1. Spin Liquid Phases ?
Houches/06//2006

2. Valence Bond Crystals Valence Bond Crystals
How to overcome effect of frustration ?
4-spin S=0 state
LRO in singlet-singlet correl. fonct.
(crystal)
Modes of gapped excitations:
integer DS=1, 0 excitations
A product of singlet wave functions
is a good app. of the N. body g.-s.
A simple way to overcome frustration
Crystal of singlets
Fully optimized bonds and absence of m.-f. interactions 03
J.B.Fouet et al. 2002, W. Brenig 2002
P. Sindzingre 2003, E. Berg et al 2003
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3. Valence Bond Crystals Quantum Spin Liquids
No spin-spin long range order
Singlet-singlet long range order
Gapful spin integer excitations
No LRO in spins
No LRO in dimers
and in any local correlation fctn
Specificity of the g.-s. w.-f.
Fractionalized spin excitations
may be gapful or gapless
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4. Resonating Valence Bond Liquids
Quantum Spin Liquids
Resonating Valence Bond Liquids
P.W. Anderson, L. Balents,
V. Elser, M.P.A. Fisher,
E. Fradkin, S. Kivelson,
C.L., G. Misguich,
R. Moessner, S. L. Sondhi
V. Pasquier, N. Read,
D. Rokhsar, D. Sutherland, S. Sachdev, S. Senthil,
D. Serban, P. Sindzingre ……. + +…
terms
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5. + +… Quantum Spin Liquids Resonating Valence Bond Liquids
terms
No LRO in any local correlation fonctions (liquid) , spin gap
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6. + +… Quantum Spin Liquids Resonating Valence Bond Liquids
By alphabetic order:
P.W. Anderson,
L. Balents,
M.P.A. Fisher,
E. Fradkin,
S. Kivelson,
N. Read,
S. Sachdev,
S. Senthil.. + +…
terms
No LRO in any local correlation fonctions (liquid), spin gap
Continuum of gapped unconfined spin ½ excitations (spinons)
Houches/06//2006

7. confined spinons in the V-B crystal
unconfined spinons in the R.V.B. Spin Liquids
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8. + +… Quantum Spin Liquids Resonating Valence Bond Liquids
terms
No LRO in any local correlation fonctions (liquid) , spin gap
Continuum of gapped unconfined spin ½ excitations (spinons)
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9. + +… Quantum Spin Liquids Resonating Valence Bond Liquids
An exactly solvable
dimer liquid model
Ising gauge theory
G. Misguich et al. ‘02 + +…
terms
No LRO in any local correlation fonctions (liquid) , spin gap
Continuum of gapped unconfined spin ½ excitations (spinons)
Subtle phase coherence properties (Quantum liquid)
and S=0 visons excitations
Houches/06//2006

10. Misguich et al dimer model (’02) only kinetic energy: sum from 3 to six dimer moves around each hexagon of the kagome lattice 3 6 4
H =  dimer moves from black to red config. + h.c.
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11. Misguich et al dimer model (’02) only kinetic energy: sum from 3 to six dimer moves around each hexagon of the kagome lattice H 3 6 4 3 6 4
H =  dimer moves from black to red config. + h.c.
Houches/06//2006

12. Coherence Effects and S=0 Visons excitations
G. Misguich V. Pasquier & D. Serban P.R.L.’02
The Resonating Valence Bond ground-state:
+ …. +
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13. - Coherence Effects and S=0 Visons excitations + + …. + ± ….
G. Misguich V. Pasquier & D. Serban P.R.L.’02
The Resonating Valence Bond ground-state:
+ …. +
An S=0 gapped excitation: the two-visons wave-function - +
± ….
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14. + +… Quantum Spin Liquids Resonating Valence Bond Liquids
Topological degeneracy + +…
terms
No LRO in any local correlation fonctions (liquid)
Continuum of gapped unconfined spin ½ excitations (spinons)
Subtle phase coherence properties (Quantum liquid)
and S=0 gapped visons excitations
Houches/06//2006

15. MSE Spin Liquid Spin gap & Topological degeneracy
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16. MSE Spin Liquid Spin gap & Topological degeneracy
Houches/06//2006

17. MSE Spin Liquid Spin gap & Topological degeneracy
Houches/06//2006

18. MSE Spin Liquid Spin gap & Topological degeneracy
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19. Topological degeneracy in SRRVB Spin Liquids
a generic g.-s. configuration
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20. Topological degeneracy in SRRVB Spin Liquids
a generic g.-s. configuration
draw an arbitrary cut
Houches/06//2006

21. Topological degeneracy in SRRVB Spin Liquids
a generic g.-s. configuration
draw an arbitrary cut
count the number of dimers
across the cut
x = 3
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22. Topological degeneracy in SRRVB Spin Liquids
Pijkl
x = 3
x = 1
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23. Topological degeneracy in SRRVB Spin Liquids
Pijkl
x = 3
x = 1
Parities of winding numbers (x, y) are good quantum numbers:
4 unconnected topological subspaces on a 2-torus
degenerate in the thermodynamic limit
Houches/06//2006

