Tutorial on frustrated magnetism

Transcription

1 Tutorial on frustrated magnetism Roderich Moessner CNRS and ENS Paris Lorentz Center Leiden 9 August 2006

2 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

3 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

4 History First system: ice Pauling, JACS s: triangular Ising magnet Wannier+Houtappel; pyrochlore Ising magnet ( spin ice ) Anderson cooperative paramagnets Villain 1977 Most complete bibliography (by Oleg Tchernyshyov) olegt/pyrochlore.html Reviews: Misguich+Lhuillier cond-mat; H.T. Diep book; R.M.+Ramirez Phys. Today

5 Frustration leads to (classical) degeneracy Consider Ising spins σ i = ±1 with antiferromagnetic J > 0: H = J ij σ i σ j? Not all terms in H can simultaneously be minimised But we can rewrite H: ( q ) H = J 2 σ i + const 2 i=1 Number of ground states: N gs = ( 4 2) = 6 for one tetrahedron Degeneracy is hallmark of frustration

6 Frustration degeneracy zero-point entropy ground-state condition: or for each triangle finite entropy in ground state: S = 0.323k B flippable spins experience vanishing exchange field What happens at low T?

7 Frustration degeneracy zero-point entropy ground-state condition: or for each triangle finite entropy in ground state: S = 0.323k B flippable spins experience vanishing exchange field lower bound on entropy S (k B /3) ln 2 Important: local d.o.f. What happens at low T?

8 Why degenerate systems are special d.o.s unfrustrated magnet ρ N 2 d.o.s frustrated magnet lnρ ~N ~N ~N ~N 1 N N E E 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)

9 The cooperative paramagnetic regime Villain Definition: regime at low temperature T J which is continuously connected to high-temperature paragmagnetic phase Properties: correlations short-ranged in space and time (?) Experiments: phase transitions occur much below the Curie-Weiss temperature: T F Θ CW Ramirez Susceptibility fingerprint of frustration χ 1 cooper- ative paramagnet CW para- magnet Θ T F Θ T CW non-generic

10 Constraint counting as a measure of frustration H = J ij S i S j (J/2)( q S i ) 2 i=1 gives ground state degeneracy: L i S i to be minimised. degeneracy grows with q Constraint counting: D = F K ground-state degeneracy D total d.o.f. F ground-state constraint K Pyrochlore antiferromagnets are particularly frustrated Units of q Heisenberg spins q=2 α φ q=3 q=4

11 Highly frustrated (corner-sharing) lattices

12 Thermodynamics: the single-unit approximation χ 1 (T) and E(T) for Heisenberg pyrochlore χ 1 /J susceptibility and energy per spin (undiluted pyrochlore) theory Monte Carlo Curie Weiss E/J T/J Natural d.o.f.: single tetrahedron spin L = i S i, with L T and L 2 at low (high) T. Solve single unit (single tetrahedron) exactly Works rather well, despite neglect of all correlations beyond nearest neighbour.

13 Order by disorder Villain, Shender basic idea: fluctuations lift degeneracy thermal obdo: F = U TS Ising spins: no low-energy fluctuations continuous spins: gapless excitations possible some soft: E η 4 S 1 η S 2 S 4 S 3 η y ground states x ordered state Where is weight concentrated? phase space

14 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 on supersolids 3d XY transition disorder by disorder (kagome)

15 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

16 The simple spinel oxides AB 2 O 4 (after Takagi) d 0.5 d 1.5 d 2.5 d 3.5 LiTi 2 O 4 LiV 2 O 4 AlV 2 O 4 LiMn 2 O 4 BCS SC heavy Fermion charge-ordered d 1 d 2 d 3 d 4 MgTi 2 O 4 {Zn,Mg,Cd}V 2 O 4 {Zn,Mg,Cd}Cr 2 O 4 ZnMn 2 O 4 valence spin+orbital spin+structural bond solid ordering phase transition ions on B-sublattice form pyrochlore lattice properties tunable by varying ions on A, B sublattices many more compounds exist LiV 2 O 4 : non-integer nominal valence; orbital d.o.f.; spin many sources of entropy at low T whence heavy Fermion behaviour?

17 Supplementary (lattice) d.o.f. in the Cr spinels nominal valence of Cr: d 3 (half-filled t 2g orbitals) isotropic S = 3/2 on pyrochlore lattice Q: Interplay of elastic degrees of freedom and frustration? magnetoelatic Hamiltonian H tot = H m + H me + H e magnetic exchange H m = J ij S i S j magnetoelastic coupling (x a... displacements) H me = aij dj ij dx a (S i S j )x a elastic energy H e = ab k abx a x b (k ab... elastic constants)

18 Unfrustrated magnetoelastics: chain in d = 1 S i S j = c nn is uniform for nearest neighbours Simplest case: dj ij /dx a = J δ a,i : H me + H e = a J c nn x a +kx 2 a minimised by x a = J c nn /(2k) = E min = (J c nn ) 2 /(4k) grows with c nn H m minimised by extremal c nn = S i S j = S 2 global minimum of H tot : only uniform contraction! quantum S = 1/2 chain: S i S j cannot independently extremised modulated S i S j modulated distortion dimerisation

19 Frustrated magnetoelastics in a nutshell Frustration degeneracy of ground states Degenerate states not symmetry equivalent S i S j can be non-uniform Distortions (strengthen)weaken (un)frustrated bonds Energy balance: distortions generally present at low T magnetic energy: linear gain (S i S j ) x elastic energy: quadratic cost kx 2 Basically: x S i S j eff. biquadratic exchange (S i S j ) 2 favours collinear states (not always seen!)

20 Collinear order by distortion in CdCr 2 O 4 Ueda et al. at plateau centre, collinear among ground states eff. biquadratic exchange leads to plateau formation details to be worked out

21 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

22 Local constraint conservation law Define flux vector field on links of the ice lattice: B i Local constraint (ice rules) becomes conservation law (as in Kirchoff s laws) gauge theory Ice configurations differ by rearranging protons on a loop Amounts to reversing closed loop of flux B Smallest loop: hexagon (six links) B = 0 = B = A

23 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 energy functional of entropic origin: Z = δ.b,0 D B δ( B) exp[ K 2 B 2 ] B Artificial magnetostatics! Resulting correlators are transverse and algebraic (but not critical!): e.g. B z (q) B z ( q) q 2 /q 2 (3 cos 2 θ 1)/r 3.

24 Bow-ties in the structure factor of ice proton distribution in water ice, I c Li et al.

25 Quantum ice : artificial electrodynamics Hilbert space: (classical) ice configurations Add coherent quantum dynamics for hexagonal loop: H RK = t + h.c. + v + Effective long-wavelength theory H q = Ẽ 2 + c 2 B 2 Maxwell This describes the Coulomb phase of a U(1) gauge theory: gapless photons, speed of light c 2 confining phases Coulomb staggered v t 0 1 v/t deconfinement microscopic model! Artificial electrodynamics with ice as ether Wen s noodle soup MF 8 TF RK

26 Summary frustration degeneracy strong fluctuations new phases/phase transitions/dynamics simple model systems many realisations materials physics nanotechnology cold atoms