Spin-frame field theory of a three-sublattice antiferromagnet

Abstract

We present a nonlinear field theory of a three-sublattice hexagonal antiferromagnet. The order parameter is the spin frame, an orthogonal triplet of vectors related to sublattice magnetizations and spin chirality. The exchange energy, quadratic in spin-frame gradients, has three coupling constants, only two of which manifest themselves in the bulk. As a result, the three spin-wave velocities satisfy a universal relation. Vortices generally have an elliptical shape with the eccentricity determined by the Lam\'e parameters.

Full text

Spin-Frame Field Theory of a Three-Sublattice Antiferromagnet
Bastian Pradenas and Oleg Tchernyshyov
William H. Miller III Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, Maryland 21218, USA
(Dated: August 1, 2023)
We present a nonlinear field theory of a three-sublattice hexagonal antiferromagnet. The order
parameter is the spin frame, an orthogonal triplet of vectors related to sublattice magnetizations and
spin chirality. The exchange energy, quadratic in spin-frame gradients, has three coupling constants,
only two of which manifest themselves in the bulk. As a result, the three spin-wave velocities satisfy
a universal relation. Vortices generally have an elliptical shape with the eccentricity determined by
the Lam´e parameters.
Theory of magnetic solids is often described as a lattice
problem, exemplified by the Heisenberg model of atomic
spins. However, discrete models are notoriously difficult
to solve outside of the simplest tasks, such as finding
the spectrum of linear spin waves. An analytic theory
of nonlinear solitons—such as domain walls or vortices—
in the framework of a lattice model is often not feasible
or cumbersone [1]. Continuum theories have the clear
advantage of being more amenable to analytic treatment.
By design, they focus on the physics of long distances
and times, capturing the universal aspects of low-energy
physics at the expense of microscopic details.
In magnetism, a well-known example is micromagnet-
ics, the continuum theory of a ferromagnet going back to
Landau and Lifshits [2]. It is usually formulated through
the equation of motion for the magnetization field mof
unit length parallel to the local direction of spins,
S∂m
∂t =−m×δU
δm.(1)
Here Sis the spin density, U[m] = RddrUis a potential-
energy functional, whose functional derivative −δU/δm
acts as an effective magnetic field. The energy density is
usually dominated by the Heisenberg exchange interac-
tion of strength A,
U=A
2∂im·∂im+. . . (2)
Doubly repeated indices imply summation. The omitted
terms represent weaker anisotropic interactions of dipolar
and relativistic origin. We use the calligraphic font to
indicate intensive quantities (densities).
The Landau-Lifshitz equation (1) provides a starting
point for understanding the dynamics of ferromagnetic
solitons. A further coarse-graining eliminates fast inter-
nal modes of a soliton and focuses on its slow collective
motion whose seminal achievements; primary examples
are Thiele’s equation of rigid motion [3] and Walker’s
dynamical model of a domain wall [4].
The continuum approach has also been applied to sim-
ple antiferromagnets, in which adjacent spins are (nearly)
antiparallel and can be split into two magnetic sublattices
1 and 2, each with its own magnetization field m1and
m2of unit length. Because at low energies, the sublattice
magnetizations are (nearly) antiparallel, both can be ap-
proximated by a single field of staggered magnetization
n≈mA≈ −mB, whose dynamics is described by an
O(3) σ-model with the Lagrangian density [5–7]
L=K − U =ρ
2∂tn·∂tn−A
2∂in·∂in−. . . (3)
The first term K=ρ
2∂tn·∂tnis the kinetic energy of
staggered magnetization and ρis a measure of inertia.
The second, potential term comes from the Heisenberg
exchange energy and has the same functional form as in
a ferromagnet (2). The omitted terms represent various
weak anisotropic interactions. Minimization of the action
with the constraint n2= 1 yields the equation of motion
ρ ∂t(n×∂tn) = −n×δU
δn.(4)
As with ferromagnets, the antiferromagnetic Landau–
Lifshitz equation (4) can be translated into equations of
motion for solitons [8,9] and extended to include the
effects of spin transfer and dissipation [10,11].
The primary goal of this paper is to introduce a univer-
sal field theory for an antiferromagnet with three mag-
netic sublattices whose magnetization fields satisfy the
relation m1+m2+m3≈0. Such magnetic states are
typically realized in antiferromagnetic solids of hexag-
onal symmetry with triangular motifs. Although such
magnets have been studied for decades [12,13], recent
experimental studies of metallic antiferromagnets Mn3Sn
and Mn3Ge [14,15] have rekindled theoretical interest in
these frustrated magnets [16–18].
The existing field theory for 3-sublattice antiferromag-
nets by Dombre and Read [12] has a couple of drawbacks.
First, it is formulated specifically for the triangular lat-
tice, which has a higher spatial symmetry than other
hexagonal lattices (such as kagome) and therefore misses
some of the universal features of 3-sublattice antiferro-
magnets. Second, their mathematical formalism repre-
sents the magnetic order parameter as a 3 ×3 rotation
matrix, an abstract, and not very intuitive mathematical
object.
arXiv:2307.16853v1 [cond-mat.str-el] 31 Jul 2023

