Order and disorder in SU(3) and SU(4) Heisenberg systems
Transcript of Order and disorder in SU(3) and SU(4) Heisenberg systems
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Order and disorder in
SU(3) and SU(4) Heisenberg systems
Correlations and coherence in quantum systemsÉvora, Portugal, 8-12 October 2012
K. PencInstitute for Solid State Physics
and Optics,Wigner Research Centre,
Budapest, Hungary
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Supported by: Hungarian OTKA and Swiss National Foundation
In collaboration with:
Miklós Lajkó ISSPO, Wigner RC, Budapest,
Tamás A. Tóth, Laura MessioFrédéric Mila
EPF Lausanne
Bela Bauer, Philippe Corboz,Matthias Troyer
ETH Zürich
Andreas M. Läuchli Uni. Innsbruck
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What are the SU(N) symmetric Heisenberg models that we are interested in?
H =�
i,j
Pi,j Pi,j is the transposition operator
i j i jPi,j | 〉→ | 〉 N species on each site
that are treated equally.
simplest example: SU(2) S=1/2 (fundamental representation) [but not the S=1 !]
Pij |βiαj� = |αiβj�
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Why do we care about SU(3) or SU(4) Heisenberg models?
(i)They are interesting and intelectually challenging ( funding agencies ?)
(ii) Nostalgia from our younghood, when we all wanted to become particle physicist (SU(3) and quarks)
(iii) Cold atomic gases
(iv) Spin models
(v) Spin-orbital models (SU(4))
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SU(2) quantum Heisenberg models
square and honeycomb (bipartite) lattices: happy (mean field) antiferromagnets
triangular lattice: frustrated antiferromagnet
?
happ
y
comp
romi
se
compromise
compromise
3-sublattice 120 degree state5Wednesday, October 10, 2012
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SU(2) quantum Heisenberg models
kagome lattice: highly frustrated antiferromagnet, mean field ground state macroscopically degenerate.
Ground state debated, likely a Z2 spin liquid
frustrated square lattice: dimerized state
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SU(2) vs. SU(3) - two sites
⊗ = ⊕22 1 3× = +
½ ⊗ ½ = 0 ⊕ 1
Addition of two S=½ SU(2) spins:
using Young diagrams:
↑ or ↓ spin
|↑↓〉−|↓↑〉 singlet, odd (anti-symmetrical)
|↑↑〉, |↑↓〉+|↓↑〉, |↓↓〉 triplet even (symmetrical)
⊗ = ⊕33 3 6× = +
Addition of two SU(3) spins:
|aa〉, |bb〉, |cc〉, |ab〉+|ba〉, |ac〉+|ca〉, and |bc〉+|cb〉.even (symmetrical)
|ab〉−|ba〉, |ac〉−|ca〉, |bc〉−|cb〉: odd (anti-symmetrical).
|a〉, |b〉, and |c〉.
H = P12
P12(|αβ〉 − |βα〉) = −(|αβ〉 − |βα〉)
P12(|αβ〉 + |βα〉) = +(|αβ〉 + |βα〉) E=+1, even wave function
E=−1 , odd wave function1 2
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SU(3) singlet
in the SU(3) singlet the spins are fully entangled: we cannot write it in a product form
= |ABC〉 + |CAB〉 + |BCA〉 − |BAC〉 − |ACB〉 − |BCA〉spins fully antisymmetrized
SU(3) irreps on 3 sites
1 2 × 83 = +
Addition of three SU(3) spins (27 states):
⊕⊕ 2×=⊗ ⊗
+ 10× ×3 3
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What methods do we use?
(i) Variational - site factorized wave function(ii) Flavor wave calculations (iii) Exact diagonalization of small clusters(iv) iPEPS: infinite project entangled pair states(variational
approach based on tensor ansatz)(v) Variational: Gutzwiller projected fermionic wave
functions
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Variational (classical) approacha site-product wave function for SU(3):
minimal, when the di and dj on the bond are orthogonal
two different colors on a bond
Evar =�Ψ|H|Ψ��Ψ|Ψ� = J
�
�i,j�
��di · dj
��2
|ψi� = dA,i|A�i + dB,i|B�i + dC,i|C�i
|Ψ� =�
i
|ψi�
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SU(3) flavour-wave theory
States in the fully symmetric multiplet can be represented by 3 Schwinger
bosons aA,aB and aC
1 2 ... M1
N. Papanicolaou, Nucl. Phys. B 305, 367 (1988) A. Joshi et al. PRB 60, 6584 (1999)
We enlarge the fundamental to the fully symmetric representation of M boxes.
