Frustrated Quantum Frustrated Quantum Magnets in StrongMagnets in Strong

Magnetic FieldsMagnetic Fields

F. MilaF. MilaInstitute of Theoretical PhysicsInstitute of Theoretical Physics

Ecole Polytechnique FEcole Polytechnique Fédérale de édérale de LausanneLausanne

SwitzerlandSwitzerland

CollaboratorsCollaboratorsTheorists

S. Miyahara (Lausanne Tokyo)F. Becca (Lausanne Trieste)B. Kumar (Lausanne)

ExperimentalistsM. Takigawa, K. Kodama (ISSP, Tokyo)C. Berthier, M. Horvatic (Grenoble)Chemists

H. Kageyama, Y. Ueda (ISSP, Tokyo)

ScopeScope• Introduction to Frustrated Quantum MagnetsIntroduction to Frustrated Quantum Magnets

• Dimers + Frustration Dimers + Frustration Magnetization plateaux Magnetization plateaux

• Other anomalies in SrCuOther anomalies in SrCu22(BO(BO33))22

Dzyaloshinskii-Moriya interactions Correlated hopping

• Conclusions/PerspectivesConclusions/Perspectives

Quantum magnets: 2 Quantum magnets: 2 paradigmsparadigms

Long-range order Spin gap

Goldstone modes: Spin waves

Singlet-triplet gap

J’J

Δ=J’+O(J) (J«J’)

H=∑(i,j) Jij Si.Sj S=1/2,1,…

Defining frustrationDefining frustrationFrustration = infinite degeneracy of classical ground state

Shastry-Sutherland ModelShastry-Sutherland Model

Effect of quantum fluctuations?

J1J2

J2<2J1 J2>2J1

Quantum approachQuantum approach1) Quantum treatment of local bricks

2) Pertubative treatment of inter-brick coupling

Coupled dimers Coupled triangles

Frustrated motion of triplets

Magnetization plateaux

SrCuSrCu2(BO(BO3))2

2 degrees of freedom

Low-lying singlets

Spin ½ kagomeSpin ½ kagome

Dimers in a magnetic Dimers in a magnetic fieldfield

Isolated dimersCoupled dimers

Magnetization of spin Magnetization of spin laddersladders

CuHpCl Chaboussant et al, EPJB ‘98

NB: CuHpCl might not be a simple ladder (Stone et al, cond-mat/0103023)

Frustrated laddersFrustrated ladders

Metal-insulator transition for V=2t (J’=J/3)

Magnetization PlateauMagnetization Plateau

D. Cabra et al, PRL ‘97K. Totsuka, PRB ‘98

T. Tonegawa et al, PRB ‘99F. Mila, EPJB ‘98

Frustration

Kinetic energy

Repulsion

Metal-insulator transition

Magnetization plateau

Frustrated Coupled Frustrated Coupled DimersDimers

Triplet Hopping Triplet Repulsion

Magnetization of Magnetization of SrCuSrCu2(BO(BO3))2

Kageyama et al, PRL ‘99

Shastry-Sutherland Shastry-Sutherland modelmodel

Ground-state Product of singlets on J-bonds (Shastry, Sutherland, ’81)

Triplets Almost immobile and repulsive (Miyahara et al, ’99)(Miyahara et al, ’00)Plateaux

J’/J ' .63

Symmetry breaking inside Symmetry breaking inside plateauxplateaux

Miyahara et al, ’00

Hard-core bosonswith repulsion

NMR at 1/8-plateauNMR at 1/8-plateau

At least 11 different sites!K. Kodama, M. Takigawa, M. Horvatic, C. Berthier, H. Kageyama,

Y. Ueda, S. Miyahara, F. Becca, F. Mila, Science ‘02

Magnetization profile at 1/8Magnetization profile at 1/8Symmetry breaking

16 sites/unit cell8-fold degenerate GS

Lattice distortionSelection of one GS

Magnetization in field direction

Magnetization opposite to field

Friedel-likeoscillationsNMR pattern

Further anomalies IFurther anomalies I

1)The magnetization raises too early2)The gap does not close (see also ESR, Nojiri et al,

1999)3) The magnetization jumps before plateau

Kodama et al, Science ‘02 Kodama et al, unpublished

Δ∕gμB

Further anomalies IIFurther anomalies II

Tsujii et al,’03

4) No Bose condensation below plateau

Possible sources of Possible sources of deviationdeviation

• Correlated hopping of triplets Correlated hopping of triplets Bound statesBound states• Spin-lattice coupling Spin-lattice coupling • Dzyaloshinskii-Moriya interactions D.Dzyaloshinskii-Moriya interactions D.

(S(SixSxSj)) Inter-dimer (D’)Inter-dimer (D’) Intra-dimerIntra-dimer (D)(D)

Evidence of intra-dimer DMEvidence of intra-dimer DM

Hext=6.9 T

Kodama et al, unpublished

Hs=H1-H2≠0

Staggeredmagnetization

Dzyaloshinskii-Moriya Dzyaloshinskii-Moriya interactionsinteractions

H

D1 2

D: intra-dimer DM

High-field effects of intra-dimer High-field effects of intra-dimer DMDM

Kodama, Miyahara, et al, unpublishedD=0.034 D’=-0.02 gs=0.023 (ED, 24 sites)

Properties around 1/8 Properties around 1/8 plateauplateau

Exact diagonalizations?

Not appropriate!

Huge finite-size effects at the plateaux!

Effective hard-core boson model

Pertubative Pertubative derivationderivation

Momoi and Totsuka, PRB’01

3rd order in J’/J

Mean-field phase diagramMean-field phase diagram

Momoi-Totsuka, ‘01

Plateaux onlyat 1/3 and 1/2

B. Kumar, F. Mila, unpublished

t’

it

Semi-phenomenological Semi-phenomenological approachapproach

Mean-field (<b>, <bMean-field (<b>, <b++>,<b>,<b++b>)b>)

Next step: include <bb> and <b+b+>

Conclusions/Conclusions/PerspectivesPerspectives

Magnetization of SrCu2(BO3)2:Remarkably rich and complex behaviour

• Triplet reduced mobility and repulsion Triplet reduced mobility and repulsion Magnetization plateausMagnetization plateaus

• Intra-dimer DM interaction Intra-dimer DM interaction Staggered magnetization Staggered magnetization Early raise of uniform magnetization Early raise of uniform magnetization Persistence of gap Persistence of gap

OpenOpen issuesissues

• Magnetization jump before 1/8 plateau?Magnetization jump before 1/8 plateau?• Difference of behaviour below and Difference of behaviour below and

above 1/8 plateau?above 1/8 plateau?• Consequences of correlated hopping?Consequences of correlated hopping? Pairing?Pairing?• Dip of the gap just before plateau?Dip of the gap just before plateau?