Helimagnons in a chiral ground state of the pyrochlore antiferromagnets

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by  Eunsong Choi

Helimagnons in a chiral ground state of the pyrochlore antiferromagnets
Eunsong Choi,1 Gia-Wei Chern,1,2 and Natalia B. Perkins1
1Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Dated: January 28, 2013)
TheGoldstonemodeinahelicalmagneticphase,alsoknownasthehelimagnon,isapropagating
mode with a highly anisotropic dispersion relation. Here we study theoretically the helimagnon
excitationsinacomplexchiralgroundstateofpyrochloreantiferromagnets suchasspinelCdCr O
2 4
anditinerantmagnetYMn . Weshowthattheeffectivetheoryofthesoftmodesinthehelicalstate
2
3 possesses a symmetry similar to that of smectic liquid crystals. We compute the low-energy spin-
1 wavespectrumbasedonamicroscopicspinHamiltonianandfindadispersionrelationcharacteristic
0 of the helimagnons. By performing dynamics simulations with realistic model parameters, we also
2 obtain an overall agreement between experiment and thenumerical spin-wavespectrum. Ourwork
n thusalso clarifies themechanisms that relive themagnetic frustration in thesecompounds.
a
J
Introduction. Therichphenomenologyassociatedwith the scenario predicted in Ref. [7] that the magnetic frus-
5
2 helicalspinorderinghasattractedconsiderableattention tration is relieved by the softening of a q = 0 optical
recently. Magneticspiralshavebeenshowntoplayacru- phonon with odd parity. A long-range N´eel order with

] cialroleininducingthespontaneouselectricpolarization ordering wavevector q = 2π(0,0,1) is stabilized by the
l
e in multiferroic materials [1]. The soft magnetic fluctua- lattice distortion. Moreover, the lack of inversion sym-
-
r tionsinahelicalspin-density-wavestateofitinerantmag- metry endows the crystalstructure with a chirality. The
t
s net MnSi also holds the key to its non-Fermi liquid be- collinearstateistransformedintoamagneticspiralasthe
t. havior [2]. Contrary to most well known long-rangespin chiralityistransferredtothespinsthroughspin-orbitin-
a
orders such as ferromagnetism and antiferromagnetism, teraction. The shifted ordering vector q = 2π(0,δ,1) is
m
spin-wave excitations in a helical magnetic order is de- consistent with the observed magnetic Bragg peaks.
-
scribed by a theory similar to the elasticity of smectic In this paper we undertake a theoretical calculation
d
n liquid crystals [3, 4]. In particular, despite the broken of the spin-wave spectrum in the helical ground state
o symmetryofthehelicalorderisdescribedbyaO(3)order of pyrochlore antiferromagnet. In particular, our micro-
c parameter,thereonlyexistsasingleGoldstonemodesim- scopiccalculationgoesbeyondthephenomenologicalap-
[
ilar to the smectic-like phonon mode which emerges on proachesadoptedinmoststudiesofmagneticexcitations

2 scaleslargerthanthehelicalpitch[5,6]. Thislow-energy in helimagnets. Based on a well established microscopic
v gapless excitations, dubbed helimagnons in Ref. [5], ex- model for the helical order, we exactly diagonalize the
5
hibit a highly anisotropic dispersion relation. linearized Landau-Lifshitz-Gilbert (LLG) equation with
2
2 The recentdiscoveryofanovelchiralspinstructurein a large unit cell; the size of the unit cell determines the
4 spinel CdCr O has generated much interest both theo- ordering wavevector of the magnetic spiral in the com-
2 4
5. reticallyandexperimentally[7–14]. Averysimilarcopla- mensuratelimit. Bycomparingresultsobtainedfromdif-
0 narhelicalorderhasalsobeenreportedinthe weakitin- ferent helical pitchs, we find a spin-wave spectrum that
2 erantantiferromagnetYMn [15,16]. The magneticions is characteristicof helimagnons. Our work thus provides
2
1 in these compounds form a three-dimensional network a solid example of helimagnons in frustrated pyrochlore
:
v of corner-sharing tetrahedra (Fig. 1), known as the py- lattice. We also performed dynamical LLG simulations
i rochlore lattice. The geometrically frustrated spin inter- on large finite lattices and found an overall agreement
X
actions in this lattice gives rise to an extensively degen- with the experimental data on CdCr2O4.