The site product wave function is the “classical” solution (no quantum entanglement between sites)
|ψi� = dA,i|A�i + dB,i|B�i + dC,i|C�i
|Ψ� =�
i
|ψi�
|ϑ,ϕ� = cosϑ
2| ↑�+ sin
ϑ
2eiϕ| ↓�
cf. SU(2) spin coherent state for SU(2)
a†A,iaA,i + a†B,iaB,i + a†C,iaC,i = M
Pij =�
µ,ν∈{A,B,C}
a†µ,ia†ν,jaν,iaµ,j
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SU(3) flavour-wave theory
a†A,j =�
µ
dµ,ja†µ,j
a†B,j =�
µ
eµ,ja†µ,j
a†C,j =�
µ
fµ,ja†µ,j
Local coordinate system
1/M expansion:
a†A,i, aA,i →�M − a†B,iaB,i − a†C,iaC,i
→√M − 1
2√M
�a†B,iaB,i + a†C,iaC,i
�+ . . .
fd
e
Holstein-Primakoff bosons
|Ψ� =�
i
�a†i
�M|0�
Pij → M�(ei · dj)a
†B,i + (fi · dj)a
†C,i + (di · ej)aB,j + (di · fj)aC,j
�
×�(ei · dj)aB,i + (fi · dj)aC,i + (di · ej)a†B,j + (di · fj)a†C,j
�−M
quadratic in operators: we know how to diagonalize it (spin wave)
H = −MJL+M�
ν
�
k
ων(k)
�α†ν(k)αν(k) +
1
2
�
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The models
(i) SU(3) model on the triangular lattice
(ii) SU(3) model on the square lattice
(iii) SU(4) model on the square and cubic lattice
(iv) SU(4) model on the honeycomb lattice
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The models
(i) SU(3) model on the triangular lattice
(ii) SU(3) model on the square lattice
(iii) SU(4) model on the square and cubic lattice
(iv) SU(4) model on the honeycomb lattice
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The fate of SU(3) on triangular lattice
“classical” solution?crystal of singlets?
SU(3) classical state is perfectly happy on the triangular lattice - the 3 mutually perpendicular dʼs form a
3 sublattice structure.
SU(2) frustrated!
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SU(3) on triangular lattice - exact diagonalization
Signature of SU(3) breaking in the excitation spectrum:
Anderson towers compatible with 3 sublattice order
C2 - Casimir operator, analog of the total spin S^2
02 6 12 20 30 42S(S+1)
16
18
20
22
24
26
28
E
0 3 6 8 12 15 18C2
Kother
E
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0 0.1 0.2 0.3 0.4 0.5
0.3
0.4
0.5
0.6
0.7
0.8m
1/D
0 0.05 0.1 0.15 0.2 0.251/L
iPEPS 3x3 unit cellDMRGLFWT
0 0.1 0.2 0.3 0.4 0.5
−0.8
−0.6
−0.4
−0.2
0
0.2
E s [J]
1/D
0 0.05 0.1 0.15 0.2 0.251/L
iPEPS 2x2 unit celliPEPS 3x3 unit cellDMRGLFWTED extrap.
Unbiased calculation: iPEPS, DMRG
B. Bauer, P. Corboz, A. M. Läuchli, L. Messio, K. Penc, M. Troyer, F. Mila:Phys. Rev. B 85, 125116/1-11 (2012)
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The models
(i) SU(3) model on the triangular lattice
(ii) SU(3) model on the square lattice
(iii) SU(4) model on the square and cubic lattice
(iv) SU(4) model on the honeycomb lattice
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SU(3) classical solutions: macroscopically degenerate
Order by disorder: quantum fluctuations select the
ground state
EZP =M
2
�
ν
�
k
ων(k)
All bonds happy at the mean field level, frustration due to abundance of choices
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(b)(a) (b)(a)
SU(3) flavour-wave: dispersions, zero point energy
(a)
−π0
πkx −π
0
π
ky 0
4
ω/J
(b)
−π0
πkx −π
0
π
ky 0
4
ω/J
(c)
−π0
πkx −π
0
π
ky 0
4
ω/J
(a)
−π0
πkx −π
0
π
ky 0
4
ω/J
(b)
−π0
πkx −π
0
π
ky 0
4
ω/J
(c)
−π0
πkx −π
0
π
ky 0
4
ω/J
EZP = 1.27 EZP = 1.68The winner ! T. A. Tóth, A. M. Läuchli, F. Mila, and K. Penc:
PRL 105, 265301 (2010).20Wednesday, October 10, 2012
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02 6 12 20 30 42S(S+1)
16
18
20
22
24
26
28
E
0 3 6 8 12 15 18C2
Kother 7
8
9
10
11
12
0 3 6 8
E/J
C2
(a)N=9
16
16.5
17
17.5
18
0 3 6 8C2
(b)N=18
W even A1
M A1 B2 E1
Unbiased calculation: ED, Anderson towerstriangular lattice square lattice
(b)(a)
additional Z2symmetry
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Unbiased calculation: iPEPS, DMRG
0 0.1 0.2 0.3 0.4 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6m
1/D
0 0.05 0.1 0.15 0.2 0.251/L
iPEPS 3x3 unit cellDMRG
0 0.1 0.2 0.3 0.4 0.5
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
E s [J]
1/D
0 0.05 0.1 0.15 0.2 0.251/L
iPEPS 2x2 unit celliPEPS 3x3 unit cellDMRGLFWTED extrap.
