f. mila- introduction au magnétisme
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Introduction au
magntismeF. Mila
Institute of Theoretical Physics
Ecole Polytechnique Fdrale de Lausanne
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Plan du cours
1) Modles de base
- Ising
- Heisenberg2) Modle de Heisenberg et ordre magntique
- Structures hlicodales
- Ondes de spin
3) Dimensionnalit rduite et fluctuations quantiques
- Gap de spin
- Ordre algbrique
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Plan du cours (suite)
4) Modle d'Ising et frustration gomtrique
- Entropie rsiduelle
- Corrlations algbriques et dipolaires5) Liquides de spin quantiques
- Frustration, modes mous et fluctuations quantiques
- Liquide RVB et modles de dimres quantiques
- Liquide de spin algbrique
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Plan du cours
1) Modles de base
- Ising
- Heisenberg2) Modle de Heisenberg et ordre magntique
- Structures hlicodales
- Ondes de spin
3) Dimensionnalit rduite et fluctuations quantiques
- Gap de spin
- Ordre algbrique
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Basic Models
Heisenberg
XXZ
Transverse field Ising model
Ising
Classical Quantum
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Plan du cours
1) Modles de base
- Ising
- Heisenberg2) Modle de Heisenberg et ordre magntique
- Structures hlicodales
- Ondes de spin
3) Dimensionnalit rduite et fluctuations quantiques
- Gap de spin
- Ordre algbrique
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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 = bosonsHolstein-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 ukand vksuch 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) Modles de base
- Ising
- Heisenberg2) Modle de Heisenberg et ordre magntique
- Structures hlicodales
- Ondes de spin
3) Dimensionnalit rduite et fluctuations quantiques
- Gap de spin
- Ordre algbrique
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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 gap
infrared singularity
Ifexcitations 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)
: 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
JJ weakly coupled chains ( Giamarchi)
Review: Dagotto and Rice , Science 96
J=0 =J
Weak coupling
JJ =J+O(J)
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Spinons
Ifthe spectrum is gapless,
low-lying excitations
cannot be spin-wavesCan 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 theory
Des Cloiseaux PearsonPRB 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 Giamarchis 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 4)
Berry phase
Evolution operator
Spin coherent state
Haldane, PRL 88Path integral formulation
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Field theory approach
Pontryagin index (integer)
1 (S integer)
1(S -integer)
In 1D antiferromagnets
Destructive interferences for -integer spins
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Plan du cours (suite)
4) Modle d'Ising et frustration gomtrique
- Entropie rsiduelle
- Corrlations algbriques et dipolaires
5) Liquides de spin quantiques
- Frustration, modes mous et fluctuations quantiques
- Liquide RVB et modles de dimres quantiques
- Liquide de spin algbrique
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AF Ising model on triangular lattice
?
Many different ways of minimizing the energy
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Residual entropy
= or
=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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Kasteleyns 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 eigenvalues
of (2x2) Kasteleyn matrix
Entropy Ising model
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Spin-spin correlations
Many ground states average overground states
Pfaffian representation inverse ofKasteleyn 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 heightvariable
simple algebraic decay
2D pyrochlore: spin= gradient of height variable
dipolar correlations
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Plan du cours (suite)
4) Modle d'Ising et frustration gomtrique
- Entropie rsiduelle
- Corrlations algbriques et dipolaires
5) Liquides de spin quantiques
- Frustration, modes mous et fluctuations quantiques
- Liquide RVB et modles de dimres quantiques
- Liquide de spin algbrique
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Strong Quantum effects in 2D
Basic 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
J1J2
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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 generallyFrustrated magnets with infinite number
of ground states and soft modes
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Quantum spin liquids
First definition (Shastry-Sutherland, 1981)
Absence of magnetic long-range order
even at T=0
Problem
Some systems have a symmetry breakingin the ground state, hence some kind of
order, 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 groundstates
which break the translational symmetry
Fouet et al, 2003; Canals, 2003
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Spin liquidsNew definition
Absence ofany kind of orderCan quantum magnets
behave as spin liquids? Spin-1/2 kagome?
Quantum Dimer models
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RVB spin liquids
Anderson, 1973
GS = + +
Restore translational invariance with resonances
between valence-bond configurations
Question: with one spin per unit cell, canwe preserve SU(2) without breaking translation?
Singlet
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Spin kagome
Proliferation of low-lying singlets
Lecheminant et al,
PRB 97
Exactdiagonalizations
of clusters
with up to 36 sites
RVB i t t ti
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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 basis
for kagome after Mambrini and Mila, 00
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Quantum Dimer Models
Square 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 lattice
Moessner 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 Heisenbergantiferromagnets with no long-range order
but algebraic correlations, hence a gaplessspectrum
Can be described using the fermionic
representation 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)