Multipolar ordering in d - and f -electron systems P. Fazekas - PowerPoint PPT Presentation

Multipolar ordering in d - and f -electron systems P. Fazekas Budapest, Hungary Workshop on Correlated Thermoelectric Materials Hvar, September 25-30 2005

Example: PrFe 4 P 12 ordering transition of Pr 3+ ions ⇒ 4f 2 shells

PrFe 4 P 12 : metamagnetic transition… Y. Aoki et al, Phys. Rev. B 65 , 06446 (2002)

PrFe 4 P 12 : metamagnetic transition antiferro-quadrupolar order replaced by large dipole polarization A. Kiss and P. Fazekas: J. Phys.: Condens. Matter 1 5, S2109 (200 3 )

AFM-looking susceptibility cusps belong often not to AFM at all, but multipolar order: The true order is hidden PrFe 4 P 12 : antiferro-quadrupolar NpO 2 : antiferro-octupolar URu 2 Si 2 (we suggest) staggered octupolar

Magnetism (1900--): the ordering of atomic dipoles (often simply spins) due to quantum mechanical exchange In addition to magnetic dipoles, the atomic shell can support electric quadrupole, magnetic octupole, etc. moments as order parameters The study of multipolar order is logically the next step in extending the discipline of magnetism. The interaction of quadrupoles, etc. is not inherently weaker than dipole coupling, their ordering is not a secondary effect compared to analogous phenomena in ordinary magnetism. 1974– orbital order in transition metal oxides quadrupolar order in rare earth and actinide systems 2000– octupole ordering in NpO 2 2002– novel heavy fermionic state and exotic superconductivity mediated by quadrupolar fluctuations in Pr-filled skutterudies 2000– re-evaluation of the role of orbital ordering in the phase transitions of titanium, vanadium, mangan, iron, Ru, etc compounds

The concept of local order parameters Simplest example: two-dimensional local Hilbert space of an S=1/2 spin Basis states | ⇑ > and | ⇓ > Local order parameters: all the linear operators (observables) | ⇑ > < ⇑ | , | ⇑ > < ⇓ | 1 = | ⇑ > < ⇑ | + | ⇓ > < ⇓ | non-trivial order parameters: | ⇓ > < ⇑ | , | ⇓ > < ⇓ | S z = | ⇑ > < ⇑ | - | ⇓ > < ⇓ | S + = | ⇑ > < ⇓ | , S - = | ⇑ > < ⇑ | , 4 independent operators Or alternatively S x , S y , S z Γ 6 ⊗ Γ 6 = Γ 1 + Γ 4 the S=1/2 local Hilbert space supports usual magnetic (3-dimensional vector) order The same recipe works in more complicated cases

The possibilities of ordering: If the local Hilbert space is n-dimensional, there are n 2 -1 independent local order parameters | φ 1 > < φ 1 | | φ 1 > < φ 2 | … | φ 1 > < φ n | | φ 2 > < φ 1 | | φ 2 > < φ 2 | … | φ 2 > < φ n | . . | φ n > < φ 1 | | φ n > < φ 2 | … | φ n > < φ n | ( | φ 1 > < φ 1 |+ | φ 2 > < φ 2 | +…+| φ n > < φ n | = 1 is trivial ) In general, the n 2 -1 operators represent magnetic dipoles, electrical quadrupoles, magnetic octupoles, etc.; multipolar order parameters

electrical quadrupole moments measure the deviation from spherical charge distribution within the given L=2 subspace, x → L x , y → L y , z → L z may be used (for f-electrons, x → J x , y → J y , z → J z ) five quadrupole operators

free ion ( l =2) cubic field 2x2 =4 2-dimensional Γ 3 orbital space quadrupolar basis 5x2 =10 d-shell 3x2 =6 3-dimensional Γ 5 orbital space or 5-dimensional orbital space residual unfrozen orbital moment “ l =1”

local order parameters in the 2-dimensional orbital space τ z = -1/2 τ z = +1/2 |x 2 -y 2 > |3z 2 -r 2 > pseudospin states related to Q zz quadrupole moment = -2 Q zz Q zz = 2

+ = 1 Non-trivial order parameters - ) = τ z ( ( ) + = ( τ z+ + τ - )/2= τ x ( ) - = ( τ z+ + τ - )/2I = τ y Γ 3 ⊗ Γ 3 = Γ 1 + Γ 2 + Γ 3 the τ =1/2 local Hilbert space supports Γ 3 quadrupolar and Γ 2 quadrupolar order

The nature of the order parameters τ x , τ y , τ z Assume τ x , τ y , τ z acts like effective fields How do they polarize? τ z = ± 1/2 |x 2 -y 2 >= |3z 2 -r 2 >= real orbital order τ x = ± 1/2 1/ √ 2(|x 2 -y 2 > + |3z 2 -r 2 >) 1/ √ 2(|x 2 -y 2 > - |3z 2 -r 2 >) also real orbital order (rotated basis). However τ y = ± 1/2 1/ √ 2(|x 2 -y 2 > - i|3z 2 -r 2 >) 1/ √ 2(|x 2 -y 2 > + i|3z 2 -r 2 >) basis states for complex orbital order

if there is an intersite interaction which acts on quadrupolar moments H 12 = λ Q zz ( R 1 ) Q zz( R 2 ) then quadrupolar order follows ferroquadrupolar order antiferroquadrupolar order

octupole moment, octupole order - - = + + I + - + octupolar state: currents flow in atomic shell, but net magnetic moment is zero 2000: observation of octupolar ordering in NpO 2 P. Santini and G. Amoretti, Phys. Rev. Lett. 8 5, 2188 (200 0 )