B. Bauer, P. Corboz, A. M. Läuchli, L. Messio, K. Penc, M. Troyer, F. Mila:Phys. Rev. B 85, 125116/1-11 (2012)
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structure of the flavor wave Hamiltonian
H = (a† + b)(b† + a)
+ (b† + c)(c† + b)
+ (c† + d)(d† + c)
H = (a† + b)(b† + a)
+ (b† + c)(c† + b)
+ (c† + d)(d† + c)
each term separately EZP = 0 the 3-site term gives EZP > 0
a b c d a b c d
nearest neighbor also of different color
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The models
(i) SU(3) model on the triangular lattice
(ii) SU(3) model on the square lattice
(iii) SU(4) model on the square and cubic lattice
(iv) SU(4) model on the honeycomb lattice
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multipletsSU(4) singlet
SU(4) singlet plaquetteentangled spins and orbitals
= −spin singlet bond:
= |↑↓〉−|↓↑〉
orbital singlet bond:
= | 〉−| 〉
= |abcd〉−|bacd〉+|badc〉−|bdac〉−...spins fully antisymmetrized
SU(4) irreps on 4 sites
1 3× 15 35
⊗ ⊕ 2×
4 × = +
Addition of four SU(4) spins (256 states):
⊕⊕ 3× ⊕ 3×=⊗ ⊗
+ + +2× 20 3× 45× ×4 4 4
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SU(4) ladder
!"#!$%&#"'%(")*+*)$$#,
#-.%/)/%("'0
−0.25 0 0.25 0.5 0.75 1 1.25 k
−16.5
−16
−15.5
−15
−14.5
−14
−13.5
−13
−12.5
E(k
)/J
even state odd state
12 1-
3 3 4 45 67
!"#!$%&#"'%(")*+*)$$#,
#-.%/)/%("'0
−0.25 0 0.25 0.5 0.75 1 1.25 k
−16.5
−16
−15.5
−15
−14.5
−14
−13.5
−13
−12.5
E(k
)/J
even state odd state
12 1-
3 3 4 45 67
!"#!$%&#"'%(")*+*)$$#,
#-.%/)/%("'0
−0.25 0 0.25 0.5 0.75 1 1.25 k
−16.5
−16
−15.5
−15
−14.5
−14
−13.5
−13
−12.5
E(k
)/J
even state odd state
12 1-
3 3 4 45 67
M. van den Bossche et al., Phys. Rev. Lett. 86, 4124 (2001).
GS twofold degenerate (translational invariance broken):
gapped excitation spectrum, situation similar to
J1-J2 SU(2) Heisenberg chain:
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SU(4) on 2D-square lattice
!"#$%"&'%"()**+,'-"./"0,$%1"2"."!34$"567778
'9$+:"&3$1)%$43;$:3)%-"<="*3:'*
>!?)4&"&'1'%'@$:'"1@)A%&"*:$:'B
CD)!&3E'%*3)%$4"*FA$@'"4$::3+'
'9+3:$:3)%*"5B8G
!!"#
!!$#
!!%#
!!&#
!!'#
()*+,-.( /0,.)1,-.(
2!
2'
23
2$
2'
2&
24
567
89:
Ground state 4-fold degenerate?
Z2 liquid ? Wang & Viswanath (PRB 2009)
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SU(4) on 2D-square lattice: breaking SU(4)?
