Anisotropic Hubbard model on a triangular lattice -- spin dynamics in Ho Mn O_3

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by  Saptarshi Ghosh

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: [e-mail] J. R. D. Copley, and H. Takagi, Phys. Rev. B 68, 014432
† Electronic address: [e-mail] (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,