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Spontaneous symmetry breaking in the Heisenberg antiferromagnet on a triangular
lattice
Basti´an Pradenas, Grigor Adamyan, and Oleg Tchernyshyov
William H. Miller III Department of Physics and Astronomy,
Johns Hopkins University, Baltimore, Maryland 21218, USA
(Dated: April 18, 2025)
We present a detailed investigation of an overlooked symmetry structure in non-collinear an-
tiferromagnets that gives rise to an emergent quantum number for magnons. Focusing on the
triangular-lattice Heisenberg antiferromagnet, we show that its spin order parameter transforms
under an enlarged symmetry group, SO(3)L×SO(3)R, rather than the conventional spin-rotation
group SO(3). Although this larger symmetry is spontaneously broken by the ground state, a residual
subgroup survives, leading to conserved Noether charges that, upon quantization, endow magnons
with an additional quantum number—isospin —beyond their energy and momentum. Our results
provide a comprehensive framework for understanding symmetry, degeneracy, and quantum num-
bers in non-collinear magnetic systems, and bridge an unexpected connection between the paradigms
of symmetry breaking in non-collinear antiferromagnets and chiral symmetry breaking in particle
physics.
I. INTRODUCTION
In quantum mechanics and quantum field theory,
quantum numbers serve as discrete labels assigned to
eigenstates of observables, characterizing intrinsic prop-
erties, underlying symmetries, and associated conserva-
tion laws. While quantum numbers typically classify
distinct states, degeneracies can arise—situations where
multiple states share identical eigenvalues, such as en-
ergy. In these cases, additional quantum numbers or
auxiliary labels are introduced to fully distinguish among
degenerate states. One famous example is the isospin,
introduced by Heisenberg [1] to account for the nearly
identical strong interactions experienced by neutrons and
protons. By regarding the neutron and the proton as
two isospin states of the same nucleon, one assigns them
the isospin projections I3=−ℏ
2and I3= +ℏ
2, re-
spectively. This idea extends to all hadrons, including
pions, which form an isospin triplet (I=ℏ). Pions
posed a puzzle due to their anomalously light masses
compared to other hadrons. The resolution came in
the 1960s with the idea of spontaneous chiral symme-
try breaking in QCD, where the underlying chiral sym-
metry, SU(2)L×SU(2)R, is broken down to its diagonal
subgroup—identified as isospin rotations—thereby ren-
dering pions as the (pseudo) Goldstone bosons in the
limit of vanishing up and down quark masses. The low-
energy physics of these Goldstone bosons is captured by
non-linear sigma model (NLSM), in which the isospin
group emerges as the residual symmetry after chiral sym-
metry breaking [2–4].
In condensed-matter systems, particularly in ordered
magnets, NLSMs serve as low-energy effective theories
for describing symmetry breaking and magnetic excita-
tions. For instance, in two-sublattice antiferromagnets,
the spontaneous breaking of the global spin-rotation sym-
metry, SO(3), down to SO(2) gives rise to two magnons
as Goldstone bosons. The unbroken SO(2), which corre-
sponds to spin rotations about the ordered spins, serves
as a residual symmetry that allows one to define the
magnon spin projection quantum number, often denoted
as Szassuming the magnetic order aligns with ez. Var-
ious authors have referred to this quantum number as
“pseudo-spin,” “polarization,” or “helicity” [5–11].
A more intriguing case emerges in non-collinear anti-
ferromagnets, such as those on triangular lattices. There,
the non-collinear spin order breaks the global spin SO(3)
symmetry, leaving no obvious axis for continuous rota-
tions and suggesting no remaining residual symmetry.
Yet one finds that two of the three magnon branches
are degenerate [12–15], hinting at an underlying residual
symmetry.
In this paper, we resolve this apparent paradox by
demonstrating that the spin order parameter of non-
collinear antiferromagnets—represented by an SO(3) ma-
trix O—actually supports a larger set of transforma-
tions than previously assumed. Specifically, Otrans-
forms under two independent rotation groups, SO(3)L
and SO(3)R, acting from the left and right, respectively.
As a result, the full symmetry group is SO(3)L×SO(3)R,
rather than a single SO(3).
A detailed analysis of the spontaneous breaking of this
group by the non-collinear spin order reveals that a resid-
ual symmetry subgroup survives. This residual symme-
try (1) preserves the non-collinear ground state, (2) un-
derlies the degeneracy observed in two of the magnon
branches, and (3) endows the magnons with an addi-
tional quantum number—isospin. We adopt this term
from chiral symmetry breaking in QCD, where a simi-
lar residual symmetry gives rise to the isospin quantum
number for pions.
The paper is structured as follows. In Sec. II, we briefly
revisit the collinear N´eel antiferromagnet in its O(3)-
NLSM formulation. We highlight how its spontaneous
symmetry breaking yields magnons with a well-defined
spin projection. In Sec. III, we move to the triangular-
lattice Heisenberg antiferromagnet, where the order pa-
rameter is naturally described by O(4)-NLSM. We then
arXiv:2504.12411v1 [cond-mat.str-el] 16 Apr 2025
2
clarify why two spin-wave branches remain degenerate
despite the complete breaking of SO(3). In Sec. IV,
we introduce the enlarged symmetry group—the chiral
group SO(3)L×SO(3)R—and describe its action on the
spin order parameter, establishing it as the full symmetry
group relevant to the problem. In Sec. V, we present the
O(4)-NLSM formulation of the triangular antiferromag-
net. This parametrization proves especially convenient,
as the chiral group transformations act directly on vec-
tors rather than on matrices. In Sec. VI, we detail how
the ground state of the non-collinear spin order sponta-
neously breaks the chiral symmetry and identify the sub-
groups that remain unbroken. In Sec. VII, we use these
residual symmetries to construct the associated Noether
charges and introduce the isospin quantum number—an
additional quantum label for magnons, beyond their en-
ergy and momentum. Finally, in Sec. VIII, we summarize
our results and comment on possible extensions to other
non-collinear magnets.
II. NLSM FOR THE N´
EEL AFM
In a two-sublattice antiferromagnet (AFM), the order
parameter is the N´eel vector n, defined as the difference
in magnetization between the two sublattices,
n=1
2(m1−m2),(1)
where m1and m2are unit vectors representing the di-
rections of the magnetization of sublattices 1 and 2, re-
spectively. Each sublattice spin is given by
Sa=Sma, a = 1,2,(2)
with Sdenoting the spin length. Additionally, a uniform
magnetization field is defined by
m=m1+m2.(3)
Although mvanishes in a ground state, it becomes
nonzero during antiferromagnetic dynamics. At low-
energies, mis a slave of the N´eel vector and is often
written as
m=χSn×∂tn,(4)
where χis the magnetic susceptibility, and S=S/Auc is
the spin density, defined by the total spin Sper magnetic
unit cell of area Auc.
The continuum field theory of the Heisenberg N´eel an-
tiferromagnet can be succinctly formulated as an O(3)
nonlinear sigma model (NLSM) [16–18], with the La-
grangian given by:
L=ρ
2∂tn·∂tn−J
2∂in·∂in,(5)
where ρ=χS2is the inertia density, Jis the exchange
stiffness [14,15], and summation over repeated indices
(a)
(b)
(c)
n
m1
m2
FIG. 1. (a) Illustration of the antiferromagnetic (N´eel) order,
where the shaded region indicates the magnetic unit cell. (b)
Schematic representation of the magnetic order parameter.
