Spin Liquid Phases ? Houches/06//2006

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 Houches/06//2006

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 Houches/06//2006

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 Houches/06//2006

+ +… Quantum Spin Liquids Resonating Valence Bond Liquids terms No LRO in any local correlation fonctions (liquid) , spin gap Houches/06//2006

+ +… 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

confined spinons in the V-B crystal unconfined spinons in the R.V.B. Spin Liquids Houches/06//2006

+ +… 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) Houches/06//2006

+ +… 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

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. Houches/06//2006

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

Coherence Effects and S=0 Visons excitations G. Misguich V. Pasquier & D. Serban P.R.L.’02 The Resonating Valence Bond ground-state: + …. + Houches/06//2006

- 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 - + ± …. Houches/06//2006

+ +… 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

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

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

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

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

Topological degeneracy in SRRVB Spin Liquids a generic g.-s. configuration Houches/06//2006

Topological degeneracy in SRRVB Spin Liquids a generic g.-s. configuration draw an arbitrary cut Houches/06//2006

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 Houches/06//2006

Topological degeneracy in SRRVB Spin Liquids Pijkl x = 3 x = 1 Houches/06//2006

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

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

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) ? Houches/06//2006

+ +… 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

+ +… 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

On the square lattice the original model due to Rokhsar & Kivelson has no real spin liquid phase only a Q.C. point Houches/06//2006

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 ? Houches/06//2006

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 Houches/06//2006

Houches/06//2006

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 Houches/06//2006

Quantum ground-state and first excitations of the Heisenberg model on the kagomé lattice have a total spin S=0 A gap < 10 -3 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 - µ ) exp( c At low temperature Cv is insensitive to large magnetic fields ( Sindzingre et al.. PRL 00 , Ramirez et al.. PRL 00) Houches/06//2006

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) Houches/06//2006

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

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/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? Houches/06//2006

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/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? Houches/06//2006

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/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 Houches/06//2006

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/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 Houches/06//2006