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Oct. 28, 2005 KIAS 1
One-dimensional approach tofrustrated magnets
Akira Furusaki (RIKEN)
Collaborators: Oleg Starykh (Utah)
Leon Balents (UCSB)PRB 72, 094416 (2005)
Oct. 28, 2005 KIAS 2
Phases of antiferromagnets Neel ordered phase
Valence Bond Solid
Spin liquid (RVB)
Tomonaga-Luttinger liquid (d=1)
2
1
algebraic correlations
No local order parameter
dimer LRObroken translational symmetry
staggered magnetization
gapless magnons, S=1
gapped magnons, S=1
deconfined massive spinons, S=1/2
deconfined massless spinons, S=1/2
Oct. 28, 2005 KIAS 3
Frustrated antiferromagnets: a route to exotic spin liquids ?
(classical) antiferromagnets on frustrated lattices haveextensive ground-state degeneracy
Lifting degeneracy by quantum fluctuations exotic phases ?
cf. fractional quantum Hall liquids Landau levels with extensive degeneracy + interactions Laughlin states, anyons,..
fractionalized spin liquids?
Oct. 28, 2005 KIAS 4
Geometric frustration
?
spins on a triangle interactingvia antiferromagnetic exchange
2-fold degeneracy
Lattices with triangles geometric frustration
2dTriangular lattice -(ET)2X, Cs2CuCl4….
Kagome lattice
3dPyrochlore lattice spinel oxides,….
Oct. 28, 2005 KIAS 5
Pyrochlore: most frustrated 3d latticeCorner-sharing 3d lattice oftetrahedra (4 spins in each)
On every tetrahedron:
Extensive degeneracy = # tetrahedra
Oct. 28, 2005 KIAS 6
2-dimensional pyrochlore = checkerboard lattice
1Jcorner-sharing geometry,but and are not equivalent2J
Y
X
J1
J2
.N.N.N
''2.N.N
'1 rrrr SSJSSJH
S=1/2 antiferromagnetic Heisenberg model
rS
on and
Oct. 28, 2005 KIAS 7
related 2d lattice models
J1-J2 model
Shastry-Sutherland model
J1: N.N. AF exchangeJ2: N.N.N. AF exchange
Oct. 28, 2005 KIAS 8
J1-J2 model
2d J1-J2 model
LRO:21 JJ ,
:21 JJ ,0&0, LRO
finite-temperature Ising transition
12 2JJ Maximal frustration
2J1J
J
Anisotropic model
Starykh & Balents, PRL (2004)
ColumnarDimer phase
Classical phase boundary J2=J1/2
J2/J
J1/J
Oct. 28, 2005 KIAS 9
Shastry-Sutherland model [Physica 108B, 1069 (1981)]
dimer plaquette singlet Neel
Koga and Kawakami, PRL 84, 4461 (2000)
21 / JJ
Oct. 28, 2005 KIAS 10
Ground-state phase diagram ofthe checkerboard model (2d pyrochlore)
J1/J2
?0
Neel (square lattice of J1 bonds)Plaquette phaseChainsJ2>>J1
1
(classicallydecoupled)
= 4-spin singlet
earlier results
?