(c) The corresponding N´eel order vector.
is assumed. The latin indices ispan over the dimen-
sions x, y. This Lagrangian is invariant under the global
spin-rotation, the SO(3) group, resulting in the Noether
current
jt=ρn×∂tn,ji=−Jn×∂in.(6)
Its conservation directly encodes the equations of motion
for this system. The time component jtis the spin den-
sity, related to the uniform magnetization,
jt=Sm.(7)
A. Spontaneous Symmetry Breaking
The ground state of the N´eel antiferromagnet spon-
taneously breaks the global SO(3) symmetry down to
SO(2), allowing the N´eel vector to point in an arbi-
trary direction in three-dimensional space. The contin-
uous family of symmetry-related ground states thus cor-
responds to points on the unit sphere S2. We denote the
ground-state orientation by n0, and without loss of gen-
erality, choose it to point along the z-axis: n0=ez. To
analyze this symmetry breaking, we introduce the gen-
erators of the Lie algebra of the rotation group SO(3),
denoted by Si, with i=x, y, z. The generators Sxand
Sycorrespond to rotations about axes perpendicular to
the ground state and are therefore spontaneously broken,
while Sz, corresponding to rotations about the ground
state direction ez, remains unbroken.
Magnons, the Goldstone bosons arising from this spon-
taneous symmetry breaking, correspond to small trans-
verse deviations of the spin order parameter from the
ground state orientation n0. These small deviations can
be parametrized as
n=nx, ny,q1−n2
x−n2
y,(8)
where the transverse components nxand nyare small
n2
x+n2
y≪1.
3
Since the vector nis constrained to lie on S2, only two
of its three components are independent. Consequently,
the broken symmetry generators Sxand Syinduce non-
linear shift transformations on the Goldstone modes to
preserve the unit-length constraint |n|= 1:
na→na+αaq1−n2
x−n2
y,(9)
where the Latin indices a=x, y denote the transverse
components, and αaare infinitesimal angles associated
with shifts in the directions orthogonal to the ground
state direction n0.
In contrast, the Goldstone modes transform linearly
under the unbroken SO(2) subgroup, corresponding to
rotations about the n0direction generated by Sz:
na→na+βϵab nb,(10)
where βis an infinitesimal rotation angle and ϵab is the
antisymmetric Levi-Civita symbol in two dimensions.
B. Linearized Theory and Noether Currents
To calculate the Noether currents associated with
these transformations, we expand the Lagrangian about
the ground state n0and introduce small-amplitude spin
waves via
δn=ϕ×n0,(11)
where the vector field doublet ϕ= (ϕx, ϕy,0) encodes the
local rotation angles about directions transverse to the
ground state. Substituting this into Eq. (5), we obtain:
L=ρ
2∂tϕ2
x+∂tϕ2
y−J
2∇ϕ2
x+∇ϕ2
y,(12)
where the gradient operator is defined as ∇= (∂x, ∂y)T.
At linear order, the transformation in Eq. (9) simplifies to
a linear shift, ϕa→ϕa+αa. The quadratic Lagrangian,
Eq. (12), remains invariant under these field shifts, which
directly lead to the conservation of Noether currents:
jt
a=ρ ∂tϕa, ji
a=−J ∂iϕa.(13)
These results are nothing but the linear-order Noether
currents of Eq. (6). Furthermore, the Lagrangian in
Eq. (12) preserves the symmetry under global SO(2) ro-
tations (Eq. (10)) of the fields ϕ= (ϕx, ϕy,0) about
the vacuum state, reflecting the degeneracy of the two
magnon branches with speed cI,II =pρ/J. This resid-
ual SO(2) symmetry ensures another conserved current:
jt
SO(2) =ρϵabϕa∂tϕb, ji
SO(2) =−Jϵabϕa∂iϕb,(14)
where a, b ∈ {x, y}and ϵab is the two-dimensional Levi-
Civita symbol. Physically, this current represents the
spin density defined in Eq. (7), projected along the
ground-state direction n0=ez.
C. Spin Pro jection as a Quantum Number
Consider circularly polarized spin waves around the
ground-state order n0. To analyze these modes system-
atically, we first introduce the complex fields
ϕ±=1
√2ϕx±i ϕy,(15)
which compactly represent the small-angle doublet de-
fined in Eq. (11)A plane-wave solution for these fields
can then be naturally expressed as
ϕ±(x) = ϕ0
√2e±i(ωt−k·x),(16)
where we have introduced the spacetime coordinate x=
(t, x), with tdenoting time and x= (x, y) the two-
dimensional spatial position. Here, ωis the angular fre-
quency, and k= (kx, ky) is the corresponding wavevec-
tor. The + sign corresponds to a right-handed circu-
lar polarization, while the −sign corresponds to a left-
handed one. Each of these modes carries a definite spin-
density projection along the ground-state axis n0
jt
SO(2) =±ρωϕ2
0,(17)
Upon canonical quantization, we promote the fields
ϕ±(x) to operators and introduce the bosonic creation
and annihilation operators a†
k, akand b†
k, bkcorrespond-
ing to the quanta of ϕ+and ϕ−, respectively. These op-
erators satisfy the canonical commutation relations The
bosonic commutation relations are
[ak, a†
k′] = δk,k′,[bk, b†
k′] = δk,k′,(18)
with all other commutators vanishing. The operators
expansions of the fields take the form
ϕ+(x) = sℏ
ρA X
k
1
√2ωbke−ikx +a†
keikx,
ϕ−(x) = sℏ
ρA X
k
1
√2ωake−ikx +b†
keikx,
(19)
where Ais the sample area, kx =ωt −k·xand ω=
pρ/J|k|.
The total spin projection Szis obtained by integrating
the conserved SO(2) current over space. Substituting the
mode expansions of ϕ±into jt
SO(2) in Eq. (14), we obtain:
Sz= :ZA
d2x jt
SO(2)(x): = ℏX
ka†
kak−b†
kbk.(20)
Here, the symbol :: denotes normal ordering of operators.
Defining the number operators for the quanta of ϕ+and
ϕ−as
N+=X
k
a†
kak, N−=X
k
b†
kbk,(21)
4
we have
Sz=ℏN+−N−.(22)
Thus, ϕ+magnon quanta carry spin projection Sz=ℏ,
while ϕ−magnon quanta carry spin projection Sz=
−ℏ. The residual SO(2) symmetry guarantees that each
magnon is labeled not only by its energy and momentum,
but also by a well-defined spin projection.
This example illustrates the intimate connection be-
tween residual symmetries and the conserved charges and
quantum numbers—a connection that we will further ex-
ploit in the following sections.
m1
m2
n
(a) (b)
FIG. 2. Trajectories of the spin order for a well-defined spin
projection, Sz, in a spin wave within the N´eel antiferromag-
net. (a) The spin directions ma(a= 1,2) for right-handed
circularly polarized magnons. (b) The N´eel vector trajectory
around its equilibrium axis, ez.