r
a erate groundstate at the classicallevel [17]. As a result, Model Hamiltonian. Our starting point is a classical
magnetic ordering at low temperatures is determined by spin Hamiltonian that includes nearest-neighbor (NN)
the dominant residual perturbations in the system. exchangeinteractionsandspin-orbitcouplingmanifested
as the Dzyaloshinskii-Moriya (DM) interaction [18, 19]
The helical magnetic order in CdCr O is stabilized
2 4
between NN spins:
by a combined effect of magnetoelastic coupling and rel-
ativistic spin-orbit interactions. Despite a rather high
= (J +K )S S +D (S S ) . (1)
Curie-Weiss temperature |ΘCW|≈70 K, the spiralmag- H Xhijih ij i· j ij · i× j i
netic order sets in only at T =7.8 K, indicating a high
N
degree of frustration. The magnetic transition at T is The isotropic NN exchange can be recast into
N
accompanied by a structural distortion which lowers the (J/2) M(r)2 up to a constant, where M(r) denotes
r| |
crystalsymmetryfromcubictotetragonal[8]. Recentin- thevePctorsumofspinsonatetrahedronatr. Minimiza-
frared absorption measurements indicated a broken lat- tion of this term requires the vanishing of total mag-
tice inversion symmetry below T [12], consistent with netic moment on every tetrahedron but leaves an exten-
N
2
sively degenerate manifold. The lattice distortion intro-
duces the exchange anisotropy K through magnetoe-
ij
lastic coupling. Here we consider tetragonal lattice dis-
tortionsthatpreservethelatticetranslationalsymmetry.
In particular,the odd-parity distortion breaks the inver-
sion symmetry and stabilizes a collinear spin order with
wavevector q = 2π(0,0,1) [20, 21]. The DM term is al-
lowedontheidealpyrochlorelattice,wherethebondsare
not centrosymmetricas requiredby the so-calledMoriya
rules[19]. Moreover,thehighsymmetryofthepyrochlore
latticecompletelydeterminesthe orientationsofthe DM
vectors up to a multiplicative factor. The explicit form
of the vectors D can be found in Ref. [22, 23].
ij
To understandhow the coplanar helicalorder is stabi-
lized, we briefly review the continuum theory basedon a
gradientexpansionoftheorderparameters[7]. Theanti- FIG.1: (Coloronline)Coplanarhelicalorderwithspinslying
ferromagneticorderonthepyrochlorelatticeischaracter- in the ac plane and rotating about the b axis. The magnetic
order corresponds to the solution nˆ = (cosθ(y),0,sinθ(y)),
ized by three staggeredorder parametersL (i=1,2,3). 1
i with θ(y) = Qy + const with Q = 2πδ. It produces a
For example, L = (S +S S S )/4 describes the
staggered spins1on tw0o opp1o−site2[0−11]3and [01¯1] bonds, Bragg peak Q = 2π(0,δ,1) in neutron scattering, consistent
with the experimental measurement on CdCr O . The ex-
2 4
herethe subscriptµ=0∼3denotes the four sublattices change anisotropies are K[0,1,±1] = K[′±1,0,1] = −2Ku, and
ofthepyrochlorelattice(Fig.1). Theorderparameterfor K[±1,0,1] = K[′0,1,±1] = K[1,±1,0] = K[′1,±1,0] = Ku, where K
the collinear order is L1(r)=nˆ1eiq·r, and L2 =L3 =0, andK′ denoteexchangeanisotropyonNNbondsofthegreen
where q=2π(0,0,1),andnˆ1 is anarbitraryunit vector. and red tetrahedra, and the subscript indicates the bond di-
Assumingaslowvariationofstaggeredmagnetizations rection.
in the spiral, we parameterize the order parameters as:
L (r) = eiq·rφ (r)nˆ (r), where nˆ are three orthogonal
i i i i nˆ (r) = cosθ(ξ)eˆ +sinθ(ξ)cˆ, where ξ r eˆ . Sub-
IunnitthveeJctors, φ1lim≈it1, v−an21i(sφh22in+gφof23)t,otaanldmφa2g,nφe3ti≪zatφio1n[o7]n. st1ituting nˆ1(r) inζto Eq. (2) yields an e≡nerg·y ξdensity:
→∞ E[θ]= D∂ θ+K (∂ θ)2/4.MinimizationofE givesa
tetrahedra requires nˆ3 ∂ynˆ1, and the φ fields can be − ξ u ξ
k helical wavenumber
analytically expressed in terms of the director fields nˆ .