24. Topological degeneracy in SRRVB Spin Liquids
Pijkl
x = 3
x = 1
Parities of winding numbers (x, y) are good quantum numbers:
4 unconnected topological subspaces on a 2-torus
degenerate in the thermodynamic limit
4-fold degeneracy of low lying singlets
Houches/06//2006

25. Topological degeneracy in SRRVB Spin Liquids
Pijkl
x = 3
x = 1
Parities of winding numbers (x, y) are good quantum numbers:
4 unconnected topological subspaces on a 2-torus
degenerate in the thermodynamic limit
4-fold degeneracy of low lying singlets
A topological quantum bit (Kitaev quant-phys/ ) ?
Houches/06//2006

26. + +… Quantum Spin Liquids Resonating Valence Bond Liquids
Topological degeneracy
A topological quantum-bit
A. Y. Kitaev 97, 03
L. Ioffe and coll. 02
G. Misguich et al ‘04 + +…
terms
No LRO in any local correlation fonctions (liquid)
Continuum of gapped unconfined spin ½ excitations (spinons)
Subtle phase coherence properties (Quantum liquid)
and S=0 gapped visons excitations
Houches/06//2006

27. + +… Quantum Spin Liquids Resonating Valence Bond Liquids
Topological degeneracy
A topological quantum-bit
What seems the most favorable
conditions to observe
Quantum Spin Liquids?
triangular geometry
importance of “effective kinetic
terms” acting coherently on
more than two spins + +…
terms
No LRO in any local correlation fonctions (liquid)
Continuum of gapped unconfined spin ½ excitations (spinons)
Subtle phase coherence properties (Quantum liquid)
and S=0 gapped visons excitations
Houches/06//2006

28. On the square lattice the original model due to Rokhsar & Kivelson
has no real spin liquid phase only a Q.C. point
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29. SR-RVB H = 2 J2 S< i,j> Si . Sj + J4 S (Pijkl + P-1ijkl )
on the triangular lattice
SR-RVB
Gapless LR-RVB ?
Kagomé-like ?
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30. Quantum behavior of models with infinite local degeneracy in the classical limit Heisenberg model on the kagomé checkerboard and pyrochlore lattices Half integer odd spins versus integer ones
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31. Houches/06//2006

32. Classical Heisenberg Hamiltonian on the kagomé lattice
An infinite number of soft modes, an infinite T=0 degeneracy
Same property on the checkerboard lattice, or the pyrochlore lattice
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33. Quantum ground-state and first excitations
of the Heisenberg model on the kagomé lattice
have a total spin S=0
A gap < in the singlet sector (if any)!
Very large extensive entropy from singlets at ultra-low T
A small spin gap (~1/20) p T C v =
but D - )
exp( c
At low temperature Cv is insensitive to large magnetic fields
( Sindzingre et al.. PRL 00 , Ramirez et al.. PRL 00)
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34. Spin-3/2 kagomé Antiferromagnet
No theoretical results
Experiments: the spin liquid picture is plausible.
Some features are very much alike the spin-1/2 system
Dynamics of spins (Uemura et al 1994)
Low lying local singlet excitations (Ramirez et al. 2000)
A very tiny spin gap if any (Ramirez et al, Bono, Mendels et al.)
Quasi-critical behavior of the spin susceptibility at intermediate temperatures (C. Mondelli, H. Mutka and coll. 2002, A. Georges and coll. 2001)
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35. Conclusion SU(2) magnets in 2d Quantum Spin Liquids
Semi-classical Néel phases
Quantum phases: Valence Bond Crystals and Spin Liquids
Quantum Spin Liquids
A Realistic Spin Liquid : MSE on triangular latt.
Open question: spin-1/2-Heisenberg model on the kagomé lattice. A true new phase or a system near a Q.C. point?
Half-odd integer spins versus integer ones
Houches/06//2006

36. A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F
A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F. Mila, C.L. cond-mat/
Z2 spin liquid on a cylinder: 2-fold degenerate g.s. a topological q-bit
protected from any local perturbations
How write and read it?
Houches/06//2006

37. A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F
A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F. Mila, C.L. cond-mat/
Z2 spin liquid on a cylinder: 2-fold degenerate g.s. a topological q-bit
protected from any local perturbations
How write and read it?
Houches/06//2006

38. A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F
A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F. Mila, C.L. cond-mat/
Z2 spin liquid on a cylinder: 2-fold degenerate g.s. a topological q-bit
protected from any local perturbations
How write and read it?
Introduce a local perturbation
which change the geometry
from a cylinder to a plane
Houches/06//2006

39. A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F
A toy model for a topological quantum-bit G. Misguich, V. Pasquier, F. Mila, C.L. cond-mat/
Z2 spin liquid on a cylinder: 2-fold degenerate g.s. a topological q-bit
protected from any local perturbations
How write and read it?
gap
1/L
exp(-L)
perturbation Vc
Houches/06//2006