III. NONLINEAR SIGMA MODEL FOR
NON-COLLINEAR AFMS
We now turn our attention to non-collinear antiferro-
magnets, focusing specifically on the triangular Heisen-
berg antiferromagnet, which exhibits magnetic order
characterized by three distinct sublattices. In this sys-
tem, each sublattice spin is defined as
Sa=Sma, a = 1,2,3,(23)
where Sis the spin length and mais a unit vector along
the direction of the sublattice magnetization. In the
ground state, the three spins within each triangle satisfy
S1+S2+S3= 0,(24)
implying that they are coplanar and mutually oriented
at 120◦, as illustrated in Fig. 3(a). In this configuration,
the magnetic order parameter may be viewed as a rigid
body formed by the unit vectors m1,m2, and m3, which
can be rotated collectively without altering the internal
structure. Since the overall orientation of a rigid body is
characterized by an SO(3) matrix, the spin order param-
eter is naturally represented as a matrix of rotations. In
fact, Dombre and Read [12] developed a nonlinear sigma
model formulation of the triangular-lattice antiferromag-
net based on rotation matrices.
An alternative, but equivalent, formulation was re-
cently introduced by two of us [15], where the order
parameter is expressed in terms of an orthogonal spin
frame. In this approach, the uniform magnetization is
defined as,
m=m1+m2+m3,(25)
while two staggered vectors are defined as
nx=1
√3(m2−m1),
ny=1
3(2m3−m2−m1),
(26)
and the vector spin chirality is given by
nz=2
3√3m1×m2+m2×m3+m3×m1.(27)
In the ground state the uniform magnetization vanishes,
m= 0, and the spin-frame vectors nx,ny, and nzform
an orthonormal set:
na·nb=δab,na×nb=ϵabc nc,(28)
with the Latin indices a, b, c assume the values x,y, and
z. The set of spin frame vectors, {nx,ny,nz}, and the
fixed global frame, {ex,ey,ez}, are connected by the
Dombre-Read rotation matrix O[12] as
nb=eaOab,ea=Oab nb.(29)
m2
m3
m1ny
nx
nz
(a)
(b)
(c)
FIG. 3. (a) Illustration of the 120◦order in a three-sublattice
antiferromagnet, with the shaded area denoting the magnetic
unit cell. (b) Schematic representation of the magnetic order
parameters m1,m2, and m3. (c) Depiction of the correspond-
ing spin frame vectors, labeled as nx,ny, and nz.
As in the conventional two-sublattice antiferromag-
nets, although the uniform magnetization mvanishes in
the ground state, it becomes nonzero as the spin frame
5
vectors evolve in time. In the low-energy limit, this be-
havior is captured by
m=1
2χSna×∂tna,(30)
where sum over repeated indices is assumed, χis the
magnetic susceptibility, and S=S/Auc denotes the spin
density, with Sbeing the total spin per magnetic unit
cell.
The continuum description of the antiferromagnet in
the spin frame is governed by the Lagrangian
L=ρ
4∂tna·∂tna−µ
2∂inj·∂inj
−λ
2ϵij nz·∂inz×∂jnz.
(31)
Here, the Greek indices i, j take the values xand y, while
the Latin index aruns over x,y, and z. The parameters
ρ=χS2and µdescribe the inertial and exchange stiff-
ness properties of the antiferromagnet, respectively. The
final term, which is topological in nature and weighted
by λ, reduces to a boundary contribution in the bulk and
is responsible for the emergence of localized helical edge
modes [19]. In what follows, we ignore this term for the
bulk analysis.
Each vector nacan be expressed in terms of small an-
gles of rotations around the ground state [15]:
na=n(0)
a+ϕ×n(0)
a+O(ϕ2),(32)
where ϕ= (ϕx, ϕy, ϕz) represents small rotation an-
gles around the spin frame vectors in the ground state,
n(0)
x,n(0)
y,n(0)
z. These angles parametrize to the Gold-
stone modes arising from the spontaneous breaking of
spin-rotation symmetry.
At low energies, the Lagrangian in Eq. (31) can be
expanded in gradients of the fields ϕ. Retaining only the
quadratic terms yields [12,14,20]:
L=ρ
2∂tϕ2
x+∂tϕ2
y+∂tϕ2
z
−µ
2∇ϕ2
x+∇ϕ2
y+ 2∇ϕ2
z.
(33)
Where the gradient operator is defined as ∇= (∂x, ∂y)T.
This low-energy effective theory predicts three magnon
branches, each with distinct propagation speeds. Among
these, two branches (cIand cII) are degenerate, while the
third branch (cIII) has a higher speed:
cI, II =rµ
ρ, cIII =r2µ
ρ.(34)
A. Apparent Excess of Symmetries
The Lagrangian (33) exhibits three approximate shift
symmetries, ϕa→ϕa+αafor a=x, y, z. These symme-
tries lead to the Noether currents:
jt
a=ρ∂tϕa, ji
x,y =−µ∂iϕx,y, j i
z=−2µ∂iϕz,(35)
where a=x, y, z and the Latin indices run over x, y. The
conservation of these currents corresponds directly to the
first-order equations of motion.
Additionally, there is an SO(2) symmetry acting on
the field doublet (ϕx, ϕy) via the transformation
ϕa→ϕa+βϵab ϕb,(36)
where ϵab is the two-dimensional Levi-Civita symbol.
This transformation gives rise to a conserved current:
jt=ρϵabϕa∂tϕb, ji=−µϵabϕa∂iϕb,(37)
for a=x, y only. Like the N´eel antiferromagnet, these
systems exhibit approximate shift symmetries for each
magnon branch and feature a single residual SO(2) sym-
metry.
However, unlike in the N´eel case where the SO(2) sym-
metry can be directly traced to a residual symmetry of
the spontaneously broken SO(3), the situation here is
more subtle. In principle, in the triangular-lattice AFM,
the non-collinear magnetic order fully breaks SO(3), leav-
ing no obvious residual symmetry. This raises a funda-
mental question: Why does the symmetry count remain
comparatively high, and how is the SO(2) symmetry as-
sociated with the degenerate ϕxand ϕymagnon modes
formally defined and preserved?
A plausible explanation is that the order parameter
in non-collinear antiferromagnets admits a larger set of
transformations than SO(3). This extended symmetry
could account for the persistence of the effective SO(2)
[Eq. (36)] symmetry in the low-energy spectrum.
