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Introduction au magnétisme
F. MilaInstitute of Theoretical Physics
Ecole Polytechnique Fédérale de Lausanne
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Plan du cours
1) Modèles de base- Ising- Heisenberg
2) Modèle de Heisenberg et ordre magnétique- Structures hélicoïdales- Ondes de spin
3) Dimensionnalité réduite et fluctuations quantiques- Gap de spin- Ordre algébrique
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Plan du cours (suite)
4) Modèle d'Ising et frustration géométrique- Entropie résiduelle- Corrélations algébriques et dipolaires
5) Liquides de spin quantiques- Frustration, modes mous et fluctuations quantiques- Liquide RVB et modèles de dimères quantiques- Liquide de spin algébrique
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Plan du cours
1) Modèles de base- Ising- Heisenberg
2) Modèle de Heisenberg et ordre magnétique- Structures hélicoïdales- Ondes de spin
3) Dimensionnalité réduite et fluctuations quantiques- Gap de spin- Ordre algébrique
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Basic Models
Heisenberg
XXZ
Transverse field Ising model
Ising
Classical Quantum
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Plan du cours
1) Modèles de base- Ising- Heisenberg
2) Modèle de Heisenberg et ordre magnétique- Structures hélicoïdales- Ondes de spin
3) Dimensionnalité réduite et fluctuations quantiques- Gap de spin- Ordre algébrique
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Magnetic long-range order
Classical GS: helix with pitch vector Q
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Quantum spins: Large S Fluctuations around classical GS = bosons
Holstein-Primakoff
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I - Local rotation
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II – Terms of order S2 and S
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III – Fourier transformation
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IV – Bogoliubov transformation
Goldstone modes
with uk and vk such that
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Physical consequences
La2CuO4
Inelastic Neutron Scattering Spin-wave dispersion
(Coldea et al,PRL 2001)
Specific heat: Cv / T D
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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)
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Plan du cours
1) Modèles de base- Ising- Heisenberg
2) Modèle de Heisenberg et ordre magnétique- Structures hélicoïdales- Ondes de spin
3) Dimensionnalité réduite et fluctuations quantiques- Gap de spin- Ordre algébrique
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Quantum Fluctuations (T=0)
diverges in 1D
No magnetic long-range orderin 1D antiferromagnets
Ground-state and excitations in 1D?
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Spin gapinfrared singularity
If excitations are spin waves,there must be a spin gap to produce
an infrared (low-energy) cut-off
First example: spin 1 chain (Haldane, 1983)
(see below)
Recent example: spin 1/2 ladders
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Spin ladders
SrCu2O3
(Azuma, PRL ’94)
Δ: spin gap
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Magnetization of spin ladders
CuHpCl Chaboussant et al, EPJB ‘98
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Origin of spin gap in ladders
Strong coupling
J’J
J’«J weakly coupled chains ( Giamarchi)
Review: Dagotto and Rice , Science ‘96
J=0 Δ=J’
Weak coupling
J«J’ Δ=J’+O(J)
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Spinons If the spectrum is gapless,
low-lying excitations cannot be spin-waves
Can the spectrum be gapless in 1D?
YES!
Example: S=1/2 chain (Bethe, 1931)
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Nature of excitations? Spinons!
S=1
Spinons
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Excitation spectrum
Stone et al, PRL ‘03
Early theoryDes Cloiseaux – Pearson
PRB ‘62
A spin 1 excitation= 2 spinons continuum
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Algebraic correlations
Jordan-Wigner transformation 1D interacting fermions
Spin-spin correlation Bosonisation
see Giamarchi’s lecture
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Unified framework
Haldane, 1983
When to expect spin-waves, and when to expect spinons?
Integer spins: gapped spin-waves
Half-integer spins: spinons
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Field theory approach
Solid angle of path (mod 4π)
Berry phaseEvolution operator
Spin coherent state
Haldane, PRL ‘88
Path integral formulation
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Field theory approach
Pontryagin index (integer)
1 (S integer)
± 1(S ½-integer)
In 1D antiferromagnets
Destructive interferences for ½-integer spins
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Plan du cours (suite)
4) Modèle d'Ising et frustration géométrique- Entropie résiduelle- Corrélations algébriques et dipolaires
5) Liquides de spin quantiques- Frustration, modes mous et fluctuations quantiques- Liquide RVB et modèles de dimères quantiques- Liquide de spin algébrique
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AF Ising model on triangular lattice
?
Many different ways of minimizing the energy
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Residual entropy
= or
Ω=2N/3
S>1/3 ln 2=0.23105… Exact entropy?
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Mapping onto dimer model - I
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Mapping onto dimer model - II
Dimer covering of honeycomb lattice
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Counting the dimer coverings
How to calculate it? Transform it into a determinant!
