Tutorial on frustrated magnetism
Roderich Moessner
CNRS and ENS Paris
Lorentz Center Leiden
9 August 2006
Tutorial on frustrated magnetism
Overview
• Frustrated magnets– What are they? Why study them?
• Classical frustration – degeneracy and instability• Order by disorder• Quantum frustration
– weak quantum fluctuation– strong quantum fluctuations, and the S = 1/2 kagome
magnet
• The spinels: experimental model systems– magnetoelastics and heavy Fermions
• Outlook
Tutorial on frustrated magnetism
Why study frustrated magnets
• Materials physics– because they exist (and may be useful)
• Conceptually important model systems – often tractable– strong correlations/fluctuations– coupled degrees of freedom– interesting (quantum) phases, including liquids Betouras,
Shtengel
– unconventional phase transitions Krüger, Vishwanath
Tutorial on frustrated magnetism
History
• First system: ice Pauling, JACS 1935
• 1950s: triangular Ising magnet Wannier+Houtappel; pyrochloreIsing magnet (‘spin ice’) Anderson
• ‘cooperative paramagnets’ Villain 1977
• Most complete bibliography (by Oleg Tchernyshyov)http://www.pha.jhu.edu/˜olegt/pyrochlore.html
• Reviews: Misguich+Lhuillier cond-mat; H.T. Diep book; R.M.+Ramirez Phys. Today
Tutorial on frustrated magnetism
Frustration leads to (classical) degeneracy
Consider Ising spins σi = ±1 with antiferromagnetic J > 0:
H = J∑
〈ij〉
σiσj
?• Not all terms in H can simultaneously be minimised• But we can rewrite H:
H =J
2
(
q∑
i=1
σi
)2
+ const
• Number of ground states: Ngs =(
4
2
)
= 6 for one tetrahedron
• Degeneracy is hallmark of frustration
Tutorial on frustrated magnetism
Frustration ⇒ degeneracy ⇒ zero-point entropy
• ground-state condition: ↑↑↓ or ↑↓↓ for each triangle• finite entropy in ground state: S = 0.323kB
• ‘flippable spins’ experience vanishing exchange field
What happens at low T?
Tutorial on frustrated magnetism
Frustration ⇒ degeneracy ⇒ zero-point entropy
• ground-state condition: ↑↑↓ or ↑↓↓ for each triangle• finite entropy in ground state: S = 0.323kB
• ‘flippable spins’ experience vanishing exchange field
⇒ lower bound on entropy
S ≥ (kB/3) ln 2
• Important: local d.o.f.
What happens at low T?
Tutorial on frustrated magnetism
Why degenerate systems are special
d.o.s – unfrustrated magnet
ρ
E1
N N
N2
d.o.s – frustrated magnet
E
ρln
~N ~N~N~N
• Ground states can exhibit subtle correlations (seen at low T )• Degenerate ground states provide no energy scale⇒ all perturbations are strong ⇒ many instabilities
• Very rich behaviour (theory+experiment) – but also hard• Cf. quantum Hall physics (degenerate Landau levels)
Tutorial on frustrated magnetism
The cooperative paramagnetic regime Villain
• Definition: regime at lowtemperature T ≪ J which iscontinuously connected tohigh-temperatureparagmagnetic phase
• Properties: correlationsshort-ranged in space andtime (?)
• Experiments: phase tran-sitions occur much belowthe Curie-Weiss tempera-ture: TF ≪ ΘCW Ramirez
‘Susceptibility fingerprint’of frustration
χ−1
magnetpara-
cooper-ative para-
magnet
CWΘ
CWΘ TF
T
non-g
eneric
Tutorial on frustrated magnetism
Constraint counting as a measure of frustration
H = J∑
ij
SiSj ≃ (J/2)(
q∑
i=1
Si)2
gives ground state degeneracy:L ≡
∑
i Si to be minimised.degeneracy grows with q
Units of qHeisenberg spins
q=2
q=3
q=4α
φ
Constraint counting: D = F − K
• ground-state degeneracy D
• total d.o.f. F• ground-state constraint K
Pyrochlore antiferromagnets areparticularly frustrated
Tutorial on frustrated magnetism
Highly frustrated (corner-sharing) lattices
Tutorial on frustrated magnetism
Thermodynamics: the single-unit approximation
χ−1(T ) and E(T ) for Heisenbergpyrochlore
0 1 2 3 4 5 6 70
1
E/J
0 2 4 6 8 10
T/J
0
5
10
15
20
25
30
35
χ−1/J
susceptibility and energy per spin (undiluted pyrochlore)
theory
Monte Carlo
Curie−Weiss
• ‘Natural’ d.o.f.:single tetrahedronspin L =
∑
i Si, withL ∝
√T and L → 2
at low (high) T .• Solve ‘single unit’
(single tetrahedron)exactly
• Works rather well,despite neglectof all correlationsbeyond nearestneighbour.
Tutorial on frustrated magnetism
Order by disorder Villain, Shender
• basic idea: fluctuations lift degeneracy• thermal obdo: F = U − TS
• Ising spins: no low-energy fluctua-tions
• continuous spins: gapless excitati-ons possible – some soft: E ∝ η4
η
η S2S1
S4S3
x
ground states
phase space
y
ordered state Where is weightconcentrated?
