

� �

�

��

© Simon Trebst

Orbital degrees of freedomOrbitals are spatially anisotropic degrees of freedom.

Point group symmetry of lattice is picked up by orbitals.

spins orbitalsfeature

coupling to lattice

symmetry of Hamiltonian

excitations

frustration

weak strong

highcontinuous

lowdiscrete

often

gaplessalmost always

gapped

sometimes almostalways

orbital frustration

© Simon Trebst

Orbital exchange

Jahn-Teller interactions

Kugel-Khomskii type interactions

classical orbital model

quantum orbital model

Jahn-Teller lattice distortionsmediate orbital-orbital exchange

Q3 Q2

induced bymagnetic superexchange

© Simon Trebst

The 120 degree modelBoth types of orbital exchanges give rise to the same Hamiltonian

for the eg orbital degrees of freedom

H120 =−�

i,γ=x,y

14

�JzT

zi T

zi+γ + 3JxT

xi T

xi+γ

±√

3Jmix(T zi T

xi+γ + T

xi T

zi+γ)

�−

�

i

JzTzi T

zi+z

Jahn-Teller distortions

Kugel-Khomskii type superexchange

pseudospins are classical O(2) spins

pseudospins are quantum SU(2) spins

© Simon Trebst

Both types of orbital exchanges give rise to the same Hamiltonian for the eg orbital degrees of freedom

The 120 degree model

H120 =−�

i,γ=x,y

14

�JzT

zi T

zi+γ + 3JxT

xi T

xi+γ

±√

3Jmix(T zi T

xi+γ + T

xi T

zi+γ)

�−

�

i

JzTzi T

zi+z

Jahn-Teller distortions

Kugel-Khomskii type superexchange

pseudospins are classical O(2) spins

pseudospins are quantum SU(2) spins

pseudospin representation

|θ� = cos(θ/2)|3z2 − r2�+ sin(θ/2)|x2 − y2�= (T z, T x)

0◦

60◦120◦

180◦

240◦300◦

30◦

90◦

150◦

210◦

270◦

330◦

|3z2 − r2�|x2 − y2�

|y2 − z2�

|z2 − x2�|3x2 − r2�

|3y2 − r2�

© Simon Trebst

The 120 degree modelOur approach: explore 120º model in an extended parameter space.

H120 =−�

i,γ=x,y

14

�JzT

zi T

zi+γ + 3JxT

xi T

xi+γ

±√

3Jmix(T zi T

xi+γ + T

xi T

zi+γ)

�−

�

i

JzTzi T

zi+z

Jx = Jz = Jmix

Jx/Jz

Jmix/Jz

‘truncated’ model

symmetricmodel

Jx = Jz Jmix = 0

H120 = −J

�

i,γ=x,y,z

(τ i · eγ)(τ i+γ · eγ)

symmetric model for equal-coupling

τ i =�[T z

i +√

3T xi ]/2, [T z

i −√

3T xi ]/2, T z

i

�

Jx = Jz = Jmix

|3z2 − r2�

|3y2 − r2�

0◦

120◦

240◦|3x2 − r2�

© Simon Trebst

Orbital-only models

Planar compass model

Kitaev honeycomb model

T zi T z

i+z

T xi T x

i+x

T yi T y

i+y

T xi T x

i+x

T zi T z

i+z

topological ordergapless spin liquid

Ir oxides, polar molecules

topological orderdual to toric code in transverse mag. field

dual to Xu-Moore model (Josephson arrays)

© Simon Trebst

The classical 120º modelpseudospins are classical O(2) spins

© Simon Trebst

Zero temperature: degenerate ground states

Emergent symmetries: U(1) and Z2 symmetries

Ground-state manifold: infinite, but sub-extensive number of states

xz plane yz plane xy plane0◦120◦ 240◦

� �

�

��

D = 23L

M. Biskup et al., Comm. Math. Phys. 255, 253 (2005).Z. Nussinov et al., Europhys. Lett. 67, 990 (2004).

© Simon Trebst

0 60 120 180 240 300 360

angle θ∗-1.73

-1.72

free

ener

gy F(θ∗

) Jmix = 1

Low temperatures: Order by disorder

Spin-wave approximation: expansion in fluctuations around ordered state with at each site.