E/N = −1.293 J E/N = −0.727 J
E/N = −1.363 J E/N = −3/2 J
(b)(a)
(c) (d)
Order by disorder:Assuming on-site long
range order, which configuration has the
lowest zero-point energy?
The number of good mean field solution is
highly degenerate.
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SU(4) on 2D-square lattice: iPEPS
Tensor network method, with D-dimensional links between sites.
E/N = −1.293 J E/N = −0.727 J
E/N = −1.363 J E/N = −3/2 J
(b)(a)
(c) (d)
D = 2 and a unit cell 4 × 4. The bond- energy pattern of the
ground state is similar to that of the plaquette state selected by
LFWT.
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SU(4) on 2D-square lattice: iPEPS
V V V V
A
B
B
A
H
HD = 12 and a unit cell 4 × 2
dimerization and Neel-like state: both spatial and the SU(4)
symmetry is broken
⊗ = ⊕44 6 10× = +
the 6 dimensional irreducible representation is realized on the
dimers, can Neel order
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SU(4) on 2D-square lattice: iPEPS
0 0.1 0.2 0.3 0.4 0.5−1.1
−1
−0.9
−0.8
−0.7
−0.6
Es
1/D
2x2 unit cell4x2 unit cell4x4 unit cellVMCED N=16ED N=20
0 0.1 0.2 0.3 0.4 0.5
−0.8
−0.6
−0.4
−0.2
Eb
1/D
H bondsV bondsA/B bonds
0
0.2
0.4
0.6
0.8
CO
P
COP=∑
k|n k − 1/4 |
(c)
(a) (b)
V V V V
A
B
B
A
H
H
bond
ene
rgy
colo
r or
der
para
met
er
ener
gy
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0 4 6SU(4) Casimir
-17.5
-17
-16.5
-16
-15.5
-15
-14.5
-14
Tota
l Energ
y [J
]
(0,0) A1
(0,0) B1
(π,0) oe
(π,π/2) e
(π/2,π/2) e
(π,0) ee
(π,π) B2
(π,π/2) o
(π/2,0), e
-π-π/2 0 π/2 π
kx
-π
-π/2
0
π/2
π
ky
(a) (b) (c) (d)
SU(4) on 2D-square lattice: ED
Color structure factor ⟨P(i, j)(a,b)P(k,l)(a,b)⟩bond energy correlations ⟨PijPkl⟩ − ⟨Pij⟩2
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SU(4) on 2D-square lattice: our scenario
Dimerization: 6-dimensional irreps are formed,
they can Néel order
P. Corboz, A. M. Läuchli, K. Penc, M. Troyer, F. Mila, PRL 107, 215301 (2011).
SU(4) can be lowered to Sp(4) in cold atoms:H. Hung, Y. Wang, and C. Wu, Phys. Rev. B 84, 054406 (2011).E. Szirmai and M. Lewenstein, EPL 93, 66005 (2011).
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The models
(i) SU(3) model on the triangular lattice
(ii) SU(3) model on the square lattice
(iii) SU(4) model on the square and cubic lattice
(iv) SU(4) model on the honeycomb lattice
34Wednesday, October 10, 2012
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(b)(a)(a)
(c) (d)
(b)
Lifting of the degeneracy in flavor wave theory
Basic building blocks:nearest (mean field) and next nearest (fluctuations) neighbor colors different.