IV. THE CHIRAL SYMMETRY
In three-dimensional space, an SO(3) ma-
trix—representing a rigid body’s orientation—can
be specified in two distinct but equivalent ways, each in-
volving three angles. In one approach, the orientation is
achieved through a sequence of extrinsic rotations about
axes fixed in the global coordinate system; alternatively,
it can be achieved via intrinsic rotations—commonly
known as Euler rotations—performed about axes at-
tached to the body (or its principal axes). In our
formulation, we consider two right-handed, orthonormal
frames. The first is the fixed, global coordinate system
with basis vectors {ex,ey,ez}, and the second is a
body-fixed frame with basis vectors {nx,ny,nz}. The
relative orientation between these frames is fully encoded
by the rotation matrix O, which can be decomposed as
O=ETN, (38)
where
E=exeyez, N =nxnynz.(39)
Here, Eand Nare 3 ×3 matrices whose columns corre-
spond to the basis vectors of the global and body-fixed
6
frames, respectively. In this representation, each element
Oab is given by the inner product:
Oab =ea·nb,(40)
so that the bth column of Ocontains the components of
nbin the global frame, while the ath row provides the
components of eain the body frame, respectively. That
is,
(nb)a=ea·nb,(ea)b=ea·nb.(41)
Next, we examine how rotations relative to the global
and body frames act on O. An extrinsic rotation, defined
relative to the global frame and represented by a rotation
matrix L∈SO(3), transforms the components of the
body-fixed vectors as
(nb)a7→ Lac (nb)c,(42)
or equivalently, the full rotation matrix transforms as
Oab 7→ LacOcb ,or O7→ LO. (43)
Thus, extrinsic rotations are implemented by left-
multiplication of O. Conversely, an intrinsic rotation,
defined in the body-fixed frame and represented by R∈
SO(3), transforms the components of the global basis vec-
tors as
(ea)b7→ Rbc (ea)c,(44)
so that the rotation matrix transforms according to
Oab 7→ OacRcb =Oac (RT)cb,or O7→ ORT.(45)
Thus, intrinsic rotations correspond to right-
multiplication of O. Moreover, these two transfor-
mations can act simultaneously, and since left and right
multiplications act on different sides of O, extrinsic
and intrinsic rotations commute. Combined, these
simultaneous transformations define the chiral group,
G = SO(3)L×SO(3)R,(46)
where SO(3)Lcorresponds to extrinsic rotations and
SO(3)Rto intrinsic rotations, as hinted by the decom-
position of Ointo the Eand Nmatrices containing the
global and body axes, respectively
Therefore, in a non-collinear antiferromagnet the
transformations acting on the spin order parameter are
not merely ordinary rotations in three-dimensional space;
they form a slightly more complex group structure. Also,
we choose to retain the name chiral because of its
close resemblance to the chiral group in particle physics,
SU(2)L×SU(2)R—even though there is no a physical chi-
ral object here, as in contrast to QCD where the left and
right SU(2) groups act on left- and right-handed fermions
of the chiral condensate, respectively.
Since the Lie algebra of a direct product group is the
direct sum of the Lie algebras of its factors, the Lie alge-
bra of the chiral group is given by
so(3)L⊕so(3)R.(47)
Reflecting the two distinct ways of rotating the spin order
orientation, and forming a six dimensional algebra.
The chiral group (46) provides a concise and uni-
fied description of the transformations that incorporate
both extrinsic and intrinsic rotations acting on the spin
order. Interestingly, the resulting symmetry group is
highly reminiscent of the principal chiral model in par-
ticle physics, which is used to describe the dynamics of
mesons, such as pions [21].
V. O(4)-SYMMETRIC NLSM PERSPECTIVE
Much like the collinear N´eel antiferromagnet, which
admits an O(3) nonlinear sigma model description, a non-
collinear AFM on a triangular (or hexagonal) lattice can
be cast in an O(4) model. Here, rather than describing
the spin order through a rotation matrix Oor a spin
frame, it is more natural to introduce an order parameter
defined as a unit vector q∈S3(with antipodal points
identified). In this formulation, the Lie algebra of the
chiral group acts directly on vectors, avoiding the less
natural two-sided matrix multiplication required in the
previous representation (see Eq. (46)). This approach
naturally exploits the isomorphism
so(4) ∼
=so(3)L⊕so(3)R,(48)
and has the added advantage of treating all orientations
of the spin order equally on the three-sphere, reflecting
the close relationship between SO(3) and its double cover,
SU(2).
By introducing the four-component unit vector qas a
set of coefficients for the SU(2) matrix
U(q) = q0I−iq1σ1−iq2σ2−iq3σ3,(49)
where σaare the Pauli matrices, and using that the cor-
responding rotation matrix can be parametrized as
Oab =2q2
0−1δab + 2qaqb−2q0ϵabcqc,(50)
where a, b, c = 1,2,3, δab is the Kronecker delta, and
ϵabc is the Levi-Civita symbol. The theory describing
the triangular-lattice AFM described by Eq. (31), takes
the NLSM form:
L= 2ρh(e1·∂tq)2+ (e2·∂tq)2+ (e3·∂tq)2i
−2µh(e1·∇q)2+ (e2·∇q)2+ 2 (e3·∇q)2i.(51)
Here, where the gradient operator is defined as ∇=
(∂x, ∂y)T. Additional orthonormal tangent vectors ea
(with a= 1,2,3) describe the local deviations around
q. Since these vectors are orthogonal to q, together they
form a complete basis in four-dimensional space.
The more symmetric O(4)-NLSM formulation natu-
rally reveals the three distinct magnon branches that
characterize the low-energy spectrum of the antiferro-
magnet on a triangular lattice. Just as in the N´eel an-
tiferromagnet, where magnons are understood as small
7
deviations of the unit vector nfrom a reference state,
here magnons are interpreted as small deviations of the
four-dimensional vector qfrom a reference configuration,
with their contributions captured by the projection onto
the tangent-space vectors.
For a universal symmetry-breaking analysis and for
pedagogical purposes, we further reduce the system to
the fully symmetric O(4) principal chiral model (PCM)
in two spatial dimensions:
L= 2ρ(∂tq)2−2µ(∇q)2,(52)
This Lagrangian is invariant under global SO(4) rotations
acting on q∈S3. Such models capture topological exci-
tations (e.g., skyrmions, baby skyrmions) in non-collinear
or non-coplanar AFMs [22] and illuminate how additional
internal symmetries arise from the geometrical structure
of S3as we will see in the next section.
We will later revisit the triangular-lattice antiferro-
magnet described by Eq. (51), which represents a system
with lower symmetry than that of the principal chiral
model.
VI. CHIRAL SYMMETRY BREAKING
In the O(4)-NLSM formulation, the infinitesimal ex-
trinsic and intrinsic rotations act directly on the spin
order parameter q, rather than through two-sided ma-
trix multiplication. This simplification follows from the
isomorphism in Eq. (48), and takes the form:
dq =dϕaλa+dψaρaq. (53)
Here, dϕaand dψaare infinitesimal rotation angles as-
sociated with the global and body-fixed frames, respec-
tively. The matrices λagenerate extrinsic (global) rota-
tions, while ρagenerate intrinsic (body-fixed) rotations.
The explicit forms of these generators are
λ1=1
2
0−1 0 0
1000
000−1
0010
, λ2=1
2
0 0 −1 0
0 0 0 1
1 0 0 0
0−100
, λ3=1
2
0 0 0 −1
0 0 −1 0
0 1 0 0
1 0 0 0
,(54)
for global rotations, and
ρ1=1
2
0−100
1 0 0 0
0 0 0 1
0 0 −1 0
, ρ2=1
2
0 0 −1 0
0 0 0 −1
1 0 0 0
0 1 0 0
, ρ3=1
2
000−1
0010
0−1 0 0
1000
.(55)
for the generators of body-fixed rotations.
The commutation relations for these generators corre-
spond to two commuting so(3) algebras:
[λa, λb] = ϵabcλc,[ρa, ρb] = −ϵabc ρc,[λa, ρb] = 0.(56)
These relations define the Lie algebra of the SO(4) group,
which describes rotations in four-dimensional space. The
resulting algebra is six-dimensional, corresponding to the
six independent planes in 4D space where rotations can
occur. Notice that the generators of the body-axis rota-
tions, ρa, satisfy what are sometimes called anomalous
commutation relations [23,24].