δθi = θi − θ∗

θi = θ∗

M. Biskup et al., Comm. Math. Phys. 255, 253 (2005).Z. Nussinov et al., Europhys. Lett. 67, 990 (2004).

0◦

60◦120◦

180◦

240◦300◦

symmetric 120º model

|3z2 − r2�|x2 − y2�

|y2 − z2�

|z2 − x2�|3x2 − r2�

|3y2 − r2�

© Simon Trebst

Moving away from symmetric point

Emergent symmetries: U(1) and Z2 symmetries

Ground-state manifold: infinite, but sub-extensive number of states

xz plane0◦120◦ 240◦

Jmix/Jz

Jx = Jz yz plane xy plane

✓✘ ✘

D = 23L → D = 2L

© Simon Trebst

Low temperatures: Order by disorderSpin-wave approximation: expansion in fluctuations around ordered state with at each site.

δθi = θi − θ∗

θi = θ∗

0 60 120 180 240 300 360

angle θ∗-1.73

-1.72

-1.71

-1.70

free

ener

gy F(θ∗

) Jmix = 0Jmix = 0.6Jmix = 0.8Jmix = 0.9Jmix = 0.95Jmix = 1

Jmix/Jz

Jx = Jz

© Simon Trebst

Low temperatures: Order by disorderSpin-wave approximation: expansion in fluctuations around ordered state with at each site.

δθi = θi − θ∗

θi = θ∗

0 60 120 180 240 300 360

angle θ∗-1.73

-1.72

-1.71

-1.70

free

ener

gy F(θ∗

) Jmix = 0Jmix = 0.6Jmix = 0.8Jmix = 0.9Jmix = 0.95Jmix = 1

Jmix/Jz

Jx = Jz

0◦180◦ |3z2 − r2�|x2 − y2�

truncated 120º model

90◦

270◦

�|3z2 − r2�+ |x2 − y2�

�/2

�|3z2 − r2� − |x2 − y2�

�/2

© Simon Trebst

Thermal phase transitions

00.5

11.5

22.5

3sp

ecifi

c he

at C v(T)

120o model truncated model (Jmix = 0)

L = 32L = 24L = 20L = 16L = 12L = 10L = 8L = 6L = 4

0 0.25 0.5 0.75 1 1.25 1.5temperature T / Jz

0

0.2

0.4

0.6

0.8

1

orde

r par

amet

er M(T)

120o model truncated model

M = max(Mxy, Mxz, Myz)

Mxy =�

z

������

x,y

Tx,y,z

�����

© Simon Trebst

Moving back to the symmetric pointFinite temperature: A line of continuous thermal phase transitions.

0.6

0.7

0.8

0.9

1

1.1te

mpe

ratu

re T

/ Jz

entropyselection

energyselection

0 0.2 0.4 0.6 0.8 1 1.2 1.4Jmix / Jz

-1.75

-1.5

-1.25

ener

gy E(T)

T / Jz = 0.05

symmetricmodel

Jmix/Jz

Jx = Jz

Zero temperature: First-order transition to ‘mixed state’.

0 0.2 0.4 0.6 0.8 1 1.2 1.4Jmix / Jz

0.6

0.7

0.8

0.9

1

1.1te

mpe

ratu

re T

/ Jz

entropyselection

energyselection

mixed statexz / yz plane ordering

θ1 + 180◦θ1

© Simon Trebst

Summary: classical 120º model

• T=0: first-order transition between family of models with subextensive ground-state manifold and a phase with unique ground state.

• low T: transition between entropically and energetically selected states.

• finite T: line of continuous thermal phase transitions with increasing Tc away from symmetric point.

