One-dimensional approach to frustrated magnets

1
One-dimensional approach tofrustrated magnets

• Akira Furusaki (RIKEN)
• Collaborators Oleg Starykh (Utah)
• Leon Balents (UCSB)

PRB 72, 094416 (2005)
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Phases of antiferromagnets

• Neel ordered phase
• Valence Bond Solid
• Spin liquid (RVB)
• Tomonaga-Luttinger liquid (d1)

staggered magnetization
gapless magnons, S1
dimer LRO broken translational symmetry
gapped magnons, S1
No local order parameter
deconfined massive spinons, S1/2
algebraic correlations
deconfined massless spinons, S1/2
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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,..
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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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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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2-dimensional pyrochlore checkerboard lattice
corner-sharing geometry, but and are
not equivalent
S1/2 antiferromagnetic Heisenberg model
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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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• J1-J2 model

LRO
LRO
finite-temperature Ising transition
Maximal frustration
2d J1-J2 model
Anisotropic model
Starykh Balents, PRL (2004)
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• Shastry-Sutherland model Physica 108B, 1069
  (1981)

dimer
plaquette singlet
Neel
Koga and Kawakami, PRL 84, 4461 (2000)
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Ground-state phase diagram ofthe checkerboard
model (2d pyrochlore)
earlier results
?
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J1gtgtJ2

• B. Canals PRB 65, 184408 (2002)
• spin wave analysis J(q)

Eigenvector of lowest eigenstate
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J1J2

• Exact diagonalization Fouet et al., PRB 67,
  054411 (2003)
• J1J2, up to N36 (N40)
• ground state S0 2-fold degeneracy (Ngtgt1)

valence bond crystal (plaquette singlet)
many low-lying excited states in the S0 sector
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• Series expansion Brenig and Grzeschik, PRB 69,
  064420 (2004)

triplet dispersion
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Quadrumer boson approximation
hard-core bosons
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• 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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Comparison with 1/S
(a)
(b)
(c)
(d)
(e)
Tchernyshyov, Starykh, Abanov Moessner, PRB 68,
144422 (2003)
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J1ltltJ2
decoupled spin chains
sliding Luttinger liquids ?
Starykh, Singh Levin, PRL 88, 167203 (2002)
AF coupling is frustrated.
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A single Heisenberg chain (S1/2)
staggered dimerization
Low-energy theory
SU(2)1 WZW theory
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Quasi LRO (algebraic correlations)
dominant correlations
staggered magnetization
Neel order
Dimer order (VBS)
staggered dimerization
uniform magnetization is subdominant
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Weak perturbation J1
Inter-chain interaction
However, the and terms
are dangerously irrelevant!
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2nd order perturbation in V
Operator Product Expansion
relevant operator allowed by symmetry
Mean-field analysis
crossed-dimer state
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comparison with exact diagonalization (36 spins)
Sindzingre, Fouet, Lhuillier, PRB 66, 174424
(2002)
dimer-dimer correlations
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Global phase diagram

• scenario I direct transition between
  crossed-dimer and plaquette VBS

? 1st order transition or intermediate
coexistence phase
Continuous transitions are forbidden by
Landau-Ginzburg-Wilson theory.
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• scenario II crossed-dimer and plaquette VBS via
  an intermediate ordered phase

? 1st order transition or intermediate
coexistence phase
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application to 3d pyrochlore
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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.

?
?
Neel
Plaquette
Crossed Dimer
PRB 72, 094416 (2005)
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appendix