The same algebra (56) appears in the quantum me-
chanical treatment of angular momentum in molecules
and more general rigid rotors, where the rotational states
of a rigid molecule are typically described in terms of its
total angular momentum and its projections onto both
globally fixed and the molecule’s internal axes (body-
fixed axes) [24].
The full invariance of the O(4)-NLSM (Eq. (52))
under global transformations of the chiral group
SO(3)L×SO(3)Rgives rise to conserved spin currents:
jα
L,a =∂L
∂(∂αq)·∂q
∂ϕa
= 4ρ ∂αq·λaq, (57)
jα
R,a =∂L
∂(∂αq)·∂q
∂ψa
= 4ρ ∂αq·ρaq. (58)
Here, the Greek index αdenotes the spacetime compo-
nents of the current, and we adopt the (+,−,−) metric
convention. Furthermore, we define the magnon speed of
propagation as c=pµ/ρ. It is important to emphasize
that the conservation of these currents, ∂αjα
(L,R),a = 0,
corresponds to the projections of the Landau–Lifshitz
equations onto the two sets of axes: the three global axes
and the three body-fixed axes.
The time components of these Noether currents rep-
resent the projections of the spin density, Sm(see
Eq. (30)), onto the fixed global and body-attached axes:
σL,a =jt
L,a =Sm·ea, σR,a =jt
R,a =Sm·na.(59)
To explore the breaking of the chiral symmetry by the
8
spin-order ground state, we take, without loss of gener-
ality, the ground state to be
qgs = (1,0,0,0).(60)
This state corresponds to full alignment between the
global frame and the spin-frame axes (i.e., ea=na). In
this configuration, both extrinsic and intrinsic rotations
are broken, as indicated by
λaqgs = 0, ρaqgs = 0.(61)
However, the breaking of the generators is not com-
plete but partial, since a particular combination of ex-
trinsic and intrinsic rotations leaves the ground state in-
variant. To isolate the unbroken part of the generators,
we define the vector and axial components of the chiral
group as
Va=λa−ρa, Aa=λa+ρa.(62)
Here, the vector generators satisfy the usual commuta-
tion relations of rotations in three dimensional space,
[Va, Vb] = ϵabcVc,(63)
while the commutators between the axial and vector gen-
erators are
[Aa, Vb] = ϵabcAc,[Aa, Ab] = ϵabc Vc.(64)
Indeed, the ground state of the O(4) nonlinear sigma
model (Eq. (52)) remains invariant under the subalgebra
generated by Va. This defines the residual group:
SO(3)V≡SO(3)L×SO(3)R=L−1.(65)
Transformations in this residual group correspond to con-
jugations (similarity transformations) of the rotation ma-
trix Oby another matrix L, explicitly given by
O→LOL−1.(66)
This implies that the residual SO(3)Vsymmetry is re-
alized through simultaneous extrinsic rotations of the
global axes and intrinsic rotations of the corresponding
body-fixed axes, performed by equal and opposite angles
(i.e., ϕa=−ψa, as introduced in Eq. (53)). In con-
trast, the axial generators are completely broken by the
ground state. Since the symmetries of the Lagrangian
differ from those of the ground state, we conclude that
the ground state of the principal chiral model breaks
SO(3)L×SO(3)Rdown to SO(3)V.
VII. THE ISOSPIN QUANTUM NUMBER
The breaking of chiral symmetry gives rise to three
massless Goldstone modes in the low-energy theory, each
associated with one of the broken axial directions Aa.
From Eq. (64), we observe that the axial generators Aa
transform as a 3-vector under the unbroken SO(3)Vsub-
group. Therefore, the Goldstone modes also transform
as a triplet under SO(3)V. Around the uniform ground
state defined in Eq. (60), these modes are parametrized
by the components q1,q2, and q3, as
q= (q0,q),(67)
where q0=p1−q2and q= (q1, q2, q3). The residual
SO(3)Vtransformations consist of rotations that act ex-
clusively on the components q1,q2, and q3, while preserv-
ing both q2
1+q2
2+q2
3and q2
0independently. This invari-
ance is manifestly reflected in the trace of the spin order
parameter O, which remains unchanged under residual
group transformations (65):
Tr[P OP −1] = Tr[O]=4q2
0−1.(68)
Here, Pis an arbitrary three-dimensional rotation ma-
trix, and we have used the relation (50). This reflects
the invariance of the Goldstone mode amplitude under
the residual SO(3)Vtransformations.
For this residual symmetry, we define a corresponding
Noether current. In particular, using Eqs. (57) and (62),
the conserved vector current is given by
jα
V,a = 4ρ ∂αq·Vaq. (69)
Its associated charge density is
σV,a =jt
V,a = 4ρϵabc qb∂tqc.(70)
Furthermore, combining Eq. (59) and (62), this charge
density can be interpreted as the difference between the
projections of the spin density onto the global and body-
fixed axes:
σV,a =Sm·ea− S m·na.(71)
We refer to Eq. (69) as the isospin (short for “iso-
topic spin”) current because the Lie algebra of the global
charges (i.e., Vagenerators) is identical to that of ordi-
nary spin. Although isospin is not related to intrinsic
angular momentum, it emerges as a conserved quantity
rooted in the symmetry properties of non-collinear anti-
ferromagnets. These charges, arising from the residual
SO(3)V, label an internal degree of freedom carried by
magnons. A similar situation occurs in particle physics,
where the spontaneous breaking of chiral symmetry by
the QCD vacuum leads to pions as Goldstone bosons,
with the corresponding isospin charge.
To see how the isospin charge can be used to label dif-
ferent magnon states—and, in particular, how it can be
promoted to a quantum label—we consider states near
the ground state (60). In this regime, where |q| ≪ 1,
the order parameter (67) takes the approximate form
q= (q0,q)≈(1,q), the chiral principal model La-
grangian (52) reduces to:
L= 2ρ(∂tq)2−2µ(∇q)2+. . . (72)
9
In the small-amplitude limit, the rotation matrix O=
eϕxLxeϕyLyeϕzLzcan be approximated by Oij ≈δij −
ϵijk ϕk. Comparing this with Eq. (50), we recognize that
the field triplet
q= (q1, q2, q3) = 1
2(ϕx, ϕy, ϕz) (73)
parametrizes the three Goldstone modes in the small-
amplitude regime, as specified in Eq. (32). Consequently,
the chiral principal model Lagrangian becomes
L=ρ
2(∂tϕ)2−µ
2(∇ϕ)2+. . . (74)
In contrast to the triangular Heisenberg antiferromagnet,
where only two of the branches are degenerate, the prin-
cipal chiral model features three Goldstone branches that
all propagate with the same velocity, cI,II,III =pµ/ρ. As
a consequence, the residual symmetry is not SO(2) but a
higher SO(3)V. Under a global SO(3)Vtransformation,
the Goldstone fields transform as
ϕa→ϕa+ϵabc βbϕc,(75)
where βbare small constant parameters and ϵabc is the to-
tally antisymmetric tensor. These transformations leave
the quadratic Lagrangian (74) invariant. The current
components in Eq. (69) become
jt
V,a =ρϵabcϕb∂tϕc, ji
V,a =µϵabcϕb∂iϕc.(76)
The above currents are the SO(3)Vanalogue of the
SO(2) charge density found in collinear Heisenberg anti-
ferromagnets (37).
