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From Néel ordered AFMs to

Quantum Spin Liquids

C. Lhuillier

Université Paris VI & IUF

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• 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)

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• 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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