Octupolar eigenstate in the Γ 3 ={x 2 -y 2 ,3z 2 -r 2 } subspace T xyz =

ANTIFERRO-OCTUPOLAR ORDER Ψ * Ψ Ψ * Ψ violates time reversal invariance without magnetism

Order parameters: Cubic symmetry Tetragonal symmetry T z β B 2u time reversal T 2u [T x b , T y b , T z b ] [J x , J y ] , [T x β , T y β ] E u odd T 1u [J x , J y , J z ] J z A 2u odd A 2u T xyz T xyz B 1u odd T 2g [O xy , O yz , O zx ] [O yz , O zx ] E g even O xy B 2g odd E g [O 2 0 , O 2 2 ] O 2 2 B 1g even O 2 A 1g even 0 J x J y (J x 2 -J y 2 ) A 2g even J x J y J z (J x 2 -J y 2 ) A 1u odd

lowest eigenstate in l =4 free g -shell hexadecapole 2 -l y 2 ) l x l y (l x shown: charge cloud and signs of lobes of the (real) eigenstates

current flow and charge contours for triakontadipole: 32-magnetic-pole lowest eigenstate in l =4 free g -shell 2 -l y 2 ) l x l y l z (l x Thanks: K. Radnóczi

Interaction = dipole-dipole (3 terms) + quadrupole-quadrupole (5 terms) + octupole-octupole (7 terms) + dipole-octupole +quadrupole-hexadecapole+ …triakontadipole ..+ … In principle, many different phases, multicritical points and lines, varied response to external fields

• Since octupolar ordering as a symmetry breaking transition violates time reversal invariance without magnetic moments • and since an external magnetic field destroys time reversal invariance • Is it possible to have spontaneous symmetry breaking by octupolar ordering in the presence of a magnetic field? magnetic field applied in certain magic directions does not interfere with the relevant octupolar currents Ordering in the Γ 5u = { (J x 2 -J y 2 )J z , (J y 2 -J z 2 )J x , (J z 2 -J x 2 )J y } octupolar OP triplet Field induces {H x , H y , H z } ⇒ {J x , J y , J z } so H||(111) induces no Γ 5u octupoles H||(123) H||(111) H||(001) transition smeared sharp transitions splitting into two sharp transitions (Annamária Kiss, P.F.) PHYSICAL REVIEW B 6 8, 174425 (200 3)

LiNiO 2 NaNiO 2 triangular layers of Ni ions interplay and mutual frustration isostructural and isoelectronic of orbital and spin degrees of freedom systems with very different phases NaNiO 2 : orbital order and low-T magnetic order LiNiO 2 : spin-orbital fluctuations prevent ordering down to T=0 ? F Vernay, K Penc, PF, F Mila: PRB 70 (2004) 014428

Symmetry constraints on intersite interaction time reversal geometry

n 14 J p double hopping J H intra-atomic exchange n 12 n 16 Higher symmetry at special parameter values J p =J H =0, t=t’ SU(4) symmetry

orbital exchange: just as important as spin exchange 1 2 the arising of the orbital singlet 4 3 effective orbital ordering interaction favours spin ferromagnetism, orbital antiferromagnetism/liquid

Two sites: parallel spins staggered orbitals antiparallel spins uniform orbitals

pair problem: exact diagonalization t b /t a Inferred extrapolation to the lattice J t /J s =(U-J H )/U J t /J s =(U-J H )/U

N=2 N=4 N=16 N=12

ferro-orbital antiferromagnetic chains

Dimer phases

Uniform orbital order Jahn-Teller active strong 1-dim AF spin correlations interchain character varying within D our alternative for NaNiO 2

Orbital correlations give rise to unfrustrated 2-dim antiferromagnet weak interchain orbital correlations non-trivial orbital character orbital mean field Ansatz not suitable

SU(4) phase possibly 12-site plaquette state

orbital singlet: analogous to spin singlet ⇑⇓ - ⇓⇑ - 2 2 1 1 for a 4-site plaquette, a spin-orbital singlet can be defined a higher (non-geometrical) symmetry makes spins and orbitals equivalent spin singlet orbital singlet

Spin ferromagnet F1 three-sublattice orbital order

Spin ferromagnet F3 uniform complex orbital order Jahn-Teller inactive pseudo-hexagonal phase

orbital spin ordering ordering real complex magnetic | S z ± 1/2 〉 | a ± i b 〉 = | L z ± 1 〉 = | τ y ± 1/2 〉 | τ z ± 1/2 〉 trigonal symmetry mixes dipoles and octupoles

Experimental results on URu 2 Si 2 T. T. M. Palstra et al., • phase transition at 17.5K PRL 55 2727 (1985) • magnetic moment ∼ 0.03 µ B question: the order parameter time-reversal invariant or breaks time-reversal invariance quadrupolar order proposed by Santini Proposed by us (A. Kiss, P.F., Phys. Rev. B 2005 P. Santini and G. Amoretti, PRL 73 1027 (1994)