Tutorial on frustrated magnetism
Quantum frustration
• used to describe many (very different) situations• simplest starting point think of transverse field Ising model
– Hilbert space spanned by class. (discrete) ground states– quantum dynamics: as local as possible
• quantum obdo– ‘maximally
flippable’(triangle)
– recent work onsupersolids
– 3d XY transition
• disorder by disorder(kagome)
Tutorial on frustrated magnetism
‘The holy grail’: S = 1/2 kagome
• kagome lattice has played important role historically– first experimental on SCGO (with kagome motif) Obradors
– kagome S = 1/2 remains a mystery
• apparently no order at all• spin gap ∆
• small singlet gap (if any)• many singlet states with
E < ƥ even more theories
Tutorial on frustrated magnetism
The ‘simple’ spinel oxides AB 2O4 (after Takagi)
d0.5 d1.5 d2.5 d3.5
LiTi2O4 LiV2O4 AlV2O4 LiMn2O4
BCS SC heavy Fermion charge-orderedd1 d2 d3 d4
MgTi2O4 {Zn,Mg,Cd}V2O4 {Zn,Mg,Cd}Cr2O4 ZnMn2O4
valence spin+orbital spin+structuralbond solid ordering phase transition
• ions on B-sublattice form pyrochlore lattice• properties tunable by varying ions on A, B sublattices• many more compounds exist
• LiV2O4: non-integer nominal valence; orbital d.o.f.; spin– many sources of entropy at low T– whence heavy Fermion behaviour?
Tutorial on frustrated magnetism
Supplementary (lattice) d.o.f. in the Cr spinels
• nominal valence of Cr: d3 (half-filled t2g orbitals)⇒ isotropic S = 3/2 on pyrochlore lattice
• Q: Interplay of elastic degrees of freedom and frustration?• magnetoelatic Hamiltonian Htot = Hm + Hme + He
– magnetic exchange Hm = J∑
〈ij〉 Si · Sj
– magnetoelastic coupling (xa ... displacements)
Hme =∑
aij
dJij
dxa
(Si · Sj) xa
– elastic energy He =∑
ab kabxaxb (kab ... elasticconstants)
Tutorial on frustrated magnetism
Unfrustrated magnetoelastics: chain in d = 1
• Si · Sj = cnn is uniform for nearest neighbours
• Simplest case: dJij/dxa = J ′δa,i:
Hme +He =∑
a J ′cnnxa +kx2
a minimised by xa = −J ′cnn/(2k)
=⇒ Emin = −(J ′cnn)2/(4k) grows with |cnn|• Hm minimised by extremal cnn = Si · Sj = −S2
• global minimum of Htot: only uniform contraction!• quantum S = 1/2 chain:
– Si · Sj cannot independentlyextremised
– modulated Si · Sj ⇒ modulateddistortion ⇒ dimerisation
f
f
f
Tutorial on frustrated magnetism
Frustrated magnetoelastics in a nutshell
• Frustration → degeneracy of ground states• Degenerate states not symmetry equivalent
⇒ Si · Sj can be non-uniform
• Distortions (strengthen)weaken (un)frustrated bonds• Energy balance: distortions generally present at low T
– magnetic energy: linear gain (Si · Sj) × x
– elastic energy: quadratic cost kx2
• Basically: x ∼ Si · Sj ⇒ eff. biquadratic exchange (Si · Sj)2
⇒ favours collinear states (not always seen!)
Tutorial on frustrated magnetism
Collinear order by distortion in CdCr 2O4 Ueda et al.
• at plateau centre, collinear ↑↑↑↓ among ground states• eff. biquadratic exchange leads to plateau formation• details to be worked out
Tutorial on frustrated magnetism
Emergent gauge structure: from spins to fluxes
• Think of spins as living on links of dual lattice• Easiest for Ising spins = 1 unit of flux• Experimental realisation: spin ice compounds
Tutorial on frustrated magnetism
Local constraint → conservation law
• Define ‘flux’ vector field on links of theice lattice: Bi
• Local constraint (ice rules) becomesconservation law (as in Kirchoff’s laws)
⇒ gauge theory
∇ · B = 0 =⇒ B = ∇× A
• Ice configurations differ byrearranging protons on a loop
• Amounts to reversing closed loop offlux B
• Smallest loop: hexagon (six links)
Tutorial on frustrated magnetism
Long-wavelength analysis: coarse-graining
• Coarse-grain B → B with ∇ · B = 0
• ‘Flippable’ loops have zero average flux:low average flux ⇔ many microstates
• Ansatz: upon coarse-graining, obtain energyfunctional of entropic origin:
Z =∑
B
δ∇.B,0 →∫
DB δ(∇ · B) exp[−K
2B
2]
• Artificial magnetostatics!• Resulting correlators are transverse and
algebraic (but not critical!): e.g.
〈Bz(q)Bz(−q)〉 ∝ q2
⊥/q2 ↔ (3 cos2 θ − 1)/r3.
Tutorial on frustrated magnetism
Bow-ties in the structure factor of ice
proton distribution in water ice, Ic Li et al.
Tutorial on frustrated magnetism
‘Quantum ice’: artificial electrodynamics
• Hilbert space: (classical) ice configurations• Add coherent quantum dynamics for hexagonal loop:
HRK = −t
| 〉〈| | + h.c.
+ v
| 〉〈 | + · · ·
• Effective long-wavelength theory Hq =∫
E2 + c2
B2 Maxwell
• This describes the Coulomb phase of a U(1) gauge theory:– gapless photons, speed
of light c2 ∝ v − t
– deconfinement– microscopic model!
RKTF
0 18−
MF
‘staggered’confining phases Coulomb
v/t
• Artificial electrodynamics with ice as ‘ether’ Wen’s noodle soup
Tutorial on frustrated magnetism
Summary
• frustration ⇒ degeneracy ⇒ strong fluctuations– new phases/phase transitions/dynamics · · ·
• simple model systems• many realisations
– materials physics– nanotechnology– cold atoms
Tutorial on frustrated magnetism
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