Introduction to Introduction to magnetismmagnetism in in condensedcondensed

mattermatter physicsphysics

F. MilaF. MilaInstitute of Institute of TheoreticalTheoretical PhysicsPhysics

Ecole Polytechnique FEcole Polytechnique Fééddééralerale de de LausanneLausanneSwitzerlandSwitzerland

First part: atoms and metalsFirst part: atoms and metals

‘‘AtomicAtomic’’ magnetism in condensed mattermagnetism in condensed matterOrbital moment, spin, crystal field, spinOrbital moment, spin, crystal field, spin--orbit orbit

coupling coupling Magnetism of itinerant electronsMagnetism of itinerant electrons

Orbital effects: De HaasOrbital effects: De Haas--Van Alphen Van Alphen oscillations, Quantum Hall effectsoscillations, Quantum Hall effects

Spin effects: Pauli susceptibility, Stoner Spin effects: Pauli susceptibility, Stoner ferromagnetism, spinferromagnetism, spin--density wavesdensity waves

Second part: localized momentsSecond part: localized moments

Localized moments in metals: Kondo, RKKY,Localized moments in metals: Kondo, RKKY,……Mott insulatorsMott insulatorsMagnetic interactions: Heisenberg, dipolar,Magnetic interactions: Heisenberg, dipolar,……Low temperature phases of Heisenberg modelLow temperature phases of Heisenberg model

LongLong--range orderrange orderAlgebraic order (1D)Algebraic order (1D)Spin liquidsSpin liquids

Electron in a Electron in a magneticmagnetic fieldfield

‘‘Orbital Orbital effectseffects’’Electron=Electron=chargedcharged particleparticle

Zeeman Zeeman couplingcouplingElectron=spinElectron=spin--1/2 1/2 particleparticle

RelativisticRelativisticspinspin--orbitorbit couplingcoupling

ElectrostaticElectrostatic potentialpotentialVectorVector potentialpotentialH=H=rr ££ AA

‘‘AtomicAtomic’’ magnetismmagnetism in a in a crystalcrystalUniform Uniform magneticmagnetic fieldfield: A=: A=½½ (r (r ££ H)H)

Larmor Larmor diamagnetismdiamagnetism

SphericalSpherical potentialpotential of ionof ion+ + distortiondistortion by by surroundingsurrounding ionsions

‘‘crystalcrystal fieldfield’’

Total Total magneticmagnetic momentmomentcoupledcoupled to the to the fieldfield

Transition Transition metalsmetals: Cu, Ni, V,: Cu, Ni, V,……

AnisotropiesAnisotropies(single ion, g(single ion, g--tensortensor,..),..)

Crystal Crystal fieldfield ÀÀ spinspin--orbitorbit

Effective spinEffective spin+ + sometimessometimes

orbital orbital degeneracydegeneracy

Crystal Crystal fieldfield effectseffects of of dd--electronselectrons

Ex: CoEx: Co4+4+ 3d3d55 withwith crystalcrystal fieldfield ÀÀ HundHund’’s s rulerule

S=5/2S=5/2

S=1/2S=1/2+ orbital + orbital degeneracydegeneracy

S=1/2S=1/2No orbital No orbital degeneracydegeneracy

LowLow--spin statesspin states

Rare Rare earthsearths: Ce, Pr, Gd,: Ce, Pr, Gd,……

Crystal Crystal fieldfield ¿¿ spinspin--orbitorbit

Lifts the Lifts the degeneracydegeneracyeffective multipleteffective multiplet

Effective momentEffective moment+ Lande + Lande ggLL factorfactor

Orbital Orbital effectseffects in in metalsmetals and and semiconductorssemiconductors: Landau : Landau levelslevels

Free Free electronelectron in a in a uniformuniform magneticmagnetic fieldfield

Landau Landau levelslevels

ConsequencesConsequences

3D 3D metalsmetals: : De Haas De Haas –– Van Van AlphenAlphenoscillations of m as a oscillations of m as a functionfunction of 1/Hof 1/H

extremalextremal sections of Fermi surface sections of Fermi surface ?? HH

2D 2D electronelectron gasgas: Quantum Hall : Quantum Hall effecteffectplateaux of Hall conductanceplateaux of Hall conductance((seesee lecture by J. Smet)lecture by J. Smet)