2
m2
m1m3
(a) (b)
nx
ny
nz
FIG. 1. (a) A ground state of the kagome antiferromagnet
and its sublattice magnetizations m1,m2, and m3. (b) The
corresponding spin-frame vectors nx,ny, and nz.
We derive a nonlinear field theory of a 3-sublattice
antiferromagnet with the order parameter represented
by a spin frame, i.e., a triad of orthonormal vectors
ˆ
n≡ {nx,ny,nz}, directly related to sublattice magne-
tizations m1,m2, and m3. At low energies, the mag-
netization dynamics reduce to rigid rotations of the spin
frame, ∂tni=Ω×ni, at a local angular frequency Ω.
One of our main results is the Landau–Lifshitz equation
for a 3-sublattice antiferromagnet,
ρ ∂tΩ=−ni×δU
δni
.(5)
A sum over doubly repeated Roman indices, i=x, y, z, is
implied hereafter. Like its analogs (1) and (4), it equates
the rate of change of the local density of angular momen-
tum with the torque density from conservative forces ex-
pressed by a potential energy functional U(ˆ
n). The trans-
parent physical meaning of the Landau–Lifshitz equation
makes it easy to add other relevant perturbations.
To define the spin frame ˆ
n, we first switch from the
three unit-vector fields of sublattice magnetizations m1,
m2, and m3to uniform magnetization mand two stag-
gered magnetizations nxand ny(Fig. 1):
m=m1+m2+m3,
nx= (m2−m1)/√3,(6)
ny= (2m3−m2−m1)/3.
To them, we add the vector spin chirality [19]
nz=2
3√3(m1×m2+m2×m3+m3×m1).(7)
As long as m= 0, sublattice fields m1,m2, and m3are
coplanar and thus define the spin plane. Staggered mag-
netizations nxand nylie in the spin plane, whereas spin
chirality nzis orthogonal to it. The three unit vectors ni
form a right-oriented orthonormal spin frame:
ni·nj=δij ,ni×nj=ϵijk nk.(8)
To derive the dynamics of the spin frame, we follow the
standard Lagrangian approach [6,12,20] and integrate
(b)
x
y
(c)
z
12
3
33
21
12
3
12312
33
21 12
3
12
FIG. 2. Hexagonal lattices and their magnetic sublattices:
(a) kagome, (b) triangular lattice. (c) Spatial coordinate axes
x,y, and z. Red, green, and blue colors indicate magnetic
sublattices 1, 2, and 3. Filled triangles and dotted lines denote
C3and C2rotation axes, respectively.
out the hard field of uniform magnetization mto obtain
the dynamics of the spin frame. Our starting point is the
Landau–Lifshitz equations for sublattice magnetizations,
S∂tm1=−m1×δV
δm1
,(9)
and similarly for sublattices 2 and 3. Like in 2-sublattice
antiferromagnets [21], the potential energy functional V
is dominated by the antiferromagnetic exchange interac-
tion imposing a penalty for m= 0 [22],
V(m,ˆ
n) = m2
2χ+U(ˆ
n),(10)
where χis the paramagnetic susceptibility. The subdom-
inant term U[ˆ
n], expressing the energy of the antiferro-
magnetic order parameter, will be discussed below.
With the aid of Eqs. (6), (9), and (10) we find that, like
in 2-sublattice antiferromagnets [5], staggered magneti-
zationss nxand nyprecess about the direction of uniform
magnetization mat the angular velocity [20]
Ω≈m
χS.(11)
The linear proportionality between the precession fre-
quency and uniform magnetization (11) can be derived
as the equation of motion for uniform magnetization m
from the following Lagrangian for fields mand ˆ
n[20]:
L(m,ˆ
n) = Sm·Ω−m2
2χ− U(ˆ
n).(12)
The angular velocity can be expressed explicitly in terms
of ˆ
nvia the kinematic identity
Ω=1
2ni×∂tni,(13)
The first term in the Lagrangian (12) is linear in the ve-
locities ∂tni, so its action represnts the spin Berry phase.
It yields the expected density of angular momentum Sm.