Considering the 120º model in an expanded parameter space,the symmetry of the original model manifests itself in various ways:

© Simon Trebst

The quantum 120º modelpseudospins are quantum SU(2) spins

© Simon Trebst

Approaching the quantum 120º model

Jx/Jz

Jmix/Jz

symmetricmodel

sign problem

‘truncated’ model

nosign problem

Truncated quantum model reminiscent of planar compass model

zero-temperature first-order phase transition

continuous thermal ordering transition (but no directional ordering)

© Simon Trebst

Thermal ordering transition (Jx=Jz)

0

0.4

0.8

1.2C v(T)

L = 20L = 16L = 12L = 10L = 8L = 6L = 4

0

0.25

0.5

M(T)

0 0.25 0.5 0.75temperature T / Jz

0

1/3

2/3

B[ M(T) ]

0.39 0.41 0.43

0.2

0.3

0.4

a) specific heat

b) magnetization

c) Binder cumulant

� �

�

��

pseudospin ordering

� �

orbital ordering

� �

�

��

© Simon Trebst

Zero-temperature phase transition

0.5 0.75 1 1.25 1.5Jx / Jz

-0.6

-0.5

-0.4

-0.3

ener

gy E(T)

T / Jz = 0.15

QMC

perturbation theory(2nd order)

QMC

� �

�

��

pseudospin ordering pseudospin ordering

� �

�

��

� �

orbital ordering

� �

�

��

orbital ordering

� �

�

��

� �

© Simon Trebst

1/S expansion

Can we understand the quantum ground states from a 1/S expansion?

J. v.d. Brink et al., Phys. Rev. B 59, 6795 (1999).K. Kubo, J. Phys. Soc. Jpn. 71, 1308 (2002).

Zero-point energy to linear order in S

E0/N = −32S

2 + ∆E0(θ) + O(S3/2, S

1/2)

H120 =−�

i,γ=x,y

14

�JzT

zi T

zi+γ + 3JxT

xi T

xi+γ

±√

3Jmix(T zi T

xi+γ + T

xi T

zi+γ)

�−

�

i

JzTzi T

zi+z

mixing terms

does not explicitly include mixing terms

Holstein-Primakoff

T zi = S − a†iai

T xi =

�S/2

�ai + a†i

�

T zi → T z

i cos θ + T xi sin θ

T xi → T x

i cos θ − T zi sin θ

© Simon Trebst

0 60 120 180 240 300 360angle

-100

-90

-80

-70

-60

zero

-poi

nt e

nerg

y E

0(θ)

Jmix = 0Jmix = 0.2Jmix = 0.4Jmix = 0.6Jmix = 0.8Jmix = 0.9Jmix = 0.95Jmix = 0.99Jmix = 1

1/S expansionZero-point energy to linear order in S

E0/N = −32S

2 + ∆E0(θ) + O(S3/2, S

1/2)

© Simon Trebst

‘Orbiton’ excitations

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Jx / Jz

0

0.5

1

1.5

2

orbi

ton

gap

Δ

� �

orbital ordering

� �

�

��

orbital ordering

� �

�

��

� �

perturbation theory(3rd order)

QMC

QMC

|3z2 − r2� ↔ |x2 − y2� ↔

© Simon Trebst

Thermal phase transitionsA line of continuous thermal phase transitions.

0.5 0.75 1 1.25 1.5Jx / Jz

0

0.2

0.4

0.6

tem

pera

ture

T

/ Jz

� �

orbital ordering

� �

�

��

orbital ordering

� �

�

��

� �

truncatedmodel

© Simon Trebst

Approaching the quantum 120º model

Jx/Jz

Jmix/Jz

‘truncated’ model

symmetricmodel

gapped stategapped state

Going off into the Jmix > 0 plane

orbiton condensation ↓

phase boundaries of gapped, polarized states

© Simon Trebst

Phase diagram from orbiton condensation

0

0.5

1

1.5

2

orbi

ton

gap

!

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Jx / Jz

0

2

4

Jmix

/ Jz

! ! ! !

! !

!

"#

! !

!

"#

T z polarized T x polarized(xy planes)

© Simon Trebst

The symmetric 120º model

Jmix/Jz

Jx = Jz

90◦

270◦

0◦180◦

0◦180◦

60◦120◦

240◦ 300◦

90◦

270◦

60◦120◦

240◦ 300◦

30◦150◦

210◦ 330◦

sixground states

twelveground states

© Simon Trebst

Summary

• Mott insulators with partially filled 3d-shellscan give rise to orbital degrees of freedom.

• Orbital-only models can be interestingin their own right, because of their highlyanisotropic and frustrated interactions.

• The 120º model for eg orbitals

• exhibits non-trivial quantum ground states

• sits in the proximity of several T=0 phase transitions and various competing phases.