Anisotropic Hubbard model on a triangular lattice — spin dynamics in HoMnO 3 Saptarshi Ghosh∗ and Avinash Singh† Department of Physics, Indian Institute of Technology Kanpur - 208016 The recent neutron-scattering data for spin-wave dispersion in HoMnO3 are well described by an anisotropic Hubbard model on a triangular lattice with a planar (XY) spin anisotropy. Best fit indicatesthatmagneticexcitationsinHoMnO3 correspondtothestrong-couplinglimitU/t>∼15, with planar exchangeenergy J =4t2/U ≃2.5meV and planar anisotropy ∆U ≃0.35meV. 6 0 0 2 There has been renewed interest in correlated elec- scale. Weexaminethebehaviourofspin-waveanisotropy tron systems on triangular lattices, as evidenced by re- gapwithspinanisotropyandalsosuggestasensitivemea- n cent studies of antiferromagnetism, superconductivity sure of finite-U, double occupancy effects. a J andmetal-insulatortransitionintheorganicsystemsκ However,theMnspinplanaranisotropyistreatedhere (BEDT TTF) X,1,2 the discoveryofsuperconductivit−y only at a phenomenological level, equivalent to the ef- 6 2 in NaxC−oO2.yH2O,3 the observation of low-temperature fective anisotropy DSi2z included in recent investigations ] insulating phases in some √3-adlayer structures such using spin models.7,8 In a detailed study of single-ion el as K on Si[111],4 and quasi two-dimensional 1200 spin anisotropy and crystal-field effects in the layered rare- - ordering and spin-wave excitations in RbFe(MoO ) earth cuprates R CuO (R=Nd,Pr,Sm), the magnetic r 4 2 2 4 t (Refs. 5,6) and the multiferroic materials YMnO and behaviour (including spin-reorientation transitions) has s 3 . HoMnO3.7,8 been attributed to coupling of Cu with the rare-earth t a Recent neutron-scattering studies of the multiferroic magnetic subsystem which exhibits a large single-ion m materialHoMnO haverevealedanon-collinear1200 an- anisotropyresultinginpreferentialorderingofrare-earth 3 - tiferromagnetic (AF) ordering below TN 72 K of the moments along specific lattice directions.12 It has been d S = 2 Mn3+ spins arranged in offset ≈layers of two- suggested that the anisotropy of Mn spins, its observed n dimensional (2D) triangular lattice.7 Measurements of temperature dependence, and the reorientation transi- o c the spin wavedispersionwere found to be well described tionsinHoMnO3alsooriginatefromasimilaranisotropic [ byanearest-neighbourHeisenbergAFwithexchangeen- exchangecouplingwiththe rare-earthHolmium,7 result- ergy J = 2.44 meV and a planar anisotropy D = 0.38 ing in magnetic behaviour as seen in layered rare-earth 3 meV at 20 K. No discernible dispersion was observed in cuprates, where frustrated interlayer coupling allows for v 5 the out-of-plane direction, indicating primarily 2D spin weaker,higher-orderinteractionstocontrolthemagnetic 7 dynamics. structure. Especially relevant for non-collinear ordering, 4 Recently spin-waveexcitationsinthe 1200 AFstateof the anisotropic exchange (Dzyaloshinski-Moriya) inter- 6 action D.(S S ) originating from spin-orbit coupling the Hubbard model on a triangular lattice were studied i j × 0 has been suggested as responsible for the clamping of within the random phase approximation (RPA) in the 5 full U range.9 The spin wave energy in the large U limit ferroelectric and antiferromagnetic order parameters in 0 YMnO .13 / was shown to asymptotically approach the correspond- 3 t Hund’s rule coupling responsible for the S = 2 spin a ing result for the Quantum Heisenberg antiferromagnet state of Mn+++ ions and crystal-field splitting have m (QHAF), thus providing a continuous interpolation be- also not been realistically incorporated here. However, tween weak and strong coupling limits. However, com- - d peting interactions and frustration were found to signifi- these realistic details do not qualitatively affect the n cantlymodifythedispersionatfiniteU,resultinginvan- spin-rotation symmetry and spin dynamics, as