Roderich Moessner- Tutorial on frustrated magnetism

Tutorial on frustrated magnetism

Roderich Moessner

CNRS and ENS Paris

Lorentz Center Leiden

9 August 2006


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


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


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


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


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?


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?


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)


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


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


Highly frustrated (corner-sharing) lattices


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.


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?


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)


‘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 < ∆• even more theories


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?


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)


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


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!)


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


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


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)


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.


Bow-ties in the structure factor of ice

proton distribution in water ice, Ic Li et al.


‘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


Summary

• frustration ⇒ degeneracy ⇒ strong fluctuations– new phases/phase transitions/dynamics · · ·

• simple model systems• many realisations

– materials physics– nanotechnology– cold atoms


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Roderich Moessner- Tutorial on frustrated magnetism