i
The effective energy functional in the gradient approxi-
Q=2πδ =dθ/dξ =2D/K . (3)
mation can be solely expressed in terms of the director u
fields nˆ and nˆ ∂ nˆ nˆ [7]:
1 2 k y 1× 1 This rotational symmetry is accidental and is lifted by
furtherneighborinteractionsandcubicsymmetryfieldof
=D dr nˆ aˆ ∂ nˆ +bˆ ∂ nˆ cˆ ∂ nˆ (2)
F Z h 1·(cid:0) × x 1 × y 1− × z 1(cid:1)i the lattice. A similar O(3) symmetry also occurs in the
spiral order of MnSi [5]. In addition, all spirals whose
+KuZ drh(∂xnˆ1)2+(∂ynˆ1)2+2(∂znˆ1)2−(nˆ2·∂znˆ1)2i. axis lies in the ab plane are also accidentally degenerate
withthe spiralinwhichspins rotate aboutthe c axis[7].
Here Ku and D denote the strengths of exchange Theexperimentallyobserved(0,δ,1)spiralisselectedby
anisotropy and DM interactions, respectively. The form alargethird-neighborexchangeJ whichfavorscoplanar
3
of the DM terms suggests spiral solutions with nˆ1 ro- spiralsrotatingabouttheaorbaxes. Theexpliciton-site
tating about one of the principal axes and staying in magnetizations of a spiral with the pitch vector parallel
the plane perpendicular to it. These special solutions to the b-axis are
were first obtained in Ref. 7. For example, the solu-
tion nˆ1 = (cosQy,0,sinQy) describes a helical order S¯µ(r)=aˆ sin(Q r+ϕµ)+ˆc cos(Q r+ϕµ), (4)
· ·
with coplanar spins lying in the ac plane; see Fig. 1.
This spiral magnetic order produces a Bragg scattering where µ=0,1,2,3 is the sublattice index, ϕ =ψ, ϕ =
0 1
atQ=2π(0,δ,1),consistentwiththeexperimentalchar- πδ+ψ, ϕ2 =π+ψ, and ϕ3 =π(1+δ)+ψ, with ψ is an
acterization of CdCr O [8]. arbitrary constant related to the U(1) symmetry of the
2 4
The energy functional actually possesses a contin- coplanar spiral, i.e. θ(r)=Q r+ψ.
F ·
uous symmetry related to the O(2) rotational invari- Helimagnons inthecontinuumlimit. Wenextconsider
ance of the helical axis. To describe the general copla- aphenomenologicalapproachtolow-energymagneticex-
nar spiral solution, we first introduce two orthogonal citations in the helical ground state. For a spiral with a
unit vectors eˆ , eˆ which lie in the xy plane. A copla- fixedaxistheobvioussoftmodesaretheU(1)phasefluc-
ξ ζ
nar spiral rotating about the eˆ direction has the form tuations ψ associatedwith the coplanardirectorfield. A
ξ
3
-3 (a) -2 (b) -2 (c)
x10 x10 x10
6 3 4
δ=0.2 δ=0.2 δ=0.2
5 δ=0.1 2.5 δ=0.1 3.5 δ=0.1
δ=0.05 δ=0.05 δ=0.05
3
4 2
2.5
εk 3 εk1.5 εk 2
1.5
2 1
1
1 0.5
0.5
0 0 0
0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015
(k00) (0k0) (00k)
(d) -1 (e) -2 (f)
x10 x10
5 1 3.5
(k00) (k00)
3
4 0.8
2.5
3 0.6
α ((000k0k)) α β 2
2 0.4 1.5
1
1 0.2
0.5
0 0 0
0.04 0.08 0.12 0.16 0.04 0.08 0.12 0.16 0.04 0.08 0.12 0.16
δ δ δ
FIG. 2: (Color online) Spin-wave dispersions along three principal axes [panels (a)–(c)] obtained from numerical exact diago-
nalization ofalargeunitcell. Theresultsarefittedtothephenomenological dispersion ω=pαk2+βk4. Themagnonenergy
is measured in unit of JS. Panels (d)–(f) show the dispersion parameters α and β as a function of the helical wavenumber δ.