discussed o below. Hund’s rule coupling in the generalized Hub- ishing spin stiffness at U 6 and a magnetic instability c ≈ bard model considered here is maximal as inter-orbital at U 7 corresponding to vanishing spin-wave energy : v at wav≈e vector q = (π/3,π/√3). The sharp fall-off of Coulomb interaction for parallel spins is dropped com- i M pletely. Including an inter-orbital density-density inter- X ωM near U 7 provides a sensitive indicator of finite-U ≈ action V and an intra-atomic exchange interaction F effects in the AF state. Indeed, recent high-resolution 0 0 r favouringparallel-spinalignment(Hund’srulecoupling), a neutron-scattering studies of the spin-wave dispersion in the more realistic orbital Hubbard model14,15 the square-latticeS=1/2AFLa CuO haverevealedno- 2 4 ticeable spin-wave dispersion along the MBZ edge,10 as- H = t a† a +U n n sociated with finite-U double-occupancy effects.11 − iγσ i+δ,γσ iγ↑ iγ↓ i,δ,γ,σ iγ In this brief report we extend the spin-wave analy- X X sis to include planar spin anisotropy, and show that the + (V0 δσσ′F0)niγσniγ′σ′ − neutron-scattering data for HoMnO3 are also well de- i,γ<Xγ′,σ,σ′ spcrroivbieddinbgyaaHmuicbrboascrodpmicoddeeslcornipatitorniaonfgutlhaerlmatotsitcee,stsheuns- + F0 a†iγσaiγ′σa†iγ′σ′aiγσ′ (1) i,γ<γ′,σ6=σ′ tial features of spin dynamics in the 1200 AF state of X HoMnO , including the spin-wavedispersionand energy remains spin-rotationally invariant under a global rota- 3 2 σ tion of the fermion spin S = Ψ† Ψ (where Ψ† for orbital γ then reduces to iγ iγ 2 iγ iγ ≡ (a† a† ) is the fermion field operator), even if orbitals iγ↑ iγ↓ Hγ = σ .∆ (5) γ are identified with the Mn orbitals (t2g,eg) resulting int − iγ i from crystal-field splitting of the atomic 3d orbitals. As Xi the intra-atomic exchange interaction F0 is much larger where the self-consistently determined mean field ∆ = than the spin excitation energy scale (∼ U+t2F0, within a U2hSiγ′iHF lies in the x −y plane in spin space. Tihe strong-couplingexpansion),allfermionspinsonasiteare HF sublattice magnetization depends only on U and is 2 effectivelycoupled,yieldingacompositequantumspinS determinedfromtheself-consistencycondition S = σ h αiHF and a corresponding multiplication by factor 2S to ob- k,l k,l intermsoftheHFstates k,l onsub- k,lh |2| iα | i tain the effective spin-wave energy scale. Therefore, or- lattice α, exactly as for the isotropiccase.9 In the strong P bital multiplicity does not change the Goldstone-mode coupling limit the sublattice magnetization S 1/2 α structure, and the spin-dynamics energy scale in the or- at the HF level, and is reduced by about 5h0%i(≈in the bital Hubbard model is essentially determined by t and isotropic case) due to quantum fluctuations, as found in Ueff =U +F0. different calculations cited in Ref. [9]. As asimplestextensionto phenomenologicallyinclude We consider the 1200 ordered AF state on the trian- spin-space anisotropy, while retaining only the relevant gular lattice, and examine transverse spin fluctuations energy scales t and Ueff, we consider the generalized - aboutthebroken-symmetrystate. AttheRPAlevel,the orbital Hubbard model16 N magnon propagator reduces to a sum of all bubble dia- grams where the interaction vertices involving S2 and H = −t a†iγσai+δ,γ,σ+ U1 a†iγ↑aiγ↑a†iγ′↓aiγ′↓ Si2y appear with interaction U2, whereas those invioxlving i,Xδ,γ,σ N iX,γ,γ′ Si2z with interaction U1. Introducing a planar spin rota- U tion, so that spins are oriented along the x′ direction for + 2 a†iγ↑aiγ′↑a†iγ′↓aiγ↓ (2) all three sublattices, we obtain for the transverse spin- N iX,γ,γ′ fluctuation propagator onatriangularlatticewithnearest-neighbour(NN)hop- [χ0(q,ω)] pingbetweensitesiandi+δ. Hereγ,γ′refertothe(ficti- [χ(q,ω)]µν = (6) αβ 1 2[U][χ0(q,ω)] tious)degenerate orbitalspersite. Thefactor 1 isin- − N N cluded to renderthe energydensity finite in the inthe2 3spin-sublatticebasisofthetwotransversespin