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 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.
We investigate the ground state phase diagram of an SU(
N)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU(
N) described by a square Young tableau of
N/2 rows and
2S columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any
S and even values of
N. In the large-
N limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large
S-limit, the semi-classical approximation points to Néel state. On a line set by
N=8S+2 in the
S versus
N phase diagram, we observe a variety of phases proximate to the Néel state. At
S=1/2 and
3/2 we observe the aforementioned four fold degenerate valence bond solid state. At
S=1 a two fold degenerate spin nematic state in which the C
4 lattice symmetry is broken down to C
2 emerges. Finally at
S=2 we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18),
S=1/2 boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by
mod(4,2S).
Motivated by near-term experiments with ultracold alkaline-earth atoms confined to optical lattices, we establish numerically and analytically the phase diagram of two-leg
SU(N) spin ladders. Two-leg ladders provide a rich and highly nontrivial extension of the single chain case on the way towards the relatively little explored two-dimensional situation. Focusing on the experimentally relevant limit of one fermion per site, antiferromagnetic exchange interactions, and $2\ensuremath{\le}N\ensuremath{\le}6$, we show that the phase diagrams as a function of the interchain (rung) to intrachain (leg) coupling ratio, ${J}_{\ensuremath{\perp}}/{J}_{\ensuremath{\parallel}}$, strongly differ for even versus odd
N. For even
N=4 and 6, we demonstrate that the phase diagram consists of a single valence bond crystal (VBC) with a spatial period of
N/2 rungs. For odd
N=3 and 5, we find surprisingly rich phase diagrams exhibiting three distinct phases. For weak rung coupling, we obtain a VBC with a spatial period of
N rungs, whereas for strong coupling we obtain a critical phase related to the case of a single chain. In addition, we encounter intermediate phases for odd
N, albeit of a different nature for
N=3 as compared to
N=5. For
N=3, we find a novel gapless intermediate phase with ${J}_{\ensuremath{\perp}}$-dependent incommensurate spatial fluctuations in a sizeable region of the phase diagram. For
N=5, there are strong indications for a narrow potentially gapped intermediate phase, whose nature is not entirely clear. Our results are based on (i) field theoretical techniques, (ii) qualitative symmetry considerations, and (iii) large-scale density matrix renormalization group (DMRG) simulations keeping beyond a million of states by fully exploiting and thus preserving the
SU(N) symmetry.
We study the effect of symmetry-breaking perturbations in the one-dimensional
SU(4) spin-orbital model. We allow the exchange in spin
(J1) and orbital
(J2) channel to be different and thus reduce the symmetry to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2).$ A magnetic field
h along the
Sz direction is also applied. Using the formalism developed by Azaria et al. [Phys. Rev. Lett.
83, 624 (1999)] we extend their analysis of the isotropic
J1=J2, h=0 case and obtain the low-energy effective theory near the
SU(4) point in the generic case ${J}_{1}\ensuremath{\ne}{J}_{2},$ $h\ensuremath{\ne}0.$ In zero magnetic field, we retrieve the same qualitative low-energy physics as in the isotropic case. In particular, the isotropic massless behavior found on the line
J1=J2<K/4 extends in a large anisotropic region. We discover, however, that the anisotropy plays its trick in allowing nontrivial scaling behaviors of the physical quantities. For example, the mass gap
M has two different scaling behaviors depending on the anisotropy. In addition, we show that in some regions, the anisotropy is responsible for anomalous finite-size effects and may change qualitatively the shape of the computed critical line in a finite system. When a magnetic field is present the effect of the anisotropy is striking. In addition to the usual commensurate-incommensurate phase transition that occurs in the spin sector of the theory, we find that the field may induce a second transition of the KT type in the remaining degrees of freedom to which it does not couple directly. In this sector, we find that the effective theory is that of an SO(4) Gross-Neveu model with an $h\ensuremath{-}\mathrm{dependent}$ coupling that may change its sign as
h varies.
We study the ground-state properties of the SU(
N) generalization of the Kondo lattice model in one dimension when the Kondo coupling
JK (both ferromagnetic and antiferromagnetic) is sufficiently strong. Both cases can be realized using alkaline-earth-metal-like cold gases in optical lattices. Specifically, we first carry out the strong-coupling expansion and identify two insulating phases [one of which is the SU(
N) analog of the well-known gapped Kondo singlet phase]. We then rigorously establish that the ground state in the low-density (for
JK<0) or the high-density (for
JK>0) region is ferromagnetic. The results are accounted for by generalizing the double-exchange mechanism to SU(
N) ``spins.'' Possible realizations of Bose-Fermi supersymmetry SU(
N∣1) in the (generalized) SU(
N) Kondo lattice model are discussed as well.