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: PrFe4P12

• rdering transition of Pr3+ ions ⇒ 4f2 shells

PrFe4P12: metamagnetic transition…

• Y. Aoki et al, Phys. Rev. B 65, 06446 (2002)

PrFe4P12: metamagnetic transition antiferro-quadrupolar order replaced by large dipole polarization

• A. Kiss and P. Fazekas:
• J. Phys.: Condens. Matter 15,

S2109 (2003)

PrFe4P12: antiferro-quadrupolar NpO2: antiferro-octupolar

AFM-looking susceptibility cusps belong often not to AFM at all, but multipolar order: The true order is hidden URu2Si2 (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

• f 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 NpO2 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)

|⇑ > <⇑ | , |⇑ > < ⇓ | |⇓ > < ⇑ | , | ⇓> <⇓ |

4 independent operators

1 = |⇑ > <⇑ | + | ⇓> <⇓ | non-trivial order parameters: Sz= |⇑ > <⇑ | - | ⇓> <⇓ | S+ = |⇑ > < ⇓ | , S- = | ⇑ > < ⇑ | , Or alternatively Sx , Sy , Sz

Γ6 ⊗ Γ6 = Γ1 +Γ4 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 n2-1 independent local order parameters

|φ1> <φ1| |φ1> <φ2| … |φ1> <φn| |φ2> <φ1| |φ2> <φ2| … |φ2> <φn| . . |φn> <φ1| |φn> <φ2| … |φn> <φn|

( |φ1> <φ1|+ |φ2> <φ2| +…+|φn> <φn| = 1 is trivial ) In general, the n2-1 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 → Lx, y → Ly, z → Lz may be used

(for f-electrons, x → Jx, y → Jy, z → Jz )

five quadrupole operators

5x2 =10 d-shell cubic field 3x2 =6 2x2 =4

Γ3 Γ5

quadrupolar basis free ion (l=2) 5-dimensional

• rbital space

3-dimensional

• rbital space

2-dimensional

• rbital space
• r

residual unfrozen orbital moment “l=1”

local order parameters in the 2-dimensional orbital space τz = +1/2 |3z2-r2> τz = -1/2 |x2-y2> pseudospin states related to Qzz quadrupole moment Qzz

= -2

Qzz = 2

• +

=τz

+

= 1 = (τz++τ-)/2= τx

Non-trivial order parameters

• = (τz++τ-)/2I =τy

( ) ( ( ) )

Γ3 ⊗ Γ3 = Γ1 +Γ2 +Γ3 the τ=1/2 local Hilbert space supports Γ3 quadrupolar and Γ2 quadrupolar order

The nature of the order parameters τx, τy, τz

Assume τx, τy, τz acts like effective fields

How do they polarize?

τz = ±1/2 |x2-y2>= |3z2-r2>= real orbital order τx = ±1/2 1/√2(|x2-y2> + |3z2-r2>) 1/√2(|x2-y2> - |3z2-r2>) also real orbital order (rotated basis). However τy = ±1/2 1/√2(|x2-y2> - i|3z2-r2>) 1/√2(|x2-y2> + i|3z2-r2>) basis states for complex orbital order

if there is an intersite interaction which acts on quadrupolar moments H12 = λ Qzz(R1) Qzz(R2) then quadrupolar order follows ferroquadrupolar

• rder

antiferroquadrupolar

• rder

+ I =

• ctupolar state:

currents flow in atomic shell, but net magnetic moment is zero

+

• +

+

• ctupole moment, octupole order

2000: observation of octupolar ordering in NpO2

• P. Santini and G. Amoretti, Phys. Rev. Lett. 85, 2188 (2000)

Octupolar eigenstate in the Γ3={x2-y2,3z2-r2} subspace Txyz =

ANTIFERRO-OCTUPOLAR ORDER Ψ Ψ* Ψ Ψ*

violates time reversal invariance without magnetism

Order parameters: Cubic symmetry Tetragonal symmetry T2u [Tx

b , Ty b , Tz b]

T1u [Jx , Jy , Jz] A2u Txyz T2g [Oxy , Oyz , Ozx] Eg [O2

0 , O2 2]

Tz

β

B2u time reversal [Jx , Jy] , [Tx

β , Ty β]

Eu odd Jz A2u odd Txyz B1u odd [Oyz , Ozx] Eg even Oxy B2g odd O2

2

B1g even O2 A1g even JxJy(Jx

2-Jy 2) A2g even JxJyJz (Jx 2-Jy 2) A1u odd

hexadecapole lxly(lx

2-ly 2)

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

triakontadipole: 32-magnetic-pole lxlylz (lx

2-ly 2)

Thanks: K. Radnóczi

current flow and charge contours for lowest eigenstate in l=4 free g-shell

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
• rdering in the presence of a magnetic field?

magnetic field applied in certain magic directions does not interfere with the relevant

• ctupolar currents

Ordering in the Γ5u = { (Jx

2-Jy 2)Jz, (Jy 2-Jz 2)Jx, (Jz 2-Jx 2)Jy } octupolar OP triplet

Field induces {Hx, Hy, Hz } ⇒ {Jx, Jy, Jz } so H||(111) induces no Γ5u octupoles

H||(111) sharp transitions H||(001) splitting into two sharp transitions H||(123) transition smeared (Annamária Kiss, P.F.)