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Kasteleyn’s method
Odd number of clockwise arrows in each even loop
Z=Pfaffian of a
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Calculation of Pfaffian
a: periodic, 2 sites per unit cell
det(a)= product on k of eigenvaluesof (2x2) Kasteleyn matrix
Entropy Ising model
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Spin-spin correlations
Many ground states average over ground states
Pfaffian representation inverse of Kasteleyn matrix
Kasteleyn matrix gapped: exponential decay
Kasteleyn matrix not gapped: algebraic decay
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Kasteleyn matrix for honeycomb
algebraic correlations
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Spin Ice Dy2Ti2O6, Ho2Ti2O6
Pyrochlore lattice Ferromagnetic exchange interactions Strong anisotropy: spins ‘in’ or ‘out’
Ground state: 2 spins in, 2 spins out
Residual entropy: the ‘ice problem’
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Pyrochlore lattice
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Residual entropy
Spin ice Ice
Pauling (1945): S/kB≈ (1/2) ln (3/2) = 0.202732
‘Exact’: S/kB≈ 0.20501(Nagle, 1966)
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Dipolar correlations
Theory:Canals-Garanin, 1999
Experiments: Fennell, 2009
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Why dipolar correlations?
Height model: algebraic correlations of height variable
Triangular lattice: spin= exponential of height variable simple algebraic decay
2D pyrochlore: spin= gradient of height variable dipolar correlations
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Plan du cours (suite)
4) Modèle d'Ising et frustration géométrique- Entropie résiduelle- Corrélations algébriques et dipolaires
5) Liquides de spin quantiques- Frustration, modes mous et fluctuations quantiques- Liquide RVB et modèles de dimères quantiques- Liquide de spin algébrique
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Strong Quantum effects in 2DBasic idea
diverges in 2D as soon as
or
since
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Defining frustration IFrustration = competition between exchange processes
Not frustrated ? Frustrated
Not enough!RVB theory of triangular lattice (Anderson)
3-sublattice long-range order (Bernu et al, 1990)
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Defining frustration IIFrustration = infinite degeneracy of classical ground state
J1-J2 model (J2>J1/2)
Effect of quantum fluctuations?
Kagome lattice
J1
J2
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J1-J2 model on square lattice
Infinite number of helical GS for J2/J1=1/2
No long-range order
More generally
Frustrated magnets with infinite numberof ground states and soft modes
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Quantum spin liquidsFirst definition (Shastry-Sutherland, 1981)
Absence of magnetic long-range ordereven at T=0
Problem
Some systems have a symmetry breakingin the ground state, hence some kind oforder, but no magnetic long-range order
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Spontaneous symmetry breaking
J1
J2
J1=2J2 2 singlet ground states(Majumdar-Ghosh, 1970)
= (l > - l >)/√2
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Checkerboard lattice
2 plaquette coverings 2 groundstateswhich break the translational symmetry
Fouet et al, 2003; Canals, 2003
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Spin liquidsNew definition
Absence of any kind of order
Can quantum magnetsbehave as spin liquids?
Spin-1/2 kagome? Quantum Dimer models
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RVB spin liquids
Anderson, 1973
GS = + + …
Restore translational invariance with resonancesbetween valence-bond configurations
Question: with one spin ½ per unit cell, canwe preserve SU(2) without breaking translation?
Singlet
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Spin ½ kagomeProliferation of low-lying singlets
Lecheminant et al,PRB ‘97
Exact diagonalizations
of clusters with up to 36 sites
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RVB interpretationResonance between Valence Bond configurations
Proposed for triangular lattice by Anderson and Fazekas (’73)
Widely accepted as a good variational basisfor kagome after Mambrini and Mila, ‘00
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Quantum Dimer ModelsSquare lattice (Rokhsar-Kivelson, ’88)
Assume dimer configurations are orthogonal
GS= superposition of all dimer coverings
V=J: Rokhsar-Kivelson point
RVB phase? No!
Kasteleyn matrix gapless algebraic correlations
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QDM on triangular latticeMoessner and Sondhi, ’01;
A. Ralko, F. Becca, M. Ferrero, D. Ivanov, FM, ‘05
Kasteleyn matrix gapped Exponential correlations RVB phase
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Algebraic spin liquids
Hypothetical ground state of Heisenberg antiferromagnets with no long-range order but algebraic correlations, hence a gapless spectrum
Can be described using the fermionicrepresentation of spin operators
Has been proposed (as basically everything else) for the spin-1/2 kagome
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Fermionic representation
Constraint
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Mean-field theory
+ constraint on average
Dimerized solution
Fluctuations beyond mean-field
Quantum Dimer Models
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Flux phase
Affleck-Marston, 1988
Square lattice
Dirac spectrum with 4 gapless points Algebraic correlations
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Stability of mean-field solutions
Compare mean-field energies not very reliable (constraint treated on average+mean-field decoupling)
Treat constraint with Gutzwiller projection
Treat as variational parameters
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Further reading
Interacting electrons and quantum magnetism
Assa Auerbach
A modern discussion of non-frustrated magnets
C. Lacroix, P. Mendels, F. Mila Eds., Springer, 2010
Introduction to Frustrated Magnetism
A recent review of all aspects of frustrated magnets(theory, experiment, synthesis)