These data are fitted to α=const, α∝δ2, and β ∝δ−2, respectively.
naive guess of the effective energy functional of the soft Lifshitz equation [4]:
modewouldbe K dr ψ 2,similartothatofpla-
u
F ∝ |∇ | ∂S (r) δ
nar magnets. However,Ra phase variation ψ = αζ corre- µ = γS (r) F , (7)
spondingtoarotationofthehelicalaxisisazeromode,a ∂t − µ × δSµ(r)
manifestationoftheU(1)symmetry. Yet,itcostsafinite
where γ is a constant. Substituting S and into the
amount of energy K α2. Consequently, there cannot µ F
be any (∂ζψ)2 term∝inuthe effective energy density. The aEnqd. (∂7)myi=eldsγc(oupcle∂d2diffcer∂e2nt+iacl e∂qu4/aQtio2)nψs.∂tAψss=umγrinmg
correct energy functional for the soft modes is [3–5] t − − k ξ − z z ⊥ ζ
a space-time dependence exp(ik r iωt), we obtain a
· −
highly anisotropic dispersion relation [5]
= 1 dr c (∂ ψ)2+c (∂ ψ)2+c⊥ ∂2ψ 2+rm2 ,(5)
F 2Z h z z k ξ Q2(cid:0) ζ (cid:1) i ω(k)=γr1/2 c k2+c k2+c k4/Q2. (8)
z z k k ⊥ ⊥
q
where c , c K , c J are elastic constants, and
k ⊥ u z 3
the true soft m∼ode is a∼generalized phase accompanied For wavevectors parallel to the spiral and the c axes the
by arotationofthe directorfield. We havealsoincluded dispersion is linear as in an antiferromagnet, while it is
a zero wavevector ferromagnetic mode m which is soft quadratic for wavevectors parallel to the a axis.
duetospinconservation. Itisworthnotingthatthefirst Exact diagonalization. At the microscopic level, the
term in Eq. (5) is absent in conventional helical order crystal symmetry anisotropy is expected to modify the
discussed in Ref. [5]. As discussed above, it originates helimagnon dispersion (8). In addition, the particular
from a rather large J3 that gives penalty to variations spiral (0,δ,1) is stabilized by a large J3 which would in-
along the z direction. Schematically the coarse-grained creasethe spinstiffness alongthe c axis [7]. To takeinto
sublattice magnetizations are accountthese effects,herewe performexactdiagonaliza-
tions with a large unit cell to investigate the low-energy
S (r) S¯ (r) + m(r)bˆ (6) acoustic-like magnons and compare the results with pre-
µ µ
∼
+ ψ(r) cos(Q r+ϕ )aˆ sin(Q r+ϕ )ˆc , dictions based on the helimagnon theory.
µ µ
(cid:2) · − · (cid:3) Tobeginwith,wefirstchooseacommensurateunitcell
The spin dynamics is governedby a generalizedLandau- of a size which contains 2 Λ tetrahedra, corresponding
×
4
to an ordering wavevector q = 2π(0,δ,1) with δ = 1/Λ. (1k0)
0.6 0.8 1 1.2 1.4
Here the factor of 2 accounts for the staggering of spins 6
(a)
along the c axis. The DM constant is then determined
5
according to the expression D = (3/√2)K tanπδ [the
u
4
discrete version of Eq. (3)] for a given fixed Ku. Next V)
e
we introduce small perturbations to spins in the ground m 3
(
state: Si = Sνˆi + σi, where νˆi is a unit vector par- εk 2
allel to the magnetic moment at site r in the ground
i
state, and σ νˆ denotes transverse deviations. Note 1
i i
⊥
that because of this constraint, there are only two de- 0
grees of freedom associated with σ at each site. As- 6
i (b)
suming a time-dependence σi exp(ik ri iωt) for 5
∼ · −
the eigenmodes,the linearizedLLGequationcanbe cast
i(nσto,σan,eige,nσva)luiseaprvoebclteomr oTf(dki)mσ~en=si−oniωnk=σ~,2whe2rΛeσ~ =4, meV) 34
1 2 n (
and T i·s·a·n n n matrix. We then numerica×lly dia×go- εk 2
nalize the matr×ix T(k) and obtainthe lowesteigenvalues
1
in the vicinity of k = 0, corresponding to the soft-mode
around the ordering vector of the magnetic spiral. 0
0.6 0.8 1 1.2 1.4
The magnon dispersions along the three principal di- (1k0)
rections in the vicinity of the ordering wavevector Q =
