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© 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.
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