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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