We study antiferromagnetic systems with enhanced “nearly-*SU*(*N*)” symmetry, which can be realized in systems of ultracold spinor atoms in optical lattices. Examples of *N* = 3 (for *S* = 1 bosons) and *N* = 4 (for $S=32$ fermions) are considered. Near the *SU*(*N*) point, the low-energy physics can be described by the *CP*^{N−1} model with an additional symmetry-breaking term lowering the symmetry down to *SU*(2) and favoring the Néel ordering. We show that the effective theory of such systems can be cast in the form of a nonlinear sigma model with the *SO*(3) matrix-valued field, which is typically obtained for frustrated magnets with non-collinear order. Further, we show that those systems possess a peculiar effect of topological binding: for a system with the underlying spin *S*, lowering of the symmetry from *SU*(2*S* + 1) to *SU*(2) leads to binding of topological unit-charge excitations of the *CP*^{2S} model (skyrmions for space dimension *d* = 2, instantons for *d* = 1, and hedgehogs for *d* = 3) into 2*S*-multiplets.

## I. INTRODUCTION

Through the past several decades, low-dimensional quantum magnets have steadily attracted attention of researchers, in particular as a convenient playground for effects involving topologically nontrivial excitations. The advent of ultracold gases has boosted this interest, as they provide access to unconventional spin states hardly achievable in crystalline materials. In particular, spinor atoms loaded into optical lattices^{1} in the Mott insulator regime are described by effective spin models with strong non-Heisenberg exchange, providing a route^{2} to the realization of spin Hamiltonians possessing enhanced *SU*(*N*) symmetries with *N* > 2. The low-energy physics of *SU*(*N*) antiferromagnets is captured by the *CP*^{N−1} model, with topological terms playing a crucial role in the case of low spatial dimensionality *d* < 3.

For such highly symmetric systems, even weak additional interactions might become important if they break the enhanced symmetry. If such a perturbation favors the Néel order, it is natural to assume that the resulting physics will be described by the standard *O*(3) nonlinear sigma model (NLSM) that is well-known to be the effective theory of “common” Heisenberg antiferromagnets. It has been argued^{3} that in a spin-1 *SU*(3) antiferromagnet perturbations that bring the symmetry down to *SU*(2) can lead to the pairing of topologically nontrivial excitations of the *CP*^{2} theory, and such a pair consisting of two excitations with a unit topological charge of *CP*^{2} corresponds precisely to a unit-charge topological excitation of the *O*(3) NLSM. At the same time, a perturbation that breaks the *SU*(*N*) symmetry down to *O*(*N*) with *N* > 3 does not affect topological excitations.^{4}

In the present paper, we consider two spin-*S* models with non-Heisenberg exchange, for *S* = 1 and *S* = 3/2, which realize *SU*(2*S* + 1) antiferromagnets on a hypercubic lattice in *d* spatial dimensions. The model with *S* = 1 is realized by the Hamiltonian^{5}

while the model with *S* = 3/2 is given by:^{6}

Here $S\u20d7\u0302j$ are spin-*S* operators at the lattice site *j*, and ⟨…⟩ denotes the sum over nearest neighbors. We perturb those *SU*(2*S* + 1)-invariant models

by the term that breaks the symmetry down to *SU*(2). We choose *λ* > 0 which favors the antiferromagnetic spin ordering.

The effective low-energy continuum theory for the unperturbed *SU*(*N*) antiferromagnets (1, 2) is given by the well-known *CP*^{N−1} model described by the following euclidean action^{7}

Here the Planck constant and the lattice spacing are set to unity, Λ is the ultraviolet momentum cutoff, the *N*-component complex vector field $z\u20d7$ is subject to the unit length constraint $z\u20d7\u2020z\u20d7=1$, $D\mu =\u2202\mu \u2212iA\mu $ is the gauge covariant derivative, and $A\mu =\u2212i(z\u20d7\u2020\u2202\mu z\u20d7)$ is the gauge field, *x*^{0} = *cτ*, *τ* = *it* is the imaginary time, the limiting velocity is $c=2d$, and the bare coupling constant $g0=d$. The topological term in the action

where the phase factors *η*_{j} = ±1 take opposite signs at lattice sites belonging to A and B sublattices, can be cast in the continuum form only for *d* = 1.^{8} Without the topological term, the action (4) can be viewed as the energy of the static (*d* + 1) dimensional “classical” spin texture.

