Introduction to Introduction to magnetism magnetism in in condensed condensed matter matter physics physics F. Mila F. Mila Institute of Institute of Theoretical Theoretical Physics Physics Ecole Polytechnique F Ecole Polytechnique F é é d d é é rale rale de de Lausanne Lausanne Switzerland Switzerland
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F. Mila- Introduction to magnetism in condensed matter physics
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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
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
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
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
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
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
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