Oct. 28, 2005 KIAS 11
J 1>>J2 B. Canals [PRB 65, 184408 (2002)]
spin wave analysis J(q)
12 5.0 JJ
)1,1( Eigenvector of lowest eigenstate
12 JJ 12 2JJ
Oct. 28, 2005 KIAS 12
J 1~ J2 Exact diagonalization [Fouet et al., PRB 67, 054411 (2003)]
J1=J2, up to N=36 (N=40)
ground state S=0: 2-fold degeneracy (N>>1)
many low-lying excited states in the S=0 sector
valence bond crystal(plaquette singlet)
Oct. 28, 2005 KIAS 13
Series expansion [Brenig and Grzeschik, PRB 69, 064420 (2004)]
triplet dispersion
Oct. 28, 2005 KIAS 14
Quadrumer boson approximation
2
42
2
31
2
43211
144332211
2SSSSSSSS
J
SSSSSSSSJ
10 2JE
11 JE
1,1,0 24131234 SSS
1,1,1 24131234 SSS
2232
10S
2
10
2
10
2
10
0
T
T
T
100
TTTTTTSS
hard-core bosons
Oct. 28, 2005 KIAS 15
rewrite the Hamiltonian in terms of the T bosons discard quartic and higher-order terms
quadratic approximation diagonalization by Bogoliubov transformation dispersion of triplet bosons
211211 coscos
3
4kkJJJJkE
(1) unstable when 21 5
8JJ bosons condense at 0k
Antiferromagnetic LRO: Neel state
(2) unstable when 21 11
8JJ
bosons condense at ,k
Magnetic LRO
Oct. 28, 2005 KIAS 16
Comparison with 1/S
Tchernyshyov, Starykh, Abanov & Moessner, PRB 68, 144422 (2003)
(a)
(b) (c) (d) (e)
Oct. 28, 2005 KIAS 17
J 1 <<J2
decoupled spin chains
AF coupling is frustrated.
sliding Luttinger liquids ?
Starykh, Singh & Levin, PRL 88, 167203 (2002)
Oct. 28, 2005 KIAS 18
A single Heisenberg chain (S=1/2)
jjj SSJH 1
Low-energy theory
SU(2)1 WZW theory
ggggggxdi
ggxdS
Tr12
Tr8
1 32
)2(SUg
xNxJaS jj
1/
xaSSSS jjjjj 211 1
jax
staggered dimerization
gTr~
giN Tr~
Oct. 28, 2005 KIAS 19
Quasi LRO (algebraic correlations)
2
1~)0(x
JxJ aa
x
x1
~0
x
NxN aa 1~0
dominant correlations
staggered magnetization
staggered dimerization
Neel order
Dimer order (VBS)
uniform magnetization is subdominant
Oct. 28, 2005 KIAS 20
Weak perturbation: J1
xNxJamnS mhn
mhh ,, 1/,2/1
yNayJxNaxJJV mvym
nvnm
mhxn
mh ,,,
,,1 1212
xNaxJa
mnSmnS
mhxn
mh
hh
,, 12
,2/1,2/1
Inter-chain interaction
vh JJ
If we keep only the term, V is not relevant sliding Luttinger liquid
However, the and terms are dangerously irrelevant!NJ
NN
Oct. 28, 2005 KIAS 21
2nd order perturbation in V
Operator Product Expansion
222
,0,,
v
xxNxJ
abb
xa
yNayJxNaxJJV mvym
nvnm
mhxn
mh ,,,
,,1 1212
uniform magnetization staggered magnetization = staggered dimerization
nm
nvmhnm manad
v
aJSV
,,,2
2212 ,,1
3
relevant operator allowed by symmetry
Mean-field analysis
mmh na 1,
nnv ma 1,
crossed-dimer state
Oct. 28, 2005 KIAS 22
comparison with exact diagonalization (36 spins)
lkjilkji SSSSSSSSklijD
;
Sindzingre, Fouet, Lhuillier, PRB 66, 174424 (2002)
dimer-dimer correlations
Oct. 28, 2005 KIAS 23
Global phase diagram scenario I: direct transition between crossed-dimer and p
laquette VBS
21 / JJNeelPlaquetteCrossed Dimer ? ?
? = 1st order transition or intermediate coexistence phase
Continuous transitions are forbidden by Landau-Ginzburg-Wilson theory.
Oct. 28, 2005 KIAS 24
scenario II: crossed-dimer and plaquette VBS via an intermediate ordered phase
? ?Crossed Dimer Neel* Plaquette NeelO(3)21 / JJ
? = 1st order transition or intermediate coexistence phase
Oct. 28, 2005 KIAS 26
Summary Global phase diagram of the AF Heisenberg model on
the 2d-pyrochlore (checkerboard lattice)
1d-approach can give some useful hints for understanding 2d and 3d frustrated magnets.
21 / JJNeelPlaquetteCrossed Dimer ? ?
PRB 72, 094416 (2005)