F. Mila- Introduction to magnetism in condensed matter physics

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Introduction to magnetism in condensed matter physicsF. Mila Institute of Theoretical Physics Ecole Polytechnique Fdrale de Lausanne Switzerland

First part: atoms and metalsAtomic magnetism in condensed matter Orbital moment, spin, crystal field, spin-orbit coupling Magnetism of itinerant electrons Orbital effects: De Haas-Van Alphen oscillations, Quantum Hall effects Spin effects: Pauli susceptibility, Stoner ferromagnetism, spin-density waves

Second part: localized momentsLocalized moments in metals: Kondo, RKKY, Mott insulators Magnetic interactions: Heisenberg, dipolar, Low temperature phases of Heisenberg model Long-range order Algebraic order (1D) Spin liquids

Electron in a magnetic fieldVector potential H=r A Electrostatic potential Relativistic spin-orbit coupling

Orbital effects Electron=charged particle

Zeeman coupling Electron=spin-1/2 particle

Atomic magnetism in a crystalUniform magnetic field: A= (r H)Total magnetic moment coupled to the field Larmor diamagnetism

Spherical potential of ion + distortion by surrounding ions crystal field

Transition metals: Cu, Ni, V,Crystal field spin-orbit

Effective spin Anisotropies + sometimes (single ion, g-tensor,..) orbital degeneracy

Crystal field effects of d-electrons

Low-spin statesEx: Co4+ 3d5 with crystal field Hunds rule


S=1/2 + orbital degeneracy

S=1/2 No orbital degeneracy

Rare earths: Ce, Pr, Gd,Crystal field spin-orbit

Lifts the degeneracy effective multiplet

Effective moment + Lande gL factor

Orbital effects in metals and semiconductors: Landau levelsFree electron in a uniform magnetic field

Landau levels

Consequences3D metals: De Haas Van Alphen oscillations of m as a function of 1/H extremal sections of Fermi surface ? H 2D electron gas: Quantum Hall effect plateaux of Hall conductance (see lecture by J. Smet)

Spin effects in metal

Zeeman term shift of up and down Fermi seas m/H Pauli susceptibility

Magnetic instabilitiesHubbard modelKinetic energy

Electron-electron interactions

instability q=0: ferromagnetism (Stoner) q0: spin-density wave

Localized moments in metalsKondo effect: screening of impurities by electron gas resistivity minimum RKKY interactions: effective interaction between moments mediated by electron gas J / cos(2kFr)/r3 Heavy fermions: periodic arrangement of localized moments flat band at Fermi level due to hybridization to electron gas

Mott insulatorsBand theoryOdd number of e-/unit cell


Strong on-site repulsion U Small bandwidth W J=4t2/U W=4t

InsulatorSpin fluctuations

E =U-W>0

Heisenberg model

Exchange mechanismsKinetic exchange: virtual hops from one Wannier function to neighbors J = 4 t2/U > 0 antiferromagnetic Superexchange: kinetic exchange through ligands antiferromagnetic Hunds rule between orthogonal ligand orbitals ferromagnetic

Anderson-Goodenough-Kanamori rules

High temperature susceptibility1/ antiferromagnet paramagnet (Curie) ferromagnet T

/ 1/(T+) / j Jij

: Curie-Weiss constant >0: AF 0Ground-state: Nel


(Antiferromagnetism) Finite temperature: molecular-field


Ordering at TN / J, flat susceptibility below TN

Quantum spinsUsual situationSome kind of helical long-range order

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

Quantum fluctuations: Large SFluctuations around classical GS = bosons Holstein-Primakoff

Linear spin-wave theory I1) Only keep terms of order S2 and S

Opposite quantization axis on sublattices A and B 2) Fourier transformation 3) Bogolioubov transformation

Linear spin-wave theory IIAnderson, 52 Kubo, 52

BosonsQuantum Fluctuations

Physical consequencesInelastic Neutron Scattering Spin-wave dispersion

(see lecture by H. Ronnow) La2CuO4(Coldea et al, PRL 2001)

Specific heat: Cv / T D

Domain of validityFluctuations around

Thermal Fluctuations (T>0)

diverges in 1D and 2D

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

Quantum Fluctuations (T=0)

diverges in 1D No magnetic long-range order in 1D antiferromagnets Ground-state and excitations in 1D?