Spin Spin effectseffects in in metalmetal

Pauli Pauli susceptibilitysusceptibility

Zeeman Zeeman termtermshift of shift of up and down Fermi up and down Fermi seasseas

m m // HH

MagneticMagnetic instabilitiesinstabilities

Hubbard modelHubbard model

ElectronElectron--electronelectroninteractionsinteractions

KineticKinetic energyenergy

instabilityinstability

q=0: q=0: ferromagnetismferromagnetism ((StonerStoner))qq≠≠0: spin0: spin--densitydensity wavewave

LocalizedLocalized moments in moments in metalsmetals

Kondo Kondo effecteffect: screening of : screening of impuritiesimpurities by by electronelectron gasgas resistivityresistivity minimumminimumRKKY interactions: effective interaction RKKY interactions: effective interaction betweenbetween moments moments mediatedmediated by by electronelectrongasgas J J // cos(2kcos(2kFFr)/rr)/r33

HeavyHeavy fermions: fermions: periodicperiodic arrangement of arrangement of localizedlocalized moments moments flat bandflat band atat Fermi Fermi levellevel due to due to hybridizationhybridization to to electronelectron gasgas

Mott insulatorsMott insulatorsBand theoryBand theory Odd number of eOdd number of e--/unit cell/unit cell MetalMetal

Strong onStrong on--site repulsion Usite repulsion UInsulatorInsulator

E =UE =U--W>0W>0

Small bandwidth WSmall bandwidth W

Spin fluctuationsSpin fluctuationsJ=4tJ=4t22/U/U

W=4tW=4t

Heisenberg modelHeisenberg model

Exchange Exchange mechanismsmechanisms

KineticKinetic exchange: exchange: virtualvirtual hopshops fromfrom one one WannierWannierfunctionfunction to to neighborsneighbors

J = 4 tJ = 4 t22/U > 0 /U > 0 antiferromagneticantiferromagneticSuperexchangeSuperexchange: : kinetickinetic exchange exchange throughthroughligands ligands antiferromagneticantiferromagneticHundHund’’s s rulerule betweenbetween orthogonal ligand orthogonal ligand orbitalsorbitals

ferromagneticferromagnetic

AndersonAnderson--GoodenoughGoodenough--KanamoriKanamori rulesrules

High High temperaturetemperature susceptibilitysusceptibility

antiferromagnetantiferromagnet

paramagnetparamagnet (Curie)(Curie)

ferromagnetferromagnet

1/1/χχ

TT

χχ // 1/(T+1/(T+θθ))

θθ // ∑∑jj JJijij

θθ:: CurieCurie--Weiss constantWeiss constant

θθ>>0: AF 0: AF θθ<0: Ferro<0: Ferro

θθ θθ

OtherOther interactionsinteractions

DipolarDipolar interactionsinteractions

MagneticMagnetic domainsdomains, , hysteresishysteresis in in ferromagnetsferromagnetsDzyaloshinskiiDzyaloshinskii--MoriyaMoriya interactionsinteractions

CantingCanting, torque, ESR , torque, ESR linewidthlinewidth,,……FourFour--spin interactions (spin interactions (higherhigher orderorder in t/U)in t/U)

nematicnematic orderorder, spin , spin liquidsliquids……

Heisenberg modelHeisenberg model

Important parametersImportant parameters

Sign of exchange integrals Sign of exchange integrals

Dimensionality of spaceDimensionality of space

FerromagneticFerromagnetic

AntiferromagneticAntiferromagnetic

1D ≠ 2D ≠ 3D !

Magnitude of spinsMagnitude of spins

Topology of exchange integralsTopology of exchange integrals

Simple topology:Simple topology:NearestNearest--neigbourneigbour on bipartite latticeon bipartite lattice

≠≠Complex topologies:Complex topologies:

-- NextNext--nearest couplings,nearest couplings,……-- NonNon--bipartite lattices: triangular, bipartite lattices: triangular, kagomekagome,,……

S=1/2 S=1/2 ≠≠ S=1 S=1 ≠≠ S=3/2 S=3/2 ……

Classical spins on bipartite latticeClassical spins on bipartite lattice

GroundGround--state: state: NNééelel orderorder

((AntiferromagnetismAntiferromagnetism))

Finite temperature: Finite temperature: molecularmolecular--field theoryfield theory

