1. One-dimensional approach to frustrated magnets
Akira Furusaki (RIKEN)
Collaborators: Oleg Starykh (Utah)
Leon Balents (UCSB)
PRB 72, (2005)
Oct. 28, 2005
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2. Phases of antiferromagnets
Neel ordered phase
Valence Bond Solid
Spin liquid (RVB)
Tomonaga-Luttinger liquid (d=1)
staggered magnetization
gapless magnons, S=1
dimer LRO
broken translational symmetry
gapped magnons, S=1
No local order parameter
deconfined massive spinons, S=1/2
algebraic correlations
deconfined massless spinons, S=1/2
Oct. 28, 2005
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3. Frustrated antiferromagnets: a route to exotic spin liquids ?
(classical) antiferromagnets on frustrated lattices have
extensive ground-state degeneracy
Lifting degeneracy by quantum fluctuations exotic phases ?
fractionalized spin liquids?
cf. fractional quantum Hall liquids
Landau levels with extensive degeneracy
+ interactions
Laughlin states, anyons,..
Oct. 28, 2005
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4. Geometric frustration
Lattices with triangles geometric frustration
spins on a triangle interacting
via antiferromagnetic exchange 2d
Triangular lattice
-(ET)2X, Cs2CuCl4….
Kagome lattice 3d
Pyrochlore lattice
spinel oxides,…. ?
2-fold degeneracy
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5. Pyrochlore: most frustrated 3d lattice
Corner-sharing 3d lattice of
tetrahedra (4 spins in each)
On every tetrahedron:
Extensive degeneracy = # tetrahedra
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6. 2-dimensional pyrochlore = checkerboard latticeX J1 J2
corner-sharing geometry,
but and are not equivalent
on and
S=1/2 antiferromagnetic Heisenberg model
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7. related 2d lattice models
J1-J2 model
Shastry-Sutherland model
J1: N.N. AF exchange
J2: N.N.N. AF exchange
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8. J1-J2 model 2d J1-J2 model J2/J (p,0) Columnar Dimer phase (p,p) J1/J
LRO
LRO
finite-temperature Ising transition
Maximal frustration
2d J1-J2 model
Anisotropic model
Columnar
Dimer phase
Classical phase boundary
J2=J1/2
J2/J
J1/J
(p,0)
(p,p)
Starykh & Balents, PRL (2004)
Oct. 28, 2005
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9. Shastry-Sutherland model [Physica 108B, 1069 (1981)]
dimer
plaquette singlet
Neel
Koga and Kawakami, PRL 84, 4461 (2000)
Oct. 28, 2005
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10. Ground-state phase diagram of the checkerboard model (2d pyrochlore)
earlier results
J1/J2 ?
Neel (square lattice of J1 bonds)
Plaquette phase
Chains
J2>>J1 1
(classically
decoupled)
= 4-spin
singlet ?
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11. J１>>J2 B. Canals [PRB 65, 184408 (2002)] spin wave analysis J(q)
Eigenvector of lowest eigenstate
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12. J１～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)
valence bond crystal
(plaquette singlet)
many low-lying excited states in the S=0 sector
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13. Series expansion [Brenig and Grzeschik, PRB 69, 064420 (2004)]
triplet dispersion
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14. Quadrumer boson approximation
hard-core bosons
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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
(1) unstable when
bosons condense at
Antiferromagnetic LRO: Neel state
(2) unstable when
bosons condense at
Magnetic LRO
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16. Comparison with 1/S (a) (b) (c) (d) (e)
Tchernyshyov, Starykh, Abanov & Moessner, PRB 68, (2003)
Oct. 28, 2005
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17. J１<<J2 decoupled spin chains sliding Luttinger liquids ?
Starykh, Singh & Levin, PRL 88, (2002)
AF coupling is frustrated.
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18. A single Heisenberg chain (S=1/2)
staggered dimerization
Low-energy theory
SU(2)1 WZW theory
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19. Quasi LRO (algebraic correlations)
dominant correlations
staggered magnetization
Neel order
Dimer order (VBS)
staggered dimerization
uniform magnetization is subdominant
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20. Weak perturbation: J1 Inter-chain interaction
If we keep only the term, V is not relevant
sliding Luttinger liquid
However, the and terms are dangerously irrelevant!
Oct. 28, 2005
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21. 2nd order perturbation in V
Operator Product Expansion
uniform magnetization staggered magnetization = staggered dimerization
relevant operator allowed by symmetry
Mean-field analysis
crossed-dimer state
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22. comparison with exact diagonalization (36 spins)
Sindzingre, Fouet, Lhuillier, PRB 66, (2002)
dimer-dimer correlations
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23. Global phase diagram
scenario I: direct transition between crossed-dimer and plaquette VBS
Neel
Plaquette
Crossed Dimer ?
? = 1st order transition or intermediate coexistence phase
Continuous transitions are forbidden by Landau-Ginzburg-Wilson theory.
Oct. 28, 2005
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24. scenario II: crossed-dimer and plaquette VBS via an intermediate ordered phase?
Crossed Dimer
Neel*
Plaquette
Neel
O(3)
? = 1st order transition or intermediate coexistence phase
Oct. 28, 2005
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25. application to 3d pyrochlore
3d crossed-dimer phase for
Oct. 28, 2005
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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. ? ?
Crossed Dimer
Plaquette
Neel
PRB 72, (2005)
Oct. 28, 2005
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27. appendix
uniform part of magnetization
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