To get the isospin quantum label, we quantize the three
fields ϕaby imposing the canonical commutator
ϕa(x), πb(y)=iℏδabδ2x−y,(77)
where the canonical momentum is defined in the standard
way as πa(x) = ∂L/∂(∂tϕa(x)). To separate left- and
right-handed polarization, we define
ϕ±=1
√2ϕx±iϕy, ϕ0=ϕz.(78)
To quantize the fields, we first introduce bosonic opera-
tors—namely, the annihilation operators ak,bk, and ck,
along with their corresponding creation operators a†
k,b†
k,
and c†
k—which satisfy the standard commutation rela-
tions (e.g., [ak, a†
k′] = δk,k′, etc.). In terms of these oper-
ators, the fields are expanded in plane-wave modes with
well-defined frequency, momentum, and polarization as
follows:
ϕ+(x) = sℏ
ρA X
k
1
√2ωbke−ikx +a†
keikx,
ϕ−(x) = sℏ
ρA X
k
1
√2ωake−ikx +b†
keikx,
ϕ0(x) = sℏ
ρA X
k
1
√2ωcke−ikx +c†
keikx,
(79)
where kx =ωt −k·x,Ais the sample area, and the
dispersion relation is given by ω=pρ/J |k|.
The polarization is directly related to the third com-
ponent of the isospin density, as given in Eq. (76). The
corresponding global isospin charge is obtained by inte-
grating this density over all space. In terms of the quan-
tized fields, becomes
QV,3= :Zd2x σV,3:= ℏX
ka†
kak−b†
kbk.(80)
Here, the symbol :: denotes normal ordering of opera-
tors. Defining the total number operators for the quanta
ϕ+and ϕ−as Naand Nb, respectively. The isospin
charge becomes
QV,3=ℏ(Na−Nb).(81)
Therefore, magnons come in three different charge states
defined by their isospin projection I3:
I3(ϕ+)=+ℏ, I3(ϕ0)=0, I3(ϕ−) = −ℏ.(82)
This represents one of the central results of this paper.
Here, we have arbitrarily chosen the third axis, taking
advantage of the full rotational SO(3)Vsymmetry of the
principal chiral model. In contrast, for an antiferromag-
net on a triangular lattice, the axis choice is not arbitrary
but is physically restricted to be associated to the direc-
tion orthogonal to the spin-plane.
A. Isospin in a Heisenberg AFM in a triangular
lattice
Even though the ground states of both the principal
chiral model and the antiferromagnet on a triangular lat-
tice, without loss of generality, defined by
O=ETN=I,(83)
are fully symmetric under SO(3)V(65)—that is, invari-
ant under transformations of the form O→P OP −1=
O—their low-energy excitations reveal a fundamental dif-
ference between the two systems.
In the principal chiral model, the residual SO(3)Vsym-
metry remains manifest in the Goldstone modes because
all three spin-wave branches are degenerate. In contrast,
for the triangular lattice antiferromagnet only two of the
three branches are degenerate, with cI,II =cIII/√2. As
a consequence, although the ground state retains full
SO(3)Vinvariance, the magnon excitations in the tri-
angular lattice antiferromagnet do not. Instead, the
residual symmetry is reduced to the smaller subgroup
SO(2)V,3⊂SO(3)V, as hinted in Sec. III A. Here, the
subscript 3 indicates that the residual isospin symme-
try for the triangular antiferromagnet consists solely of
matrix conjugation by Rz, with the z-axis defined as per-
pendicular to the spin plane in the ground state.
10
Solutions that are SO(2)V,3symmetric correspond to
circularly polarized magnons. For instance, magnons
with right-handed circular polarization and finite ampli-
tude Θ0around the ground state (83) have the following
spin order parameter:
O=Rz(Φ) Rx(Θ0)R−1
z(Φ),(84)
where Φ = ωt −k·x. The resulting matrix describes
a rotation by Θ0about the axis n=Rz(Φ)(1,0,0)T=
(cos Φ,sin Φ,0)T. Alternatively, the above rotation ma-
trix is equivalent to an Euler rotation in the (3-1-3) con-
vention with Euler angles (ϕ, θ, ψ) = (Φ,Θ0,−Φ).
The resulting dynamics of the spin order Eq.(84) corre-
spond to a periodic “wobbling” motion of the spin plane
around the ground state, see Fig.(4)—akin to a coin wob-
bling on a table, but without an overall rotation of the
coin. In this picture, the angular displacement Θ0acts
as the coin’s tilt, while the phase Φ causes the spin plane
to oscillate about its equilibrium. Unlike a coin that
might roll or spin, the spin order remains anchored to
the ground state, with only its orientation undergoing
oscillatory deviations.
nx
ny
nz
m1
m2
m3
(a) (b)
FIG. 4. Trajectory of the spin order for a well-defined
isospin. Panel (a) shows the trajectories of the spin directions
ma, with a= 1,2,3, for right-handed circularly polarized
magnons. Panel (b) displays the corresponding trajectories
of the spin-frame vectors. Notably, the xand ycomponents
of nxand nyoscillate at twice the frequency of the other com-
ponents, producing Lissajous-like patterns on the unit sphere.
In contrast, nzundergoes uniform precession around its equi-
librium axis, ez, at frequency ω.
This finite amplitude oscillatory behavior has a direct
impact on the magnon dispersion. In fact, it modifies the
dispersion relation for circularly polarized magnons to
ω2=2−cos Θ0c2
I|k|2,(85)
where cI=pµ/ρ. Thus, the effective “stiffness” is mod-
ified by the finite wave amplitude Θ0.
Furthermore, using Eqs. (59) and (62), the axial and
vector (isospin) charge densities are given by
σA,3= 0, σV,3= 4ρω sin2(Θ0/2) .(86)
For small amplitudes, we recover that σV,3is equiva-
lent to the charge density in (37) for circularly polarized
magnons, where the amplitude is related to the infinites-
imal angle of rotations by Θ0=qϕ2
x+ϕ2
y.
VIII. DISCUSSION
We have presented a detailed theoretical analysis of the
symmetry group of systems described by an SO(3) order
parameter, with particular emphasis on the non-collinear
magnetic order of the Heisenberg antiferromagnet on a
triangular lattice. Using the principal chiral model as a
guiding framework, we demonstrated that the full sym-
metry group of these systems is, in fact, the chiral group
SO(3)L×SO(3)R—encompassing both extrinsic (glob-
ally fixed) and intrinsic (body-fixed) rotations—rather
than the simpler, originally assumed SO(3).
The chiral group naturally decomposes into a vector
part and an axial part, each consisting of specific com-
binations of extrinsic and intrinsic rotations. The mag-
netic order fully breaks the axial part, giving rise to three
magnons associated with the three broken axial genera-
tors. Meanwhile, the vector subgroup remains unbroken
and defines the residual symmetries of the vacuum.
Using this framework, we clarify the internal SO(2)
symmetry responsible for the degeneracy of two spin-
wave branches in the triangular-lattice Heisenberg anti-
ferromagnets as the residual SO(2)V,3symmetry emerges
from a particular combination of joint rotations about the
global axis ezand the local spin-frame axis nz, inherited
from the larger chiral group.
