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

StrongStrong couplingcouplingapproachesapproaches andand

effective effective HamiltonianHamiltonianF. F. MilaMila

InstituteInstitute ofof TheoreticalTheoretical PhysicsPhysicsEcoleEcole Polytechnique FPolytechnique Fééddééralerale de de LausanneLausanne

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

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?

StrongStrong couplingcoupling II

1) Suppose

2) Write with

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!)

ExamplesExamples

JJ’

Weakly coupled triangles

Trimerized kagome

Heisenberg model with Ising anisotropy

Perturbation around Ising limit

DegenerateDegenerate Perturbation Perturbation TheoryTheory

Second Second orderorderExtremely useful, yet not totally standard

HigherHigher orderorder II

HigherHigher orderorder IIII

Standard Standard exampleexample: : superexchangesuperexchange

Hubbard model

½-filling 1 e-/ site

2nd order perturbation

Heisenberg model

One triangleOne triangle

CoupledCoupled trianglestriangles

JJ’

Pseudo-spin‘chirality’

Spin

First-order perturbation in J’

Triangular lattice

SpinSpin--chiralitychirality meanmean--fieldfield decouplingdecoupling

Relevant degrees of freedom selected

Reasonable to use mean-field

Degenerate GS

Dimer coverings oftriangular lattice

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

DimersDimers in a magnetic fieldin a magnetic fieldIsolated dimers

Coupled dimers

Magnetization of spin laddersMagnetization of spin ladders

CuHpCl Chaboussant et al, EPJB ‘98

DimersDimers in a magnetic fieldin a magnetic field

Isolated dimersCoupled dimers

2 states/rung degenerate GS

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

Magnetization of SrCuMagnetization of SrCu2(BO(BO3))2

Kageyama et al, PRL ‘99

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

Frustrated Coupled Frustrated Coupled DimersDimers

Triplet Hopping Triplet Repulsion

Quantum Quantum DimerDimer ModelsModelsSquare lattice (Rokhsar-Kivelson, ’88)

Assume dimer configurations are orthogonal

RK ’88Leung et al, ‘96

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

FullyFully FrustratedFrustrated IsingIsing modelmodel

Ground states: 1 unsatisfied bond/hexagon

ExampleExample ofof a a groundground statestate

Bond unsatisfied if J>0

For that ground state, unsatisfied bondslie on dashed lines

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

AddAdd a transverse a transverse fieldfield

One spin flip outside ground state manifold

TwoTwo spin flipsspin flips

Stay in ground state manifold

Flip two spinson neighboring

unsatisfied bonds

In In termsterms ofof dimersdimers

Flip dimersaround flippable

plaquette

2nd order perturbation theory in Γx

Ising model , QDM with flip term / Γx2/J

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

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

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