The allowed 9-spin configurations, not lifted in harmonic approximation:
iPEPS, D=6
A linear defect
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0 0.1 0.20
0.1
0.2
0.3
0.4
0.5
! E
b
1/D
iPEPS 2x2 unit cell
0 0.1 0.20
0.1
0.2
0.3
0.4
0.5
1/D
m
iPEPS 2x2 unit celliPEPS 4x4 unit cell
(c) (d)
(a) (b)
(b)(a)(a)
(c) (d)
(b)(b)(a)(a)
(c) (d)
(b)
iPEPS - local magnetization vanishes
4x4 unit cell, D=6
2x2 unit cell,D=6
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0 0.1 0.20
0.1
0.2
0.3
0.4
0.5
! E
b
1/D
iPEPS 2x2 unit cell
0 0.1 0.20
0.1
0.2
0.3
0.4
0.5
1/D
m
iPEPS 2x2 unit celliPEPS 4x4 unit cell
(c) (d)
(a) (b)
(b)(a)(a)
(c) (d)
(b)
iPEPS - dimerization vanishes
2x2 unit cell,D=6
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flavor liquid?local magnetization vanishes, no translational symmetry breaking
we use the fermionic representation:
Pij =�
µ,ν∈colors
f†α,ifβ,if
†β,jfα,j
PMFij =
�
α,β∈colors
�fβ,if†β,j�f
†α,ifα,j
= −�
α∈colors
tαijf†α,ifα,j
Using different Ansätze for the hoppings, we evaluate the expectation value of the Hamiltonian
Mean-field decoupling of the fermionic Hamiltonian gives a hopping Hamiltonian and a variational wave function|Ψvari� = PGutzwiller|ΨFS�
Evari =�Ψvari|H|Ψvari��Ψvari|Ψvari�
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The fermionic wave function of the pi-flux state
Dirac points
1/4
1/2
3/4
two-fold degenerate bands
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96-site cluster - real space correlations from Gutzwiller projected wavefunction
9
26
�13
2
�491
46
�15
9
26
�13
2
9
�11
232
�14
11
�1693
�45
�18
�5
18
�2
Pi-flux
E/N = −0.89466
�1
6
�5
0
�326
5
0
�2
6
�5
0
�1
�14
�188
�18
0
�1597
11
�2
�3
�19
�2
Majorana 0-flux
E/N = −0.822479
�24
�13
�1
1
�324
�73
�1
3
�13
�1
1
�24
�23
�36
�59
�11
�1518
52
�4
2
�6
�15
0-flux
E/N = −0.763304
�4
�12
�5
�1
�266
�10
�1
�2
�12
�5
�1
�4
�33
70
�45
4
�1506
�157
�7
�7
13
�7
Majorana Pi-flux
E/N = −0.754662
marked differences in 3rd neighbor correlations
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24-site cluster - real space correlations
Pi-flux
Dimension of the Hilbert space is 24!/(6!)4 = 2 308 743 493 056using symmetries makes it tractable
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10-6
10-5
10-4
10-3
10-2
10-1
1
2 4 6 8 10 12 14 16 1820
⟨ P
0δ -
1/4
⟩ (
-1)δ
/2
δ
∼δ−4
∼δ−3
64839220072
60038421696
-0.02
-0.01
0
0.01
0.02
0.03
0 4 8 12 16
×0.2
⟨ P0δ -1/4 ⟩
648chain
algebraic correlations and structure factor
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Ground state energy from different methods
0 0.05 0.1 0.15 0.2 0.25
−1.05
−1
−0.95
−0.9
−0.85
−0.8
−0.75
E s
1/D or 1/N
VMC Majorana π−fluxVMC 0−fluxVMC Majorana 0−fluxVMC π−fluxiPEPS 2x2 unit celliPEPS 4x4 unit cellED
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SU(N) on honeycomb
SU(4) is most probably an algebraic flavor liquid[P. Corboz, M. Lajkó, A. M. Läuchli, K. Penc, F. Mila, arXiv:1207.6029]
SU(6) is likely a chiral flavor liquid [G. Szirmai E. Szirmai, A. Zamora, and M. Lewenstein, Phys. Rev. A 84, 011611 (2011)],
similarly, SU(N) is also a chiral liquid [extending the results of M. Hermele, V. Gurarie, & A. M. Rey, Phys. Rev. Lett. 103, 135301 (2009)]
honeycomb optical lattices can be realized, see poster of Johannes Hecker-Denschlag
SU(2) is a Néel state
SU(3) is a plaquette state[Y.-W. Lee and M.-F. Yang, Phys. Rev. B 85, 100402 (2012). H.H.Zhao,C.Xu,Q.N.Chen,Z.C.Wei,M.P.Qin,G.M. Zhang, and T. Xiang, Phys. Rev. B 85, 134416 (2012).]
P. Corboz, unpublished
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Conclusions
(b)(a)
Phase Diagram
J’/J
1
0
3 sublattice LRO ?
spin-orbital liquid
4 sublattice LRO
QPT
QPT
QPT
spin-orbital liquid?
AB CB
CB
C
A
B C AA
A
A
A
BAB
C
B
D
A
C
B
D C
?resonating
liquid
(b)(a)
SU(2) SU(3) SU(4)
happy
happyfrustrated
fluctuation stabilized dimerization+Neel
squa
retri
angu
lar
45Wednesday, October 10, 2012
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Many happy returns of the day to you, Jose!
Many happy workshops in Evora!
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the end
47Wednesday, October 10, 2012