Furthermore, these results reveal a new quantum
observable that distinguishes left- and right-circular
magnons, going beyond the usual classification by mo-
mentum and energy. Classically, these modes are simply
oppositely polarized spin waves. Once quantized, how-
ever, they form an isospin doublet with total isospin
I=ℏ, split into projections I3=±ℏ. This additional
quantum label thus complements the standard momen-
tum and energy labels, reflecting the deeper symmetry
structure inherent to non-collinear antiferromagnets.
The control and manipulation of the emergent isospin
of magnons opens up exciting prospects for engineer-
ing interactions that depend solely on the isospin charge
states. For example, a recent study of a fully symmet-
ric O(4)-NLSM model [25]––which serves as a concep-
tual framework for both spin glass and frustrated an-
tiferromagnet systems––revealed a distinctive Hall-like
response arising when magnons in different polariza-
tion states scatter off various types of three-dimensional
topological solitons. Under the lens of our work, these
magnons can be understood as distinct isospin charge
states that interact with topological solitons via an
isospin-dependent coupling.
We surmise that this higher-symmetry framework
could be extended to other families of non-collinear an-
tiferromagnets with more intricate symmetry-breaking
patterns. In such systems, we anticipate that analogous
11
residual symmetries will play a fundamental role in defin-
ing similar quantum numbers and imposing additional
conservation laws.
ACKNOWLEDGMENTS
We thank Boris Ivanov, Ibrahima Bah, and especially
Hua Chen for a series of stimulating and insightful dis-
cussions. This research was supported by the U.S. De-
partment of Energy under Award No. DE-SC0019331
and the U.S. National Science Foundation under Grant
No. NSF PHY-2309135.
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Appendix A: SO(3)L×SO(3)R
Consider two right-handed orthonormal frames: a global coordinate system with basis vectors {ex,ey,ez}, and
abody-attached frame with basis vectors {nx,ny,nz}. The relative orientation between these two frames is fully
specified by a rotation matrix O, which can be decomposed as
12
O=ETN, (A1)
where
E=exeyez, N =nxnynz.(A2)
Explictly:
O=
ex·nxex·nyex·nz
ey·nxey·nyey·nz
ez·nxez·nyez·nz
.(A3)
Now, we assume that the global (lab) frame is defined by the standard Cartesian axes, so that the matrix Eis the
identity, i.e., E=I. In this case, the rotation matrix relating the body frame to the global frame simplifies to
O=ETN=ITN=N. (A4)
Now, consider multiplying Ron the left by a rotation about the z-axis, denoted by
Rz(θ) =
cos θ−sin θ0
sin θcos θ0
0 0 1
.(A5)
The new rotation matrix is given by
O′=Rz(θ)O=Rz(θ)N. (A6)
Since left-multiplication by Rz(θ) corresponds to rotating the global coordinate system, this operation effectively
re-expresses the body frame axes (originally given by the columns of N) in a new, globally rotated frame. In other
words, each column of O′is obtained by applying the rotation Rz(θ) to the corresponding body axis:
Extrinsic rotations: O′=Rz(θ)nxRz(θ)nyRz(θ)nz.(A7)
Thus, the effect of the left multiplication is to modify the representation of the body frame by rotating the lab frame
about the z-axis by an angle θ. The new rotation matrix is then given by
O′=O Rz(θ) = N Rz(θ).(A8)
Right multiplication by Rz(θ) applies the rotation in the body frame. That is, the operation rotates the body axes
while keeping the global frame fixed. In particular, if the columns of Nare the original body axes nx,ny, and nz,
then the updated axes (the columns of O′) are given by
Intrinsic rotations: O′=cos θnx−sin θnycos θny+ sin θnxnz.(A9)
Thus, the new rotation matrix O′encapsulates the effect of rotating the body frame about its own z-axis by an
angle θ, while the global frame remains unchanged. Note that the rotation acts clockwise on the axes nx,ny.
Appendix B: Rotations in four dimensions
In four-dimensional space, rotations of a vector can be classified into three main types, each with distinct charac-
teristics:
Simple Rotations: A simple rotation is essentially a two-dimensional rotation embedded in 4D. Only one of the
two orthogonal planes is actively rotated by a nonzero angle, while the perpendicular plane remains fixed. In other
words, the rotation occurs in only one plane.
Double Rotations: A double rotation involves simultaneous rotations in two orthogonal planes by different angles
(say, θand ϕ). This can be viewed as two independent rotations occurring at the same time. Unlike in 3D, there is
13
no single axis of rotation; instead, the motion is distributed across two independent 2-planes. Double rotations are
unique to four dimensions and higher. Schematically, they take the form:
R(θ) 0
0R(ϕ).(B1)
Isoclinic Rotations: An isoclinic rotation is a special case of a double rotation where the two rotation angles
are equal in magnitude (or equal up to a sign), i.e., θ=ϕor θ=−ϕ. This means that every vector in R4is
rotated through the same angle, even though there is no unique fixed axis. Schematically, an isoclinic rotation can be
represented as:
R(θ) 0
0R(θ)or R(θ) 0
0R(−θ).(B2)
These two cases correspond to left and right isoclinic rotations, respectively. They are identified with the two
independent actions of
so(4) ∼
=so(3)L⊕so(3)R.(B3)
Appendix C: so(4) Algebra
A standard way to describe the Lie algebra so(4) is to introduce six generators JAB (with A, B = 0,1,2,3 and
A<B) that represent an infinitesimal rotation in the (xA, xB)-plane. Their entries are defined by
(JAB )CD =δAC δB D −δADδB C .(C1)
These generators are antisymmetric (JAB =−JBA) and satisfy the usual so(4) commutation relations.
It is convenient to separate these generators into two groups:
•The three generators J0a(with a= 1,2,3) that mix the 0-component with the “spatial components”.
•The three generators Jab (with a, b = 1,2,3) that generate rotations in the three-dimensional “spatial” subspace.
Explicitly, the “boost-like” generators, which mix the 0 component with a “spatial” index, are given by:
J01 =
0−100
+1 0 0 0
0 0 0 0
0 0 0 0
, J02 =
0 0 −1 0
0000
+1 0 0 0
0000
, J03 =
0 0 0 −1
0 0 0 0
0 0 0 0
+1 0 0 0
.(C2)
The spatial generators, which generate rotations in the three-dimensional subspace, are:
J12 =
0 0 0 0
0 0 −1 0
0 +1 0 0
0 0 0 0
, J13 =
0000
000−1
0000
0 +1 0 0
, J23 =
0 0 0 0
0 0 0 0
0 0 0 −1
0 0 +1 0
.(C3)
All these generators correspond to the simple rotations discussed above. In particular, the J0aare associated
with the axial generators, Aa, while the J1aare associated with the vector generators, Va, as defined in the main text.
On the other hand, a double rotation is generated by a linear combination of two commuting generators that act
in two orthogonal planes. For instance, if we consider the (x0, x1) and (x2, x3) planes, a typical generator is
G=θ J01 +ϕ J23,(C4)
where J23 is defined analogously to J01 for the (x2, x3)-plane.