N →∞ limit. The two correlation terms involve density-density directio⊗ns µ,ν = y′,z′ and the three sublattices α,β = and exchange interactions with respect to the orbitalin- A,B,C. The sublattice-diagonal interaction matrix dices. The Hartree-Fock (HF) approximation and Ran- dom Phase Approximation (RPA) are of O(1) whereas U 1 0 [U]= 2 (7) quantumfluctuationeffectsappearathigherorderwithin 0 U 1 1 the inverse-degeneracyexpansionandthus 1/ ,inanal- (cid:20) (cid:21) N ogywith1/S forquantumspinsystems,playsthe roleof in the y′,z′ basis, and the bare particle-hole propagator ¯h. The key feature of spin-rotationsymmetry of the gen- [χ0(q,ω)]µν αβ eralizedHubbardmodelishighlightedbywritingthetwo interaction terms as = 1 hσµi−α+hσνiβ−+∗ + hσµi+α−hσνi+β−∗ H = U2 S .S + U2−U1 S2 (3) 4kX,l,m"Ek+−q,m−Ek−,l+ω Ek+,m−Ek−−q,l−ω# int − i i iz (8) N i N i X X involves integrating out the fermions in the broken- in terms of the total spin operator symmetry state. In the particle-hole matrix elements σ σ Si = ψi†γ 2ψiγ ≡ 2iγ (4) hσµi−α+ ≡hk−q,m|σµ|k,liα (9) γ γ X X oftherotatedspins,thespinorientationanglesφ inthe α where ψ† (a† a† ). An Ising (uniaxial) anisotropy fermion states k,l are transformed out. The numerical is obtainiγed≡foriUγ↑ >iγ↓U , a planar (XY) anisotropy for evaluationof[χ|0(qi,ω)]intermsoftheHF-levelAF-state 1 2 U >U , and full spin-rotation symmetry for U =U . energies and amplitudes is exactly as for the isotropic 2 1 1 2 AsappropriateforHoMnO ,weconsiderthecaseU > case studied earlier.9 3 2 U corresponding to preferential ordering of spins in the The spin-wave energies ω are then obtained from the 1 q x y plane in spin space and an anisotropy gap for out- poles1 λ (ω )=0ofEq. (6)intermsoftheeigenvalues q q of−-planeexcitations.7Magneticexcitationswereanalyzed λ (ω) −of the matrix 2[U][χ0(q,ω)]. As ω corresponds q q intermsofaHeisenbergmodelwithasimilaranisotropy to spin 1/2, it is scaled by the factor 2S for arbitrary term in Ref. [7]. At the HF level, the interaction term spinS.17 Asexpectedforplanaranisotropy,thereisonly 3 1 2200 q = (π/3,π/√3) M mmooddee 11 22 33 eexxpptt 1155 0.8 )) VV ee mm 1100 (( ωω 0.6 K M 3 55 2, ω (cid:0) / 1 ω ∆U/t = 0 0.4 0.005 00 11 00..88 00..66 00..44 00..22 00 0.015 ((22--KK//22,,KK)) 0.025 MM KK ΓΓ 0.035 0.045 2200 0.2 0.055 mmooddee 11 22 33 eexxpptt 1155 )) 0 VV ee 6 8 10 12 14 16 18 20 mm 1100 (( U/t ωω 55 FIG. 2: (color online) The ratio ω1/ω2,3 at wave vector qM provides a sensitive indicator of finite-U, double-occupancy effects, shown for different valuesof spin anisotropy ∆U. 00 1 11 11..22 11..44 11..66 11..88 22 22..22 22..44 ΓΓ MM ((HH,,00)) ΓΓ MM 0.8 FIG. 1: (color online) Spin-wave dispersion for the three 0.6 modescalculatedfromEq. (5)withJ =4t2/U ≈2.5meVand ap g ∆U ≈0.35meV,alongwithneutronscatteringdatapointsfor ω 0.4 HoMnO3 at 20K from Ref. [7]. 0.2 one Goldstone mode correspondingto planar rotationof spins, and the two out-of-plane modes become massive 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 with an anisotropy gap ω . gap ∆U/U Spin-wave dispersion ω for the three modes is shown q in Fig. 1 alongtwo symmetry directions in the magnetic Brillouinzone (MBZ), along with the neutron-scattering FIG. 3: Spin-wave anisotropy gap corresponding to out-of- planefluctuationsshowsa∼(∆U/U)1/2 behaviourwithspin dataforHoMnO at20KfromRef. [7]. Theanisotropic 3 anisotropy. Here U/t=16. Hubbard model provides a remarkably good description of the spin dynamics. We find that best fits with the magnetic excitations in HoMnO are only obtained in 3 the strong-coupling limit U/t > 15, with a planar ex- U/t = 40. Spin-wave dispersion calculated in the inter- change energy J = 4t2/U 2∼.5meV and anisotropy mediate coupling regime cannot be fitted with the neu- U U ∆U 0.35meV,≃the individual values of t tron scattering data, indicating no evidence of finite-U 2 1 and−U no≡t being≃resolvable