Ordering at TOrdering at TN N // J, flat susceptibility below TJ, flat susceptibility below TNN

J>0J>0

Quantum spinsQuantum spinsUsual situationUsual situation

Some kind of helical longSome kind of helical long--range orderrange order

up to Tup to TNN>0 in 3D, at T=0 in 2D>0 in 3D, at T=0 in 2D

Quantum fluctuations: Large SQuantum fluctuations: Large S

Fluctuations around classical GS = bosonsFluctuations around classical GS = bosons

HolsteinHolstein--PrimakoffPrimakoff

Linear spinLinear spin--wave theory I wave theory I 1) Only keep terms of order S1) Only keep terms of order S22 and Sand S

Opposite quantization axisOpposite quantization axison on sublatticessublattices A and BA and B

3) 3) BogolioubovBogolioubov transformationtransformation

2) Fourier transformation2) Fourier transformation

Linear spinLinear spin--wave theory II wave theory II

BosonsBosons

Anderson, Anderson, ‘‘52 52 Kubo, Kubo, ‘‘5252

Quantum FluctuationsQuantum Fluctuations

Physical consequencesPhysical consequences

LaLa22CuOCuO44

Inelastic Neutron Scattering SpinSpin--wavewave dispersion

((ColdeaColdea et al,et al,PRL 2001)PRL 2001)

Specific heat:Specific heat: CCvv // T T DD

((seesee lecture by H. lecture by H. RonnowRonnow))

Domain of validityDomain of validityFluctuations aroundFluctuations around

Thermal Fluctuations (T>0)Thermal Fluctuations (T>0)

diverges in 1D and 2Ddiverges in 1D and 2D

No LRO at T>0 in 1D and 2DNo LRO at T>0 in 1D and 2D ((MerminMermin--Wagner theorem)Wagner theorem)

Quantum Fluctuations (T=0)Quantum Fluctuations (T=0)

diverges in 1Ddiverges in 1D

No magnetic longNo magnetic long--range orderrange orderin 1D in 1D antiferromagnetsantiferromagnets

GroundGround--state and excitations in 1D?state and excitations in 1D?

Spin gapSpin gapIfIf excitations are excitations are spin wavesspin waves,,

there there mustmust be a be a spin gapspin gap to produce to produce an infrared cutan infrared cut--off in the integraloff in the integral

First example:First example: spin 1 chain (Haldane, 1981)spin 1 chain (Haldane, 1981)

Recent example:Recent example: spin 1/2 laddersspin 1/2 ladders

Spin laddersSpin ladders

SrCuSrCu22OO33

(Azuma, PRL (Azuma, PRL ’’94)94)

ΔΔ:: spin gapspin gap

Magnetization of spin laddersMagnetization of spin ladders

CuHpCl Chaboussant et al, EPJB ‘98

Recent developments: TlCuCl3 (Rüegg et al, 2002-2006)

OriginOrigin of spin gap in of spin gap in laddersladders

StrongStrong couplingcoupling

J’J

JJ’’««J J weakly coupled chainsweakly coupled chains

Review: Review: DagottoDagotto and Rice , Science and Rice , Science ‘‘9696

J=0 J=0 ΔΔ=J=J’’

WeakWeak couplingcoupling

JJ««JJ’’ ΔΔ=J=J’’+O(J)+O(J)

Algebraic order Algebraic order

IfIf the spectrum is the spectrum is gaplessgapless, , lowlow--lying excitations lying excitations cannotcannot be spinbe spin--waveswaves

Can the spectrum be gapless in 1D?Can the spectrum be gapless in 1D? YES!YES!

Example: S=1/2 chain (Bethe, 1931)Example: S=1/2 chain (Bethe, 1931)

Correlation function: decays algebraicallyCorrelation function: decays algebraically

Nature of excitations? Nature of excitations? SpinonsSpinons!!

S=1S=1

SpinonsSpinons

Excitation spectrumExcitation spectrum

Stone et al, PRL Stone et al, PRL ‘‘0303

Early theoryEarly theoryDes Des CloiseauxCloiseaux –– PearsonPearson

PRB PRB ‘‘6262

A spin 1 excitationA spin 1 excitation= 2 = 2 spinonsspinons

continuumcontinuum

Unified frameworkUnified framework

Haldane, 1981Haldane, 1981

When to expect When to expect spinspin--waveswaves, , and when to expect and when to expect spinonsspinons??