PHYSICAL REVIEW B 68, 174425 (2003)

LiNiO2 NaNiO2 triangular layers of Ni ions isostructural and isoelectronic systems with very different phases NaNiO2: orbital order and low-T magnetic order LiNiO2: spin-orbital fluctuations prevent ordering down to T=0 ? interplay and mutual frustration

• f orbital and spin degrees of freedom

F Vernay, K Penc, PF, F Mila: PRB 70 (2004) 014428

Symmetry constraints on intersite interaction geometry time reversal

Jp double hopping JH intra-atomic exchange

n12 n14 n16

Higher symmetry at special parameter values Jp=JH=0, t=t’ SU(4) symmetry

• rbital exchange: just as important as spin exchange

1 2 3 4 effective orbital ordering interaction favours spin ferromagnetism,

• rbital antiferromagnetism/liquid

the arising of the

• rbital singlet

Two sites: parallel spins staggered orbitals antiparallel spins uniform orbitals

Jt/Js=(U-JH)/U

tb/ta pair problem: exact diagonalization

Jt/Js=(U-JH)/U

Inferred extrapolation to the lattice

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

ferro-orbital antiferromagnetic chains

Dimer phases

Uniform orbital order Jahn-Teller active strong 1-dim AF spin correlations interchain character varying within D

• ur alternative

for NaNiO2

Orbital correlations give rise to unfrustrated 2-dim antiferromagnet weak interchain orbital correlations non-trivial orbital character

• rbital mean field Ansatz

not suitable

SU(4) phase possibly 12-site plaquette state

• 1

2 1 2

• rbital singlet: analogous to spin singlet

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

spin singlet

• rbital singlet

Spin ferromagnet F1 three-sublattice orbital order

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

• rbital spin
• rdering
• rdering

real complex magnetic |a ± ib〉 = |Lz ± 1〉 = |τy ± 1/2〉 |Sz ± 1/2〉 |τz ± 1/2〉

trigonal symmetry mixes dipoles and octupoles

Experimental results on URu2Si2

• phase transition at 17.5K
• magnetic moment ∼ 0.03 µB

question: the order parameter time-reversal invariant

• r breaks

time-reversal invariance quadrupolar order proposed by Santini

• T. T. M. Palstra et al.,

PRL 55 2727 (1985)

• P. Santini and G. Amoretti, PRL 73 1027 (1994)

Proposed by us (A. Kiss, P.F.,

• Phys. Rev. B 2005

In all samples, two-sublattice atiferromagnetism, but “micromoments” 0.02-0.04 µB Correspondence to hidden order imprecise Now understood to be extrinsic Hidden order is non-magnetic, confined to a low-pressure phase Current understanding: micromoments are in fact ordinary large moments in a small volume fraction Extrinsic in a sense which is still unclear

symmetry change (however, the same superstructure) Scenario A Scenario B now confirmed now dropped

Hidden order and antiferromagnetism

• f the same symmetry

Order parameters: Cubic symmetry Tetragonal symmetry T2u [Tx

b , Ty b , Tz b]

T1u [Jx , Jy , Jz] A2u Txyz T2g [Oxy , Oyz , Ozx] Eg [O2

0 , O2 2]

Tz

β

B2u time reversal [Jx , Jy] , [Tx

β , Ty β]

Eu odd Jz A2u odd Txyz B1u odd [Oyz , Ozx] Eg even Oxy B2g odd O2

2

B1g even O2 A1g even JxJy(Jx

2-Jy 2) A2g even JxJyJz (Jx 2-Jy 2) A1u odd

major evidence of time reversal invariance breaking hidden order: uniaxial stress induces large-amplitude antiferromagnetism Yokoyama et al (2003)

Tz

β

Mz [µB]

T=0 (ground state) Octupolar scenario: uniaxial stress induces a quadrupole density which combines with staggered octupolar order to give similarly staggered dipoles This is our explanation for the results found by Yokoyama et al (2003) A Kiss, PF: PRB 71 (2005) 2209

χ1

χ3

current flow for lowest octupolar eigenstate in l=4 free g-shell

current flow for lowest octupolar eigenstate for {t1, t3} subspace in the l=4 g-shell

Conclusions from NQR measurement:

• no quadrupolar ordering at T0 = 17K
• electric field gradient at Ru site (interpreted as quadrupolemoment at U site)

appears at T*=13.5K The possibility of two consecutive hidden orders?

symmetry lowering in a Txyz molecular field Possibilities: Oxy, Jz {Oxz, Oyz} , { Jy , Jx} Jx

2-Jy 2 , JxJyJz(Jx 2-Jy 2)

JxJy(Jx

2-Jy 2) , Jz(Jx 2-Jy 2)

A combination of 4 Ru hexadecapoles can imitate a quadrupole at the central U site

PF, A Kiss, K Radnóczi: cond-mat/0506504

Scenario for two hidden order transitions: The appearance of coupled dipole-quadrupole order

• n the background of occtupolar order

Thank you for your attention