3
Lagrangian (12) is quadratic in uniform magnetization
m. Integrating out this field with the aid of its equation
of motion (11) yields an effective Lagrangian for the re-
maining fields ˆ
nendowed with kinetic energy of rotation
with the inertia density ρ=χS2:
L(ˆ
n) = ρΩ2
2− U(ˆ
n) = ρ
4∂tni·∂tni− U(ˆ
n).(14)
Minimizing the action in the presence of holonomic
constraints (8) yields equations of motion with undeter-
mined Lagrange multipliers [23] Λij = Λji:
I
2∂2
tni=−δU
δni−Λij nj.(15)
Finally, we take a cross product with ni, sum over i, and
use Eq. (13) to obtain the Landau–Lifshitz equation (5).
The energy functional U[ˆ
n] is usually dominated by
the Heisenberg exchange interaction. The latter respects
the SO(3) symmetry of global spin rotations and there-
fore depends not on the orientation of the spin frame ˆ
n
but rather on its spatial gradients. Like in ferromagnets
and 2-sublattice antiferromagnets, the exchange energy
is quadratic in ∂αnβ, where Greek indices take on values
α=xand yonly. The form of these quadratic terms
is restricted by the D3point-group rotational symmetry
of a hexagonal lattice (Fig. 2), including ±2π/3 spatial
rotations about a C3axis normal to the xy plane and
πspatial rotations about C2axes lying in the xy plane.
Under these transformations, the staggered magnetiza-
tions (nx,ny) transform in terms of each other in the
same way as the in-plane components (kx, ky) of a spatial
vector kdo; chirality nztransforms as kz. This obser-
vation helps to construct energy terms quadratic in the
gradients of staggered magnetizations and invariant un-
der both spin rotations and lattice symmetries. To that
end, we may start with a rank-4 spatial tensor and spin
scalar ∂αnβ·∂γnδand contract its spatial indices pair-
wise to form a spatial scalar. This procedure yields our
second main result, the three possible gradient terms for
the exchange energy density,
U=λ
2∂αnα·∂βnβ+µ
2∂αnβ·∂αnβ+ν
2∂αnβ·∂βnα.(16)
This expression resembles the elastic energy density of
an isotropic solid [24], albeit with 3 Lam´e constants.
A triangular lattice has an extra spatial symmetry.
Under primitive lattice translations, sublattice indices
undergo cyclic permutations, see Fig. 2(b). Staggered
magnetizations nαeffectively undergo ±2π/3 spatial ro-
tations, whereas gradients ∂αdo not. Thus translational
symmetry forbids the λand νterms for a triangular lat-
tice.
The Landau–Lifshitz equation (5) for a 3-sublattice
Heisenberg antiferromagnet reads
ρ ∂tΩ= (λ+ν)nα×∂α∂βnβ+µnα×∂β∂βnα.(17)
Note that the exchange coupling constants λand νenter
the equation of motion through a combination λ+ν,
rather than individually. More on that below.
An antiferromagnet with nearest-neighbor exchange
interaction Jhas λ+ν= 0 and µ=JS2√3/4 on a
triangular lattice; on kagome, λ+ν=√3JS2/4 and
µ= 0 on kagome. See Supplemental Material [25] for
contributions of further-neighbor interactions.
In what follows, we use the spin-frame formulation of
the field theory to obtain the properties of excitations:
spin waves and vortices.
Spin waves. Linear spin waves on top of a uniform
ground state ni= const can be parametrized in terms
of infinitesimal rotations of the spin frame, δni=δϕ×
ni. Here δϕ=niδϕiis a triplet of infinitesimal rotation
angles δϕiabout the corresponding spin axes; ∂tδϕ=Ω.
Assuming a plane wave with wavenumber ktravelling
along the xdirection, δϕ(t, x) = eδϕ ei(kx′−ωt), we ob-
tain three spin waves with ω=ck and the following
polarizations eand velocities c:
eI=nx, cI=pµ/ρ,
eII =ny, cII =p(λ+µ+ν)/ρ,
eIII =nz, cIII =p(λ+ 2µ+ν)/ρ.
(18)
The velocities satisfy the identity
c2
I+c2
II =c2
III .(19)
Modes I, II are analogs of transverse and longitudinal
sound in a hexagonal solid in two dimensions.
Vortices. The existence of topologically stable point
defects—vortices—in a 3-sublattice Heisenberg antifer-
romagnet was first pointed out by Kawamura and
Miyashita [19]. A 2πrotation of the spin frame corre-
sponds to a loop in the order-parameter space that can-
not be continuously deformed to a point. It is convenient
to parametrize the orientation of the spin frame by start-
ing with a reference uniform configuration nx= (1,0,0),
ny= (0,1,0), nz= (0,0,1), and applying consecutive
Euler rotations through angles ϕabout nz,θabout ny,
and ψabout nz. On a triangular lattice (λ+ν= 0), a
vortex configuration with the lowest energy is described
by the Euler angles ϕ,θ, and ϕgiven by
eiϕ =x+iy
|x+iy|, θ =π
2, ψ = const.(20)
This expression agrees well with a numerically obtained
vortex configuration for a triangular lattice, Fig. 3(a).