Integer spins: gapped spinInteger spins: gapped spin--waveswaves

HalfHalf--integer spins: integer spins: spinonsspinons

Field theory approachField theory approach

Solid angle of path (mod 4Solid angle of path (mod 4ππ))

Berry phaseBerry phaseEvolution operatorSpin coherent stateSpin coherent state

Haldane, PRL Haldane, PRL ‘‘8888

Path integral formulationPath integral formulation

Field theory approachField theory approach

PontryaginPontryagin index (integer)index (integer)

1 (S integer)1 (S integer)

±± 11(S (S ½½--integer)integer)

In 1D In 1D antiferromagnetsantiferromagnets

Destructive interferences for Destructive interferences for ½½--integer spinsinteger spins

Spin liquidsSpin liquidsShastryShastry--Sutherland, 1981Sutherland, 1981

even at T=0 !

No magnetic long-range order

ExampleExample: spin: spin--1/2 1/2 laddersladders

Spin liquids in 2D?Spin liquids in 2D?Basic ideaBasic idea

diverges in 2D as soon asdiverges in 2D as soon as

oror

sincesince

Competing interactionsCompeting interactions

Classical GS: helix with pitch vector QClassical GS: helix with pitch vector Q

DispersionDispersion

Frustrated magnetsFrustrated magnetsFrustration = infinite degeneracy of classical ground stateFrustration = infinite degeneracy of classical ground state

J1-J2 model (J2>J1/2)

Exotic ground states?Exotic ground states?

Kagome latticeKagome lattice

J1

J2

SymmetriesSymmetries

SU(2)SU(2)

U(1)U(1) spin rotation around z

++ spatial spatial symmetriessymmetries(translations and point group)

Standard casesStandard cases

MagneticMagnetic longlong--range range orderorder: : brokenbroken SU(2)SU(2)SpontaneousSpontaneous dimerizationdimerization: : brokenbrokentranslationtranslationIntegerInteger spin/unit spin/unit cellcell: no : no brokenbroken symmetrysymmetry(e.g. spin 1 (e.g. spin 1 chainchain, spin, spin--1/2 1/2 laddersladders))

Alternatives?Alternatives?

More More ‘‘exoticexotic’’ alternativesalternatives

BrokenBroken SU(2) SU(2) symmetrysymmetry withoutwithout magneticmagneticLRO: LRO: quadrupolarquadrupolar orderorderRVB spin RVB spin liquidsliquids withwith halfhalf--integerinteger spin per spin per unit unit cellcell: : topologicaltopological orderorder

More More ‘‘exoticexotic’’ alternativesalternatives

BrokenBroken SU(2) SU(2) symmetrysymmetry withoutwithoutmagneticmagnetic LRO: LRO: quadrupolarquadrupolar orderorderRVB spin RVB spin liquidsliquids withwith halfhalf--integerinteger spin per unit spin per unit cellcell: : topologicaltopological orderorder

BrokenBroken SU(2) =>SU(2) => magneticmagnetic LRO?LRO?

AnyAny linearlinear combinationcombination of l of l ½½ > and l > and l -- ½½ > > cancan bebe obtainedobtained by a certain rotation of l by a certain rotation of l ½½ > >

aroundaround somesome axis axis

<Sα>=1/2 for a certain direction α

Any local state is magnetic

S=1/2: YES S=1/2: YES (if (if purelypurely local local orderorder parameterparameter))

Spin 1: NO!Spin 1: NO!

Consider

True for any α

Not magnetic

Broken SU(2) symmetry

QuadrupoleQuadrupole states and states and directorsdirectors

«« directordirector »»

Rotation of l Sz=0>

S=1 S=1 withwith biquadraticbiquadratic interactioninteraction

QuadrupolarQuadrupolar HamiltonianHamiltonian

Pure quadrupolar Hamiltonian for J1=J2/2

Quadrupolarorder

Order parameter: rank 2 tensor

S=1 on S=1 on triangulartriangular latticelattice

Antiferroquadrupolar

Directors mutuallyperpendicular on 3

sublattices(see also Tsunetsugu-Arikawa, ’06)

Ferroquadrupolar

Parallel directorsA. Läuchli, FM, K. Penc, PRL (2006)

NiGaNiGa22SS44

S. Nakatsuji et al, Science 2005

Cv/ T2

No Bragg peaks

Quadrupolarorder?