The perturbed effective action can thus be cast in the form

where $m02=2g0\lambda /(c\Lambda d\u22121)>0$ is proportional to the perturbation strength, $\u27e8S\u20d7\u27e9=z\u20d7\u2020Saz\u20d7$ is the spin average, S^{a} being spin-*S* matrices, *a* = 1, 2, 3.

Naively, one would expect that the effective theory for *perturbed* systems (3) is the *O*(3) NLSM, since this perturbation favors the antiferromagnetic ordering, and the *O*(3) NLSM^{8} is well-known to be the effective theory describing collinear antiferromagnets (see, e.g., Ref. 9 for a review). We will show that this is not the case. The *O*(3) NLSM might be visualized as a theory describing the dynamics of the “infinitely thin arrow” (the Néel vector) whose inertia momentum with respect to the arrow axis is zero, and is the usual theory for antiferromagnets that classically have collinear order. In contrast to that, frustrated antiferromagnets with non-collinear (classical) order are described by more than one Néel vector, so the resulting theory can be visualized as describing the dynamics of a rigid top with all three inertia momenta being nonzero; the field in such a theory is the *SO*(3) rotation matrix.

In the case of weakly perturbed *SU*(*N*) antiferromagnets on bipartite lattices, the induced Néel order is collinear. However, we will see that the third inertia momentum is universally generated in such systems by fluctuations of massive fields that correspond to quadrupolar degrees of freedom, and the resulting effective theory is not the NLSM of a unit-vector field, but the NLSM of a *SO*(3) matrix field. Its homotopy groups are different from those of the *O*(3) NLSM. One can still define the unit Néel vector, but the known textures of the *O*(3) NLSM such as skyrmions or hedgehogs will not, strictly speaking, be topologically protected. Nevertheless, the topological charge of skyrmions or instantons of the *CP*^{2S} model remains well-defined. We will show that, quite generally, the *SU*(2*S* + 1)↦*SU*(2) perturbation leads to binding of unit-charge topological configurations of the *CP*^{2S} model into 2*S* multiplets.

## II. THE EFFECTIVE THEORY OF THE PERTURBED *CP*^{2S} MODEL FOR *S* = 1, $32$

In order to effectively describe the perturbed theory, we separate modes that became massive under the perturbation. For this purpose, we present the vector field in the form

where $Dmm\u2032(j)$ is the Wigner matrix that represent a rotation in (2*j* + 1)-dimensional space and depends on the three Euler angles as^{10}

where $Pj\u2212m\u2032(m\u2032\u2212m,m\u2032+m)(x)$ are the Jacobi polynomials. The spin state $\psi \u20d7$ is chosen in the form that renders diagonal the on-site spin quadrupolar tensor

and its length $\rho =|\psi \u20d7|$ satisfies the normalization constraint *ρ* = 1. The gauge is later fixed by setting the overall phase *γ* to zero.

### A. *S* = 1

For *S* = 1, the condition of (9) being diagonal yields

It is convenient to introduce fields *h*, $h\u0303$ as $h+ih\u0303=sin\u2061\beta ei\gamma $. The change of variables (7) results in the transformation Jacobian $DRe(z\u20d7),Im(z\u20d7)D\alpha ,\theta ,\phi ,\rho ,h,h\u0303=\rho 5\u2061sin\u2061\theta $, which is the part of the usual measure in the integration over the rotation group space (*α*, *θ*, *φ*). Variables *ρ* = 1, $h\u0303=0$ are fixed by the constraints.

Retaining up to quadratic terms in massive field *h*, one obtains the action in the following form:

While the first term in the last equation corresponds to the well-known *O*(3) NLSM action, one can see that the presence of quadrupolar fluctuations (the massive field *h*) dynamically generates the third inertia momentum. Integrating out the massive field, one obtains the action

where Γ = 2*g*_{0} and the dynamically generated coupling *G* is given by

The action (12) can be conveniently rewritten in the form of the *SO*(3) NLSM whose field is the rotation matrix *R* ∈ *SO*(3):

where *P* = diag(1, 1, *ζ*) and *ζ* = Γ/*G*. The matrix *R* connects to the fields *α*, *θ*, *φ* by the relations

where *ω*_{μa} are the rotation “frequencies” defined as

### B. *S* = 3/2

Antiferromagnetic perturbation (3) favors field configurations with small *ϑ*, *β*. It is convenient to introduce real fields $(hx,hy,hz,h\u0303z)$:

The change of variables (7) results in the transformation Jacobian $DRe(z\u20d7),Im(z\u20d7)D\alpha ,\theta ,\phi ,\rho ,hx,hy,hz,h\u0303z=916\rho 7\u2061sin\u2061\theta $. Again, *ρ* = 1 and $h\u0303z=0$ are fixed by the constraints, and sin *θ* goes into the usual integration measure for the rotation group.