within experimental resolu- double occupancy effects, as discussed below. tion. ToestimatetheorderofmagnitudeoftheratioU/t, For the square-lattice S = 1/2 AF La CuO , finite- 2 4 ifwe nominallytakeU =1eV,weobtaint=25meVand U,double-occupancyeffectsassociatedwithhigher-order 4 (t4/U3)spincouplingsaremanifestedinnoticeablespin- wavedispersionwithRPAcalculationsallowsforaquan- wavedispersionalongthe MBZboundary.10,11 Similarly, titative determination of the effective anisotropy, in ad- for the isotropic triangular-lattice AF, the ratio ω /ω dition to highlighting any finite-U, double occupancy ef- 1 2,3 ofthe non-degenerateanddegeneratespin-waveenergies fects as seen in La CuO . The recentneutron-scattering 2 4 at wave vector q = (π/3,π/√3) is a sensitive mea- data for spin-wave dispersion in HoMnO are found to M 3 sure of the U/t ratio,9 which asymptotically approaches be well described by an anisotropic Hubbard model on 2/√10 = 0.632 in the strong-coupling (U/t ) a triangular lattice with a planar (XY) spin anisotropy. → ∞ limit, decreases monotonically with U/t, and eventually We find that the twin constraints ωgap/ω1,2 1/3 and ≈ vanishes at U/t ≈ 7.9 Variation of ω1/ω2,3 with U/t ω1/ω2,3 ≈ 2/3 in the neutron scattering data cannot be is shown in Fig. 2 for different values of anisotropy satisfied in the intermediate coupling regime, and best ∆U/t. Neutron-scattering data for HoMnO3 shows that fit indicates that magnetic excitations in HoMnO3 cor- ω /ω 11meV/16meV 0.7, which yields a lower respond to the strong-coupling limit U/t > 15, with 1 2,3 ≈ ≈ ∼ bound ( 15) on the ratio U/t from Fig. 2. J 2.5meV and ∆U 0.35meV. The ratio ω1/ω2,3 of ∼ ≃ ≃ Thespin-waveanisotropygapω /t,correspondingto thenon-degenerateanddegeneratespin-waveenergiesat gap out-of-planefluctuationswithq =0,variesas(∆U/U)1/2 wave vector qM = (π/3,π/√3) is suggested as a sen- withspinanisotropy(Fig. 3),whichtranslatestoω sitive measure of finite-U, double occupancy effects in gap √J∆U, the geometric mean of the two energy scales. ∝ a triangular-lattice antiferromagnet. Finally, in view of the formal resemblance with the orbital Hubbard model Inconclusion,theanisotropicHubbardmodelprovides (1),ourRPAcalculationprovidesastepforwardtowards a simple extension to phenomenologically include spin- investigating magnetic excitations using realistic models space anisotropy and study spin excitations in the full includingHund’srulecoupling,crystal-fieldsplittingetc. rangefromweaktostrongcoupling. Comparisonofspin- ∗ Electronic address: [email protected] J. R. D. Copley, and H. Takagi, Phys. Rev. B 68, 014432 † Electronic address: [email protected] (2003). 1 K. Kanoda, Physica C 282-287, 299 (1997); K. Kanoda, 9 A. Singh,Phys. Rev.B 71, 214406 (2005). HyperfineInteract. 104, 235 (1997). 10 R.Coldea, S.M. Hayden,G. Aeppli,T. G. Perring, C. D. 2 R. H.McKenzie, Science, 278, 820 (1997). Frost,T.E.Mason,S.-W.Cheong,andZ.Fisk,Phys.Rev. 3 K. Takada et al.,Nature422, 53 (2003). Lett. 86, 5377 (2001). 4 H. H. Weitering, X. Shi, P. D. Johnson, J. Chen, N. J. 11 A.SinghandP.Goswami,Phys.Rev.B66,92402(2002). Dinardo,andS.Kempa,Phys.Rev.Lett.78,1331(1997). 12 R.Sachidanandam,T.Yildirim,A.B.Harris,A.Aharony, 5 T. Inami, Y. Ajiro, and T. Goto, J. Phys. Soc. Jpn. 65, and O. E.-Wohlman, Phys. Rev.B 56, 260 (1997). 2374 (1996). 13 E. Hanamura, K. Hagita, and Y. Tanabe, J. Phys.: Con- 6 L. E. Svistov, A. I. Smirnov, L. A. Prozorova, O. A. Pe- dens. Matter 15, L103 (2003). trenko, L. N. Demianets, and A. Ya. Shapiro, Phys. Rev. 14 L. M. Roth, Phys.Rev. 149, 306 (1966). B 67 094 434 (2003). 15 K. Held and D. Vollhardt, Eur. Phys.J. B 5, 473 (1998). 7 O. P. Vajk, M. Kenzelmann, J. W. Lynn, S. B. Kim, and 16 A. Singh,Phys. Rev.B 43, 3617 (1991). S.-W. Cheong, Phys. Rev.Lett. 94, 087601 (2005). 17 A. Singh and P. Sen,Phys. Rev.B 57, 10598 (1998). 8 T. J. Sato, S. -H. Lee, T. Katsufuji, M. Masaki, S. Park,