Quantum Spin Liquids

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Spin Liquid Phases ?
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Valence Bond Crystals
Valence Bond Crystals
How to overcome effect of frustration ?

• LRO in singlet-singlet correl. fonct.
• (crystal)
• Modes of gapped excitations
• integer DS1, 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
J.B.Fouet et al. 2002, W. Brenig 2002 P.
Sindzingre 2003, E. Berg et al 2003
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Quantum Spin Liquids
Valence Bond Crystals

• 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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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 .
5

Quantum Spin Liquids Resonating Valence Bond
Liquids

• No LRO in any local correlation fonctions
  (liquid) , spin gap

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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..

• No LRO in any local correlation fonctions
  (liquid), spin gap
• Continuum of gapped unconfined spin ½
  excitations (spinons)

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confined spinons in the V-B crystal
unconfined spinons in the R.V.B. Spin Liquids
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Quantum Spin Liquids Resonating Valence Bond
Liquids

• No LRO in any local correlation fonctions
  (liquid) , spin gap
• Continuum of gapped unconfined spin ½
  excitations (spinons)

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Quantum Spin Liquids Resonating Valence Bond
Liquids
An exactly solvable dimer liquid
model Ising gauge theory G. Misguich et al.
02

• 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 S0 visons excitations

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Misguich et al dimer model (02)only kinetic
energy sum from 3 to six dimer moves around each
hexagon of the kagome lattice
H ? dimer moves from black to red config.
h.c.
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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
H ? dimer moves from black to red config.
h.c.
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• Coherence Effects and S0 Visons excitations
• G. Misguich V. Pasquier D. Serban P.R.L.02
• The Resonating Valence Bond ground-state

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• Coherence Effects and S0 Visons excitations
• G. Misguich V. Pasquier D. Serban P.R.L.02
• The Resonating Valence Bond ground-state

• An S0 gapped excitation the two-visons
  wave-function

-
.

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Quantum Spin Liquids Resonating Valence Bond
Liquids

• Topological degeneracy

• No LRO in any local correlation fonctions
  (liquid)
• Continuum of gapped unconfined spin ½
  excitations (spinons)
• Subtle phase coherence properties (Quantum
  liquid)
• and S0 gapped visons excitations

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MSE Spin Liquid Spin gap Topological degeneracy
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MSE Spin Liquid Spin gap Topological degeneracy
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MSE Spin Liquid Spin gap Topological degeneracy
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MSE Spin Liquid Spin gap Topological degeneracy
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Topological degeneracy in SRRVB Spin Liquids

• a generic g.-s. configuration

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Topological degeneracy in SRRVB Spin Liquids

• a generic g.-s. configuration
• draw an arbitrary cut

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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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Topological degeneracy in SRRVB Spin Liquids
Pijkl
?x 3
?x 1
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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
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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
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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/9707021) ?
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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

• No LRO in any local correlation fonctions
  (liquid)
• Continuum of gapped unconfined spin ½
  excitations (spinons)
• Subtle phase coherence properties (Quantum
  liquid)
• and S0 gapped visons excitations

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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

• No LRO in any local correlation fonctions
  (liquid)
• Continuum of gapped unconfined spin ½
  excitations (spinons)
• Subtle phase coherence properties (Quantum
  liquid)
• and S0 gapped visons excitations

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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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H 2 J2 Slt i,jgt Si . Sj J4 S (Pijkl
P-1ijkl ) on the triangular lattice
Kagomé-like ?
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Quantum behavior of models with infinite local
degeneracy in the classical limit Heisenberg
model on the kagomé checkerboard and pyrochlore
latticesHalf integer odd spins versus integer
ones
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(No Transcript)
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Classical Heisenberg Hamiltonian on the kagomé
lattice
An infinite number of soft modes, an infinite T0
degeneracy Same property on the checkerboard
lattice, or the pyrochlore lattice
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Quantum ground-state and first excitations of
the Heisenberg model on the kagomé lattice

• have a total spin S0
• A gap lt 10 -3 in the singlet sector (if any)!
• Very large extensive entropy from singlets at
  ultra-low T
• A small spin gap (1/20)

At low temperature Cv is insensitive to large
magnetic fields ( Sindzingre et al.. PRL 00 ,
Ramirez et al.. PRL 00)
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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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Conclusion

• SU(2) magnets in 2d
• 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

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A toy model for a topological quantum-bitG.
Misguich, V. Pasquier, F. Mila, C.L.
cond-mat/0410693

• 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?

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A toy model for a topological quantum-bitG.
Misguich, V. Pasquier, F. Mila, C.L.
cond-mat/0410693

• 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?

38
A toy model for a topological quantum-bitG.
Misguich, V. Pasquier, F. Mila, C.L.
cond-mat/0410693

• 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
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A toy model for a topological quantum-bitG.
Misguich, V. Pasquier, F. Mila, C.L.
cond-mat/0410693

• 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