Retaining up to quadratic terms in powers of the massive fields $h\u20d7=(hx,hy,hz)$, we obtain the action with the structure very similar to the expression (11) for *S* = 1:

where $ANLSM$ is the action of the *O*(3) nonlinear sigma-model,

$Am$ is the quadratic action of the massive field,

and $Aint$ describes the interaction,

Again, we see that fluctuations of the massive field $h\u20d7$ dynamically generate the third inertia momentum. Integrating out the massive field, one readily obtains the effective action of the same *SO*(3) NLSM form (14), with the couplings given by

Renormalization properties of the *SO*(3) NLSM have been studied by several authors.^{11,12} For *d* = 1, the ratio *ζ* flows to the *O*(4) fixed point *ζ* = 1, while Γ flows to infinity, indicating the dynamic generation of a finite correlation length. For *d* = 2, the *SO*(3) NLSM possesses long-range AF order, and couplings Γ, *ζ* flow to finite values.

## III. CLUSTERING OF TOPOLOGICAL EXCITATIONS IN *SU*(2)-PERTURBED SPIN-*S SU*(2*S* + 1) ANTIFERROMAGNET

The effective theory that we obtained, the NLSM with matrix field *R* ∈ *SO*(3), may be visualized as rotating axisymmetric top, whose axis coincides with the direction $n\u20d7(\theta ,\phi )$ of the local spin average, and the extra angular variable *α* corresponds to the angle of rotation of that top around its symmetry axis. This model has a trivial second homotopy group, *π*_{2}(*SO*(3)) = 0. However, one can still define the Néel unit vector $n\u20d7(\theta ,\phi )$, and introduce the fictitious topological charge of the *O*(3) NLSM with unit vector field $n\u20d7(\theta ,\phi )$, namely

At the same time, the topological charge in the underlying *CP*^{2S} model,

remains well defined.

The antiferromagnetic *SU*(2) perturbation favors field configurations with the maximal spin length. Imposing that condition of maximizing the spin length amounts to the following ansatz describing an arbitrary rotation of the maximum-weight spin state:

Plugging that into (25), one obtains the relation that is satisfied for the “restricted” field configurations of (26):

This suggests that the *SU*(2) AF perturbation leads to clustering of skyrmions of the *CP*^{2S} model into groups with the total charge $qCP2S$ being a multiple of 2*S*. For the *S* = 1 *SU*(3) case, this argument can be corroborated by direct energetic considerations:^{4} for example, it is possible to construct $qCP2=2$ skyrmion solutions of the *CP*^{2} model that remain exact solutions even in presence of the symmetry-breaking perturbation (3), fully minimizing the energy contribution from the perturbation term. At the same time, the energy of a $qCP2=1$ skyrmion under the perturbation obtains contribution proportional to the square of its size, indicating instability.

## IV. SUMMARY

We have studied spin-*S* antiferromagnets with non-Heisenberg exchange on a bipartite lattice, in the vicinity of highly symmetric *SU*(2*S* + 1) points, for *S* = 1 and *S* = 3/2. We have shown that if the *SU*(2*S* + 1) symmetry gets broken down to *SU*(2) by the perturbation favoring antiferromagnetic ordering, the effective theory of such a perturbed system, is the *SO*(3) nonlinear sigma model (NLSM), though there is no non-collinearity in the Néel order. The dynamic generation of the third inertia momentum, which leads to the transformation of the effective theory into the *SO*(3) NLSM, is caused by fluctuations of massive fields that correspond to non-axisymmetric deformations of the quadrupolar tensor. We argue that the perturbation leads to the clustering of topological excitations of the *CP*^{2S} model into groups with the total charge $qCP2S$ being a multiple of 2*S*.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.