More More ‘‘exoticexotic’’ alternativesalternatives

BrokenBroken SU(2) SU(2) symmetrysymmetry withoutwithout magneticmagnetic LRO: LRO: quadrupolarquadrupolar orderorder

RVB spin RVB spin liquidsliquids withwith halfhalf--integerinteger spin spin per unit per unit cellcell: : topologicaltopological orderorder

RVB spin RVB spin liquidsliquids

Anderson, 1973

GS = + + …

Restore translationaltranslational invarianceinvariance with resonancesresonancesbetween valencevalence--bondbond configurations

Question:Question: with one spin ½ per unit cell, canwe preserve SU(2) without breaking translation?

Quantum Quantum DimerDimer ModelsModelsRokhsar-Kivelson, 1988

Assume dimer configurations are orthogonal

Rokhsar-Kivelson 1988; Leung et al, 1996

Broken translation

DegenerateDegenerate GSGS

QDM on triangular latticeQDM on triangular latticeMoessner and Sondhi, ‘01

RVB spin liquidNo broken translational symmetry

LowLow--lyinglying excitation on a excitation on a cylindercylinder??

Topological sectorsTopological sectorsNumber of Number of dimersdimers cutting a given linecutting a given line

Parity conserved 2 topological sectors (N even or N odd)Cylinder: two topological sectors

Torus: four topological sectors (two cuts)

N=1 N=3

Topological degeneracyTopological degeneracyTopological sectors Topological sectors

Portions of Hilbert space not connected by local operators like the Hamiltonian

Topological degeneracy (Topological degeneracy (WenWen, 1988, 1988--90) 90) GS of ≠ topological sectors degenerate

Numerical proof in RVB phase of QDMNumerical proof in RVB phase of QDM

A. Ralko, M. Ferrero, F. Becca, D. Ivanov, FM, PRB 2005

Green’s function Quantum Monte Carlo

Topological degeneracy Topological degeneracy ≠≠ broken symmetrybroken symmetry

Strongbond

Example: spin-Peierls 4 ground states

Non-degenerateground state

TopologicalTopological orderorderNo local No local orderorder parameterparameter: no local : no local operatoroperator cancan have have differentdifferent expectation expectation values in the values in the twotwo GSGSNonNon--local string local string orderorder parameterparameter: :

nnll =1 if bond =1 if bond occupiedoccupied, 0 if bond , 0 if bond emptyempty

ElementaryElementary excitations = `visonsexcitations = `visons´́

l l ii >> == ++ + + ……

((--1)1)# # dimersdimers = = --11 ((--1)1)# # dimersdimers = 1= 1Dual Dual latticelattice

PeriodicPeriodic boundaryboundary conditions: pairs of visonsconditions: pairs of visons

fractionalfractional excitationsexcitations

Read-Chakraborty ’89, Senthil-Fisher ’00,’01

Applications of topological degeneracy?Applications of topological degeneracy?

h ψeven j Ô j ψodd i =0 for any local operator Ô

EGS(even) = EGS(odd)

Well protected qubits

Kitaev, ’97; Ioffe et al, ‘01

Topological degeneracy

FurtherFurther topicaltopical problemsproblems

DopedDoped MottMott insulatorinsulatorInterplayInterplay of of magnetismmagnetism and and superconductivitysuperconductivity((seesee lectures by Keimer and lectures by Keimer and HinkovHinkov))

Spin Hall Spin Hall effecteffectMagnetizationMagnetization plateaux plateaux Orbital Orbital degeneracydegeneracyMultiferroicsMultiferroics……

Conclusions/PerspectivesConclusions/Perspectives

SolidSolid state state magnetismmagnetism: : amazinglyamazingly richrich fieldfield!!FundamentalFundamental aspects:aspects:

FantasticFantastic playgroundplayground for for theoreticaltheoreticalphysicistsphysicists

New New systemssystems and and propertiesproperties regularlyregularlydiscovereddiscoveredApplications: lots of Applications: lots of ideasideas to to bebe furtherfurtherinvestigatedinvestigated and and developeddeveloped