houches/06//2006 from néel ordered afms to quantum spin liquids c. lhuillier université paris vi...
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Houches/06//2006
From Néel ordered AFMs to
Quantum Spin Liquids
C. Lhuillier
Université Paris VI & IUF
Houches/06//2006
• B. Bernu (Paris VI)• J.C. Domenge (Rutgers)• H.U. Everts (Hannover)• J.B. Fouet (Lausanne)• A. Laeuchli (Lausanne)• P. Lecheminant
(Cergy-Pontoise)
• W. Liming (Canton)• F. Mila (Lausanne)• G. Misguich (Saclay)• L. Pierre (Paris X)• P. Sindzingre (Paris VI)• C. Waldtmann
(Hannover)
Houches/06//2006
• Introduction: – Miscellaneous questions around R.V.B. w.-f.– From Néel states to purely Quantum phases
• Valence Bond Crystals and Valence Bond Solids
• True Spin Liquid phases
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g.s. wave-functions and spectral properties of quantum magnets
• RVB w.f. (Anderson 72, Liang, Douçot, Anderson 88)
• huge dimension of the Hilbert space of V.B. (singlets).. • BUT what about spectra of excitations?
• In most of 2- and 3-d cases because of spontaneous symmetry breaking, these singlets do not dominate the low energy physics, and semi-classical pictures are usually OK!
• What can favor exotic quantum phases in 2d?
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Bipartite latt. Frustrated latt.
Néel order versus quantum pairing : coordination number and geometry
-0,1
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0 1 2 3 4 5 6 7
nonfrustrating latticesfrustrating lattices
M / Msat
coordinance
Reduction of Néel order parameter by quantum fluct.
Coordination number
H = i,j 2 Si . Sj
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• Center of mass kinetics– Eigenstates: |0, k >– Scaling as 1/N
• G.S. + 1 phonon Q– Eigenstates: |1, k + Q>
– E |1, k + q> - E |0, k > scales as 1/L
• Translation symmetry breaking in 2 and 3D
Broken symmetry in mesoscopic solids
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• Euler triedra kinetics– Eigenstates: |0, S, k0 >
– Scaling as 1/N
• |0, S , k0 > g.s. + 1 mag. Q – Eigenstates: |1, S+1, k+ Q> E scales as 1/L
• SU(2) symmetry breaking
in 2 and 3D
Broken Symmetry in Néel A.F.
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A simple exactly solvable modeland
adiabatic continuation from the Ising antiferromagnet to the
Heisenberg Néel antiferromagnet
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Partial restoration of symmetrythrough quantum fluctuations:
“order by disorder phenomenon”
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J1-J2 model on the triangular lattice
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When the adiabatic scenario fails more severely…
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Semi-classical Néel AFM
H = < i,j> 2 Si . Sj
• Uniaxial magnet
• A soft mode at (, )
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Néel ordererd AFMs
H = < i,j> Si . Sj
H = J < i,j> Si . Sj
+ K (Pijkl + P-1ijkl )
• Uniaxial magnet
• A soft mode at (, )
Pijkl
i
l k
j
Pij = 1/2 + 2 Si . Sj = 1/2 + 2 (i,j)
Pijkl + P-1ijkl =i,kj,l ]
+2i,jj,kk,ll,i) ]
i,jk,li,lj,ki,k) (j,l) ]
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A.Laeuchli, J.C. Domenge, P.Sindzingre, C.L., M.Troyer (PRL 05..)
J4
J2
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H = J < i,j> Si . Sj
+ K (Pijkl + P-1ijkl )
When K=1, J=-2:
H - C 2
C : vector chirality
C= Si x Sj + Sj x Sk
+ Sk x Sl + Sl x Si
Both spin and chirality are
classically ordered
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• Low energy effective dynamics of a quantum top,
scaling as S(S+1)/N
biaxial magnet
• 3 soft modes at (, ), (0, ), (, 0)
• 4-sublattice symmetry
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K=1, J=-2
H - C 2
C= Si x Sj + Sj x Sk
+ Sk x Sl + Sl x Si
JFrustration ot the 4-
sublattice order increases
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Low energy effective dynamics of a quantum top
biaxial magnet 3 soft modes at (, ),
(0, ), (, 0)
Low energy effective dynamics of a rigid rotator:
uniaxial magnet1 soft mode at (, ),
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biaxial magnet Partial restoration of SU(2) symmetry by quantum
fluctuations: uniaxial magnet
Spin Nematic
SB
SD
SC
SA
C
C
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Overcoming a further increase in frustration:
Valence Bond Crystals
Singlet: 1 fully optimized bond and absence of m-f interaction
with other spins
Staggered VBCColumnar VBC
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Overcoming a further increase in frustration: Valence Bond Crystals
Singlet: 1 fully optimized bond and absence of m-f interaction
with other spins
Staggered VBCColumnar VBC
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More exotic phases?
? Critical phase with deconfined
spinons excitations:Senthil et al
Science 2004
?
n-nematic stateShannon et al. PRL 2006
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Valence Bond Crystals and
Valence Bond Solids
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Valence-Bond CrystalsShort range order in spin-spin correlations &
L.R.O in dimer singlets or larger S=0 plaquettes!J1-J4 model on the square latticeJ1-J2 model on the hexagonal latticeShastry-Sutherland model (SrCuBO)Heisenberg on the 2D pyrochlore lattice
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Heisenberg on the 2D pyrochlore lattice
Palmer et Chalker 02, Fouet et al. 03,
Berg et al. 03Brenig et al.
The classical problem has a residual entropy per spin at T=0
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Heisenberg on the 2D pyrochlore latticeGap to the first triplet Gaps in the singlet subspace
Fouet et al. 2003
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Heisenberg on the 2D pyrochlore lattice
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Heisenberg on the 2D pyrochlore lattice
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Heisenberg on the 2D pyrochlore lattice
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Heisenberg on the 2D pyrochlore latticeDispersion curves in the singlet subspaceDispersion curves in the triplet subspace
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Summary of the properties of VBC
• A spin gap and short range spin-spin correlations
• LRO order in singlet –singlet correlations
• Gapped excitations: single modes and continuums… all these excitations have integer spins (confinement of spinons)
• Up to now all known examples with spins are on bipartite lattices
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Valence-Bond Solids (A.K.L.T. 87)
• When individual spins obey: 2S=0 mod z• A unique singlet g.s. w.-f.• No lattice symmetry breaking• A gap, short range spin-spin correlations but
order in a non local order parameter?• Integer spin excitations in the bulk• Fractionalized degrees of freedom at the hedge
of the sample• A plausible 2D candidate with S=1
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The spin-1 Heisenberg antiferromagnet
Hida J. Phys. Soc. Jpn 69, 4003, 2000
A Valence Bond Solid:
• 6 spin-1/2 components of the original spin-1 build a singlet state on each hexagon.
• A large spin gap & a large gap in the singlet sector
Observation on m-MPYNN.BF4N. Wada et al. J. Phys. Soc. Jpn 66, 1997
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