4
FIG. 3. Vortices in 3-sublattice abtiferromagnets. (a) Triangular lattice with nearest-neighbor interactions only. (b) kagome
lattice with first and third-neighbor interactions, J3=J′
3=−J1/20. See Supplemental Material [25] for the definition of
further-neighbor interactions. Red, green, and blue arrows are spins of the three magnetic sublattices. The blue spins point
away from the viewer; the red and green ones have components pointing toward the reader. The circle and ellipse reflect the
expected shape of the vortex with the major axis ratio b=p(λ+µ+ν)/µ.
Although we have not been able to find an exact vor-
tex solution for a generic hexagonal antiferromagnet, we
can understand the effect of the λ+νterm on the vortex
shape perturbatively. Starting with the isotropic solu-
tion (20) for λ+ν= 0, we keep θ=π/2 and choose
for simplicity ψ=π/2 to obtain the following energy
density:
U=λ+µ+ν
2(∂xϕ)2+µ
2(∂yϕ)2.(21)
The vortex acquires an elliptical shape with the major
axis ratio b=p(λ+µ+ν)/µ:
eiϕ =bx +iy
|bx +iy|, θ =π
2, ψ =π
2.(22)
Although this result is obtained in the limit λ+ν≪µ,
our numerical calculations on a kagome lattice demon-
strate its accuracy even in the opposite limit. The right
panel of Fig. 3(b) shows a vortex in a kagome antifer-
romagnet with the ratio of first and third-neighbor ex-
change interactions J1/J3=−20, or (λ+ν)/µ = 5/3.
Its shape agrees with the extrapolated semiaxis ratio
b=p8/3.
Discussion. In this paper, we have presented a uni-
versal field theory of a hexagonal antiferromagnet with
3 magnetic sublattices. The order parameter is a spin
frame constructed from the sublattice magnetizations
and vector chirality. Its mechanics is fully specified by
the inertia of the spin frame Iand three Lam´e constants
λ,µ, and ν. The simple and versatile field theory enabled
us to establish a Pythagorean identity for the three spin-
wave velocities (19) and to predict a generally elliptical
shape for vortices (22).
It is worth noting that the three Lam´e constants enter
the equations of motion (17) in the form of two linear
combinations, λ+νand µ. As a result of that, the three
spin-wave velocities are constrained by a Pythagorean
identity (19). The origin of this behavior can be under-
stood by examining the exchange energy density (16).
The λand νterms in it can be obtained from one an-
other through integration by parts. Thus their infinitesi-
mal variations are the same—up to boundary terms—and
so they make identical contributions to classical dynam-
ics. The difference λ−νis a “silent” coupling constant
that does not manifest itself in the classical dynamics of
magnetization. It can be seen that it has an intriguing
topological nature with the aid of the identity
1
2(∂αnα·∂βnβ−∂αnβ·∂βnα) = nz·(∂xnz×∂ynz).(23)
We now see that the energy term (23) is a topological
quantity proportional to the skyrmion density of the spin
chirality (7). Its topological nature assures that it gives a
quantized contribution to the exchange energy that does
not change under continuous variations of the order pa-
rameter and therefore does not affect classical dynamics.
However, it may lead to interesting boundary effects such
as the presence of edge modes, as discussed recently in
a different context by Dong et al. [26]. We will address
the topological aspects of this field theory in a separate
publication [27].
Acknowledgments. We thank Boris Ivanov and Se
Kwon Kim for useful discussions. The research has been
supported by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences under Award No. DE-
SC0019331.