2π(0,δ,1) are shown in Fig. 2(a)–(c). The number of FIG. 3: (Color online) Simulated spectra of (a) N´eel state
spins in the unit cell corresponding to used spiral pitch with q = 2π(0,0,1) and (b) helical order with q shifted to
2π(0,δ,1) where thehelical pitch δ =0.1 dueto a finiteD=
δ is N = 40, 80, 160. To incorporate the anisotropy
s 0.14 meV. Other parameters used in the simulation are J =
due to the cubic symmetry of the lattice, the obtained
1.35 meV, Ku = 0.21 meV, and J3 = 0.28 meV. (b) Fitted
spectra are fitted to the phenomenological dispersion:
spin wave spectrum to the neutron scattering measurement
ω(k) = αk2+βk4. The dispersion along the b and c (whitecircles)[8]. ThefiniteDMinteractionsplitseachband
axescanpbewellapproximatedbyalinearrelation,while into three as shown in panel (b). The vertical peaks at the
the spectrum along the a-axis exhibits a predominant ordering wavevectors are numerical artifacts.
quadratic behavior at small δ similar to that of a fer-
romagnet. The dispersion parameters α and β obtained
fromthefittingareshowninFig.2(d)–(f)asafunctionof using inelastic neutron scattering in Ref. [8]. Here we
the helical wavenumber δ. The large spin-wave velocity present our numerical calculations and compare the re-
along the c axis can be attributed to a large J , consis- sultswiththe experimentalfindings. The measuredheli-
3
tent with the analysis of gradient expansion [7]. More calpitchδ 0.1canbeusedtofixtheratiooftheDMin-
≈
importantly, the δ-dependence of parameters for disper- teractiontotheexchangeanisotropyK . Thentheabso-
u
sion along the a-axis exhibits a scaling relation α δ2 lutevalueofK canbeestimatedbythemagnonenergies
u
∼
and β 1/δ2, reminiscent of that of a helimagnon [5]. atthezonecenter. Asforfurther-neighborexchanges,ab
∼
Magnons in CdCr O . Although the exact diagonal- initio calculations find a negligible J and a quite large
2 4 2
ization method discussed above provides a rather accu- J with antiferromagnetic sign [7, 25]. Using the follow-
3
ratemagnonspectrum,thecomplicatedband-foldingdue ing set of parameters: J = 1.35 meV, K = 0.21 meV,
u
tolargeunitcellmakesitdifficulttocomparetheresults D = 0.14 meV, and J = 0.28 meV, we obtained very
3
with experiment. Instead, here we employ a less accu- good agreementwith the experimentally measuredspec-
rate but more direct approach based on simulating the trum, as shown in Fig. 3(b).
real-time LLG equations in a large finite lattice [24]: Concluding discussions. We have studied the spin-
wave excitations in a complex helical magnetic order on
∂S ∂ α ∂S
i =S H + GS i, (9) the pyrochlore lattice. Our exact diagonalization calcu-
∂t i× ∂S S i× ∂t
i lation of the low-energy dispersion in the commensurate
where α is the dimensionless Gilbert-damping coeffi- limit of the long-period spiral clearly demonstrates the
G
cient. The dynamics simulation is initiated by a short existence of helimagnonexcitations in the frustrated py-
pulse localizedatr =0. Since the systemis drivenby a rochloreantiferromagnet. Thisnovelmagneticexcitation
i
white source with flat spectrum, we expect magnetic ex- canbeobservedinspinelCdCr O anditinerantmagnet
2 4
citationsofvariousenergyandmomentumaregenerated YMn . We have performed dynamical LLG simulations
2
in our simulations. The spin-wave spectrum is then ob- withparametersinferredfromstructuraldataandabini-
tained via the Fourier transform of the simulation data. tiocalculations. The numericalmagnonspectrumagrees
ThemagnonspectrumofCdCr O hasbeenmeasured wellwiththemeasureddispersioninCdCr O . Ourwork
2 4 2 4
5
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