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 fermions) are considered. Near the SU(N) point, the low-energy physics can be described by the CPN−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(2S + 1) to SU(2) leads to binding of topological unit-charge excitations of the CP2S model (skyrmions for space dimension d = 2, instantons for d = 1, and hedgehogs for d = 3) into 2S-multiplets.
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 CPN−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 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 CPN−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 is subject to the unit length constraint , is the gauge covariant derivative, and is the gauge field, x0 = cτ, τ = it is the imaginary time, the limiting velocity is , and the bare coupling constant . The topological term in the action
where the phase factors ηj = ±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 is proportional to the perturbation strength, is the spin average, Sa 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 CP2S 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 CP2S model into 2S multiplets.
II. THE EFFECTIVE THEORY OF THE PERTURBED CP2S MODEL FOR S = 1,
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 is the Wigner matrix that represent a rotation in (2j + 1)-dimensional space and depends on the three Euler angles as10
where are the Jacobi polynomials. The spin state is chosen in the form that renders diagonal the on-site spin quadrupolar tensor
and its length 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, as . The change of variables (7) results in the transformation Jacobian , which is the part of the usual measure in the integration over the rotation group space (α, θ, φ). Variables ρ = 1, 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 ωμa 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 :
The change of variables (7) results in the transformation Jacobian . Again, ρ = 1 and 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 , we obtain the action with the structure very similar to the expression (11) for S = 1:
where is the action of the O(3) nonlinear sigma-model,
is the quadratic action of the massive field,
and describes the interaction,
Again, we see that fluctuations of the massive field 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(2S + 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 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 , and introduce the fictitious topological charge of the O(3) NLSM with unit vector field , namely
At the same time, the topological charge in the underlying CP2S 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:
This suggests that the SU(2) AF perturbation leads to clustering of skyrmions of the CP2S model into groups with the total charge 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 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 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 CP2S model into groups with the total charge being a multiple of 2S.
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