Finally, an isoclinic rotation is a special case of a double rotation where the two rotation angles are equal in
magnitude (or equal up to a sign), i.e., θ=ϕor θ=−ϕ. In such rotations, every vector in R4is rotated through the
same angle, even though there is no unique fixed axis. Using the same two planes as before, an isoclinic rotation is
generated by
G±=θ(J01 ±J23).(C5)
14
For G+=θ(J01 +J23), exponentiation yields a rotation by θin both planes; for G−=θ(J01 −J23), the rotation is
by θin one plane and by −θin the other.
To formally describe the isoclinic rotations, we “split” so(4) into two parts:
so(4) ∼
=so(3)L⊕so(3)R.(C6)
This is accomplished by forming the (anti)self-dual combinations of the generators:
λa=1
2J0a+1
2ϵabc Jbc, ρa=1
2J0a−1
2ϵabc Jbc,(C7)
where ϵabc is the Levi-Civita symbol. (Alternatively, one may define ρawith an overall minus sign; the choice here is
conventional so that the λaand ρasatisfy)
[λa, λb] = ϵabc λc,[ρa, ρb] = −ϵabc ρc,(C8)
and
[λa, ρb]=0.(C9)
In other words, the combination J0a+1
2ϵabc Jbc generates the left-action (associated with λa), while the combination
J0a−1
2ϵabc Jbc generates the right-action (associated with ρa).
For example, for a= 1 we have
λ1=1
2J01 +J23=1
2
0−1 0 0
+1 0 0 0
000−1
0 0 +1 0
.(C10)
Similarly, one finds
λ2=1
2J02 +J31=1
2
0 0 −1 0
0 0 0 +1
+1 0 0 0
0−1 0 0
,(C11)
and
λ3=1
2J03 +J12=1
2
000−1
0 0 −1 0
0 +1 0 0
+1 0 0 0
.(C12)
For the right-sector generators, we define
ρ1=1
2J01 −J23=1
2
0−1 0 0
+1 0 0 0
0 0 0 +1
0 0 −1 0
,
ρ2=1
2J02 −J31=1
2
0 0 −1 0
000−1
+1 0 0 0
0 +1 0 0
,
ρ3=1
2J03 −J12=1
2
000−1
0 0 +1 0
0−1 0 0
+1 0 0 0
.
(C13)
15
Appendix D: Isospin spin-wave solutions
In this subsection, we derive an exact solution for waves with zero axial charge density and nonzero vector charge
density. We begin by expanding the Lagrangian density from Eq. (31) in terms of Euler angles (following convention
“313”). The resulting Lagrangian is
L=ρ
2h˙
Θ2+ sin2Θ˙
Φ2+˙
Ψ + cos Θ ˙
Φ2i−µ
2h(∇Θ)2+ sin2Θ (∇Φ)2+ 2∇Ψ + cos Θ∇Φ2i.(D1)
The axial and vector charge densities are given by
σA,3= 2ρcos2Θ
2(∂tΦ + ∂tΨ) , σV,3= 2ρsin2Θ
2(∂tΦ−∂tΨ) .(D2)
Zero axial charge density implies that σA,3= 0, which in turn requires that both Θ and Φ + Ψ remain constant.
Indeed, for circularly polarized solutions with a uniform isospin (vector) charge density, the trace of the spin order
parameter
Tr[O] = ea·na= 4q2
0−1 = 4 cos2Θ
2cos Φ+Ψ
2−1 (D3)
remains constant.
To obtain plane-wave solutions that reduce to the circular polarization ϕx±i ϕyin the small-amplitude limit, we
adopt the following ansatz:
Θ(t, r)=Θ0,
Φ(t, r) = ωt −k·r+ Φ0,
Ψ(t, r) = −ωt +k·r+ Ψ0,
(D4)
where Θ0, Φ0, and Ψ0are constants.
ρ¨
Θ + sin Θ ˙
Φ˙
Ψ−µ∇2Θ + µsin Θ cos Θ(∇Φ)2−2µsin Θh(∇Ψ + cos Θ∇Φ) · ∇Φi= 0 ,
ρ∂th∂tΦ + cos Θ∂tΨi−µ∇ · hsin2Θ∇Φ + 2 cos Θ∇Ψ + cos Θ∇Φi= 0,
ρ∂th∂tΨ + cos Θ∂tΦi−2µ∇ · ∇Ψ + cosΘ∇Φ= 0.
(D5)
Since Φ and Ψ are linear in time, their equations of motion are automatically satisfied; they reduce to the conser-
vation of currents. Notice that
∂tΦ + cos Θ0∂tΨ = (1 −cos Θ0)ω, sin2Θ∇Φ + 2 cos Θ∇Ψ + cosΘ ∇Φ=−(1 −cos Θ0)2k.(D6)
and
∂tΨ + cos Θ0∂tΦ = (cosΘ0−1)ω, ∇Ψ + cos Θ0∇Φ = (1 −cos Θ0)k.(D7)
The only no trivial equation corresponds to the equation of motion for Θ, which reduces to
ρ ω2sin Θ0−µ|k|2sin Θ02−cos Θ0= 0.(D8)
For nontrivial solutions (with sin Θ0= 0), we obtain the dispersion relation:
ω2=2−cos Θ0c2
I|k|2,(D9)
where c2
I=µ/ρ. The effective “stiffness” is modified by the wave amplitude Θ0. In the limit of vanishing amplitude,
Θ0→0, we recover
ω2=c2
I|k|2.(D10)
16
In the language of rotation matrices, the order parameter for these solutions is constructed as the conjugation of a
rotation about the x-axis, Rx(Θ0), by a rotation about the z-axis:
O=Rz(Φ) Rx(Θ0)R−1
z(Φ).(D11)
This operation rotates the x-axis into a new direction. Specifically, applying Rz(Φ) to the unit vector (1,0,0)Tyields
ˆ
n=Rz(Φ)(1,0,0)T= (cos Φ,sin Φ,0)T,(D12)
so that the net effect is equivalent to a rotation by an angle Θ0about the axis n= (cos Φ,sin Φ,0)Twithin the spin
plane defined by the ground state. Since Φ = ωt −k·x, this axis rotates in the spin plane with a constant angular
velocity ω.
A useful identity connecting SO(3) matrices and SU(2) is given by
Oab =2q2
0−1δab + 2qaqb−2q0ϵabcqc,(D13)
where a, b, c = 1,2,3, δab is the Kronecker delta, and ϵabc is the Levi-Civita symbol. More explicitly, the rotation
matrix can be written as
O(q) =
2q2
0+q2
1−1 2 (q1q2−q0q3) 2 (q1q3+q0q2)
2 (q1q2+q0q3) 2 q2
0+q2
2−1 2 (q2q3−q0q1)
2 (q1q3−q0q2) 2 (q2q3+q0q1) 2 q2
0+q2
3−1
.(D14)
Using the above identity, the trace of the spin order parameter Ris given by
Tr[O]=4q2
0−1 = 4 cos2Θ
2cos Φ+Ψ
2−1,(D15)
which remains constant for solutions with well-defined isospin density. Assuming that Φ0+ Ψ0= 0 (or, equivalently,
absorbing any constant phase into the redefinition of Θ0), we have
Tr[O] = ea·na= 1 + 2 cos Θ0
≈3−Θ2
0,(D16)
for small-amplitude spin waves. Therefore, near the ground state, the trace of Oprovides a direct measure of the
spin-wave amplitude.