5
[1] I. G. Gochev, Zh. Eksp. Teor. Fiz. 85, 199 (1983), [Sov.
Phys. JETP 58, 115–119 (1983)].
[2] L. Landau and E. Lifshits, Phys. Z. Sowjet. 8, 153 (1935),
reprinted in Collected Papers of L. D. Landau, Pergamon,
New York, pp. 101-114 (1965).
[3] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).
[4] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406
(1974).
[5] I. V. Bar’yakhtar and B. A. Ivanov, Fiz. Nizk. Temp. 5,
759 (1979), [Sov. J. Low Temp. Phys. 5, 361 (1979)].
[6] A. F. Andreev and V. I. Marchenko, Usp. Fiz. Nauk 130,
39 (1980), [Sov. Phys. Usp. 23, 21–34 (1980)].
[7] F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983).
[8] I. V. Bar’yakhtar, B. A. Ivanov, and A. L. Sukstanskii,
Zh. Eksp. Teor. Fiz. 51, 757 (1979), [Sov. Phys. JETP
78, 1509–1522 (1979)].
[9] V. G. Bar’yakhtar, B. A. Ivanov, and M. V. Chetkin,
Usp. Fiz. Nauk 146, 417 (1985), [Sov. Phys. Usp. 28,
563–588 (1985)].
[10] K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Phys.
Rev. Lett. 106, 107206 (2011).
[11] E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, and
A. Brataas, Phys. Rev. Lett. 110, 127208 (2013).
[12] T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989).
[13] A. B. Harris, C. Kallin, and A. J. Berlinsky, Phys. Rev.
B45, 2899 (1992).
[14] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212
(2015).
[15] Y. Chen, J. Gaudet, S. Dasgupta, G. G. Marcus, J. Lin,
T. Chen, T. Tomita, M. Ikhlas, Y. Zhao, W. C. Chen,
M. B. Stone, O. Tchernyshyov, S. Nakatsuji, and C. Bro-
holm, Phys. Rev. B 102, 054403 (2020).
[16] J. Liu and L. Balents, Phys. Rev. Lett. 119, 087202
(2017).
[17] Y. Yamane, O. Gomonay, and J. Sinova, Phys. Rev. B
100, 054415 (2019).
[18] S. Dasgupta and O. Tchernyshyov, Phys. Rev. B 102,
144417 (2020).
[19] H. Kawamura and S. Miyashita, J. Phys. Soc. Jpn. 53,
4138 (1984).
[20] H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Phys. Rev.
B98, 054424 (2018).
[21] S. K. Kim, Y. Tserkovnyak, and O. Tchernyshyov, Phys.
Rev. B 90, 104406 (2014).
[22] B. I. Halperin and W. M. Saslow, Phys. Rev. B 16, 2154
(1977).
[23] L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed.,
Course of Theoretical Physics, Vol. 1 (Butterworth-
Heinemann, New York, 1976).
[24] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd
ed., Course of Theoretical Physics, Vol. 7 (Butterworth-
Heinemann, New York, 1986).
[25] Supplemental Material.
[26] Z. Dong, O. Ogunnaike, and L. Levitov, Phys. Rev. Lett.
130, 206701 (2023).
[27] B. Pradenas, E. G. Ozaktas, and O. Tchernyshyov (un-
published), in preparation.
Supplemental Material
Further neighbors exchange interactions
Convention for further-neighbor interactions is shown in Fig. S1.
J2
J3
J1J1
J3J2J′
3
FIG. S1. First, second, and third-neighbors exchange interactions. The kagome lattice has two distinct types of third neighbors.
In antiferromagnets with up to third-neighbor exchange interactions, the field-theory parameters are as follows. For
the triangular lattice,
λ+ν= 0, µ =√3
4(J1−6J2+ 4J3)S2, ρ−1=9√3
2(J1+J3)a2.(S1)
Note that λ+νvanishes in accordance with the higher spatial symmetry of the triangular lattice.

6
For kagome,
λ+ν=√3
4(J1−3J2−2J3−2J′
3)S2, µ =3√3
4(J2−J3−J′
3)S2, ρ−1= 4√3(J1+J2)a2.(S2)
In both cases, ais the distance between first neighbors.
For a triangular lattice with first-neighbor interactions only, we obtain
cI=cII =3√3
2√2J1Sa, cIII =3√3
2J1Sa, (S3)
in agreement with Dombre and Read [12]. For a kagome antiferromagnet with J′
3= 0, we find
cI= 3p(J1+J2)(J2−J3)Sa, cII =p3(J1+J2)(J1−5J3)Sa, cIII =p3(J1+J2)(J1+ 3J2−8J3)Sa, (S4)
in agreement with Harris et al. [13].