Quantum antiferromagnets near SU(N) symmetry

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 lattices¹ in the Mott insulator regime are described by effective spin models with strong non-Heisenberg exchange, providing a route² 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 argued³ 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 CP² theory, and such a pair consisting of two excitations with a unit topological charge of CP² 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(2S + 1) antiferromagnets on a hypercubic lattice in d spatial dimensions. The model with S = 1 is realized by the Hamiltonian⁵ 

while the model with S = 3/2 is given by:⁶ 

Here $S⃗̂j$ are spin-S operators at the lattice site j, and ⟨…⟩ denotes the sum over nearest neighbors. We perturb those SU(2S + 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 action⁷ 

Here the Planck constant and the lattice spacing are set to unity, Λ is the ultraviolet momentum cutoff, the N-component complex vector field $z⃗$ is subject to the unit length constraint $z⃗†z⃗=1$⁠, $Dμ=∂μ−iAμ$ is the gauge covariant derivative, and $Aμ=−i(z⃗†∂μz⃗)$ is the gauge field, x⁰ = 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 $m02=2g0λ/(cΛd−1)>0$ is proportional to the perturbation strength, $⟨S⃗⟩=z⃗†Saz⃗$ 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) NLSM⁸ 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(2S + 1)↦SU(2) perturbation leads to binding of unit-charge topological configurations of the CP^2S model into 2S multiplets.

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 $Dmm′(j)$ is the Wigner matrix that represent a rotation in (2j + 1)-dimensional space and depends on the three Euler angles as¹⁰ 

where $Pj−m′(m′−m,m′+m)(x)$ are the Jacobi polynomials. The spin state $ψ⃗$ is chosen in the form that renders diagonal the on-site spin quadrupolar tensor

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

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

It is convenient to introduce fields h, $h̃$ as $h+ih̃=sin⁡βeiγ$⁠. The change of variables (7) results in the transformation Jacobian $DRe(z⃗),Im(z⃗)Dα,θ,φ,ρ,h,h̃=ρ5⁡sin⁡θ$⁠, which is the part of the usual measure in the integration over the rotation group space (α, θ, φ). Variables ρ = 1, $h̃=0$ 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 = 2g₀ 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

For $S=32$⁠, the condition of (9) being diagonal yields⁶ 

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

The change of variables (7) results in the transformation Jacobian $DRe(z⃗),Im(z⃗)Dα,θ,φ,ρ,hx,hy,hz,h̃z=916ρ7⁡sin⁡θ$⁠. Again, ρ = 1 and $h̃z=0$ 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 $h⃗=(hx,hy,hz)$⁠, 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 $h⃗$ 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.

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 $n⃗(θ,φ)$ 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, π₂(SO(3)) = 0. However, one can still define the Néel unit vector $n⃗(θ,φ)$⁠, and introduce the fictitious topological charge of the O(3) NLSM with unit vector field $n⃗(θ,φ)$⁠, 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 2S. 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 CP² 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.

We have studied spin-S antiferromagnets with non-Heisenberg exchange on a bipartite lattice, in the vicinity of highly symmetric SU(2S + 1) points, for S = 1 and S = 3/2. We have shown that if the SU(2S + 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 2S.

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