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1855-15
School and Workshop on Highly Frustrated Magnets and StronglyCorrelated Systems: From Non-Perturbative Approaches to
Experiments
Frederic Mila
30 July - 17 August, 2007
Institute of Theoretical PhysicsEcole Polytechnique Fédérale de Lausanne,
Switzerland
Strong coupling approaches and effective Hamiltonian
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StrongStrong couplingcouplingapproachesapproaches andand
effective effective HamiltonianHamiltonianF. F. MilaMila
InstituteInstitute ofof TheoreticalTheoretical PhysicsPhysicsEcoleEcole Polytechnique FPolytechnique Fééddééralerale de de LausanneLausanne
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ScopeScope•• Introduction: Introduction: whywhy strongstrong couplingcoupling??•• BriefBrief reviewreview ofof degeneratedegenerate perturbation perturbation
theorytheory•• FrustratedFrustrated magnetsmagnets in in zerozero fieldfield: : couplingcoupling
triangles, triangles, tetrahedratetrahedra,,……•• LaddersLadders andand coupledcoupled dimersdimers in a in a fieldfield•• PerturbingPerturbing IsingIsing modelsmodels•• Alternatives to Alternatives to degeneratedegenerate perturbationperturbation
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Heisenberg modelHeisenberg modelStandard analytical approach: expansion in 1/S
Spin-waves around classical GS
Problem: frustration highly degenerate GS
No natural starting point for 1/S
Alternative?
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StrongStrong couplingcoupling II
1) Suppose
2) Write with
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StrongStrong couplingcoupling IIII
3) If GS of H0 degenerate, treat V withdegenerate perturbation theory
Effective Hamiltonian
Small parameters:
NB: often useful to assume some parameters are small even if they are not (like 1/S for S=1/2!)
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ExamplesExamples
JJ’
Weakly coupled triangles
Trimerized kagome
Heisenberg model with Ising anisotropy
Perturbation around Ising limit
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DegenerateDegenerate Perturbation Perturbation TheoryTheory
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Second Second orderorderExtremely useful, yet not totally standard
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HigherHigher orderorder II
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HigherHigher orderorder IIII
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Standard Standard exampleexample: : superexchangesuperexchange
Hubbard model
½-filling 1 e-/ site
2nd order perturbation
Heisenberg model
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One triangleOne triangle
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CoupledCoupled trianglestriangles
JJ’
Pseudo-spin‘chirality’
Spin
First-order perturbation in J’
Triangular lattice
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SpinSpin--chiralitychirality meanmean--fieldfield decouplingdecoupling
Relevant degrees of freedom selected
Reasonable to use mean-field
Degenerate GS
Dimer coverings oftriangular lattice
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OtherOther examplesexamples
•• TetrahedraTetrahedra: GS : GS isis a a twotwo--foldfold degeneratedegeneratesinglet singlet pure pure ‘‘chiralitychirality’’ effective effective modelmodel
•• OddOdd rings: rings: likelike for a triangle, GS for a triangle, GS consistsconsistsofof twotwo doublets doublets spinspin--chiralitychirality modelmodel
•• UnitsUnits withwith nonnon--degeneratedegenerate singlet GS in a singlet GS in a magneticmagnetic fieldfield singlet singlet degeneratedegenerate withwithlowestlowest triplet triplet atat criticalcritical fieldfield
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DimersDimers in a magnetic fieldin a magnetic fieldIsolated dimers
Coupled dimers
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Magnetization of spin laddersMagnetization of spin ladders
CuHpCl Chaboussant et al, EPJB ‘98
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DimersDimers in a magnetic fieldin a magnetic field
Isolated dimersCoupled dimers
2 states/rung degenerate GS
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Frustrated laddersFrustrated ladders
Metal-insulator transition for V=2t (J’=J/3)
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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
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Magnetization of SrCuMagnetization of SrCu2(BO(BO3))2
Kageyama et al, PRL ‘99
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ShastryShastry--Sutherland modelSutherland model
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
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Frustrated Coupled Frustrated Coupled DimersDimers
Triplet Hopping Triplet Repulsion
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Quantum Quantum DimerDimer ModelsModelsSquare lattice (Rokhsar-Kivelson, ’88)
Assume dimer configurations are orthogonal
RK ’88Leung et al, ‘96
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QDM on triangular latticeQDM on triangular lattice
Spin liquid with gapped spectrumand topological degeneracy
Equivalent spin model? Yes! The QDM is theeffective model of an Ising model in transverse field
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FullyFully FrustratedFrustrated IsingIsing modelmodel
Ground states: 1 unsatisfied bond/hexagon
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ExampleExample ofof a a groundground statestate
Bond unsatisfied if J>0
For that ground state, unsatisfied bondslie on dashed lines
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MappingMapping onto QDM Hilbert onto QDM Hilbert spacespace
Hexagonal lattice= dual lattice oftriangular lattice
Draw a bond on the triangular lattice acrossall unsatisfied bonds dimer covering
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AddAdd a transverse a transverse fieldfield
One spin flip outside ground state manifold
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TwoTwo spin flipsspin flips
Stay in ground state manifold
Flip two spinson neighboring
unsatisfied bonds
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In In termsterms ofof dimersdimers
Flip dimersaround flippable
plaquette
2nd order perturbation theory in Γx
Ising model , QDM with flip term / Γx2/J
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Alternative I : Canonical TransformationAlternative I : Canonical Transformation
Advantages
- Possible for non degenerate GS- Systematic expansion- Gives access to full spectrum
Drawbacks
- Not universal: needs to find S - Necessary condition: all linear
corrections must vanish
Standard example: Hubbard Heisenberg
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Alternative II: COREAlternative II: CORE
CORE: Contractor Renormalization
1) Find numerically low-energy spectrum of largestaccessible system
2) Fit it numerically with effective Hamiltonian of smallersub-units
Advantages
- Very flexible- Universal- Best effective model from
quantitative point of view
Drawbacks
- Not well suited for analyticaltreatment of effective model
- Physics of effective model nottransparent
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ConclusionsConclusions
•• Effective Effective HamiltonianHamiltonian: : oftenoften a a goodgood wayway to to understandunderstand physicsphysics atat a qualitative a qualitative levellevel
•• If If naturalnatural smallsmall parameterparameter, THE , THE methodmethod to to bebe recommendedrecommended
•• SeveralSeveral waysways ofof derivingderiving an effective an effective HamiltonianHamiltonian. . DegenerateDegenerate perturbation perturbation theorytheory veryvery simple simple andand recommendedrecommended if if firstfirst or second or second orderorder sufficientsufficient