Spin gapIf excitations are spin waves, there must be a spin gap to produce an infrared cut-off in the integral

First example: spin 1 chain (Haldane, 1981) Recent example: spin 1/2 ladders

Spin laddersSrCu2O3(Azuma, PRL 94)

: spin gap

Magnetization of spin ladders


Chaboussant et al, EPJB 98

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

Origin of spin gap in laddersReview: Dagotto and Rice , Science 96

Strong couplingJ J

J=0 JJWeak coupling

=J =J+O(J)


weakly coupled chains

Algebraic orderIf the spectrum is gapless, low-lying excitations cannot be spin-waves Can the spectrum be gapless in 1D?


Example: S=1/2 chain (Bethe, 1931) Correlation function: decays algebraically

Nature of excitations? Spinons!S=1


Excitation spectrumA spin 1 excitation = 2 spinons


Early theory Des Cloiseaux Pearson PRB 62 Stone et al, PRL 03

Unified frameworkWhen to expect spin-waves, and when to expect spinons?

Haldane, 1981 Integer spins: gapped spin-waves Half-integer spins: spinons

Field theory approachHaldane, PRL 88 Path integral formulation

Evolution operator Spin coherent state

Berry phase

Solid angle of path (mod 4)

Field theory approachIn 1D antiferromagnets 1 (S integer) 1 (S -integer)Pontryagin index (integer)

Destructive interferences for -integer spins

Spin liquidsShastry-Sutherland, 1981

even at T=0 ! No magnetic long-range order Example: spin-1/2 ladders

Spin liquids in 2D?Basic idea diverges in 2D as soon as or


Competing interactions

Classical GS: helix with pitch vector Q Dispersion

Frustrated magnetsFrustration = infinite degeneracy of classical ground state

J1-J2 model (J2>J1/2)

Kagome lattice

J1 J2

Exotic ground states?


SU(2) U(1) spin rotation around z + spatial symmetries (translations and point group)

Standard casesMagnetic long-range order: broken SU(2) Spontaneous dimerization: broken translation Integer spin/unit cell: no broken symmetry (e.g. spin 1 chain, spin-1/2 ladders)


More exotic alternativesBroken SU(2) symmetry without magnetic LRO: quadrupolar order RVB spin liquids with half-integer spin per unit cell: topological order

More exotic alternativesBroken SU(2) symmetry without magnetic LRO: quadrupolar orderRVB spin liquids with half-integer spin per unit cell: topological order

Broken SU(2) => magnetic LRO?(if purely local order parameter) Any linear combination of l > and l - > can be obtained by a certain rotation of l > around some axis =1/2 for a certain direction Any local state is magnetic

S=1/2: YES

Spin 1: NO!Consider

True for any

Broken SU(2) symmetry

Not magnetic

Quadrupole states and directors

Rotation of l Sz=0>


S=1 with biquadratic interaction

Quadrupolar Hamiltonian

Pure quadrupolar Hamiltonian for J1=J2/2 Quadrupolar order Order parameter: rank 2 tensor

S=1 on triangular latticeAntiferroquadrupolar Directors mutually perpendicular on 3 sublattices

(see also Tsunetsugu-Arikawa, 06)

FerroquadrupolarA. Luchli, FM, K. Penc, PRL (2006)

Parallel directors

NiGa2S4S. Nakatsuji et al, Science 2005

Cv/ T2 No Bragg peaks Quadrupolar order?

More exotic alternativesBroken SU(2) symmetry without magnetic LRO: quadrupolar order

RVB spin liquids with half-integer spin per unit cell: topological order

RVB spin liquidsQuestion: with one spin per unit cell, can we preserve SU(2) without breaking translation?Anderson, 1973

GS =



Restore translational invariance with resonances between valence-bond configurations

Quantum Dimer ModelsRokhsar-Kivelson, 1988

Assume dimer configurations are orthogonal Broken translation Degenerate GSRokhsar-Kivelson 1988; Leung et al, 1996

QDM on triangular latticeMoessner and Sondhi, 01

No broken translational symmetry RVB spin liquid

Low-lying excitation on a cylinder?

Topological sectorsNumber of dimers cutting a given line



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

Topological degeneracyTopological sectorsPortions of Hilbert space not connected by local operators like the Hamiltonian

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

Numerical proof in RVB phase of QDMGreens function Quantum Monte CarloA. Ralko, M. Ferrero, F. Becca, D. Ivanov, FM, PRB 2005

Topological degeneracy broken symmetryExample: spin-Peierls 4 ground states

Strong bond

Non-degenerate ground state

Topological orderNo local order parameter: no local operator can have different expectation values in the two GS Non-local string order parameter:

nl =1 if bond occupied, 0 if bond empty

Elementary excitations = `visonsDual lattice (-1)# dimers = -1 (-1)# dimers = 1




Periodic boundary conditions: pairs of visons fractional excitationsRead-Chakraborty 89, Senthil-Fisher 00,01