We generalize the construction of the spin-1/2 SU(2) resonating valence bond (RVB) state to the case of the self-conjugate
6 representation of SU(4). As for the case of SU(2) [J.-Y. Chen and D. Poilblanc, Phys. Rev. B 97, 161107(R) (2018)], we use the projected entangled pair state (PEPS) formalism to derive a simple (two-dimensional) family of generalized SU(4) RVB states on the square lattice. We show that, when longer-range SU(4)-singlet bonds are included, a local gauge symmetry is broken down from U(1) to
Z2, leading to the emergence of a short-range spin liquid. Evidence for the topological nature of this spin liquid is provided by the investigation of the Renyi entanglement entropy of infinitely long cylinders and of the modular matrices. Relevance to microscopic models and experiments of ultracold atoms is discussed.
The isotropic limit of spin systems with orbital degeneracy has global SU(4) symmetry. On many 2D lattices, the ground state does not possess long-range order, which may explain the observed spin liquid properties of
LiNiO2. In the SU(4) N\'eel-ordered state, spin-spin correlations can be antiferromagnetic between two neighboring sites with parallel magnetic moments.
To begin with, we introduce several exact models for SU(3) spin chains: First is a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation
3 of SU(3) if the original spins of the model transform under representation 3. Second is a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and, hence, a Haldane gap in the excitation spectrum. We generalize some of our models to
SU(n). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of
SU(n) spins, in general, possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes $\ensuremath{\lambda}$ which is divisible by
n. If $\ensuremath{\lambda}$ and
n have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If $\ensuremath{\lambda}$ and
n have a common divisor different from
n, it will depend on the specifics of the model including the range of the interaction.
Anderson's idea of a (short-ranged) resonating valence bond (RVB) spin liquid has been the first ever proposal of what we now call a topologically ordered phase. Since then, a wealth of exactly solvable lattice models have been constructed that have topologically ordered ground states. For a long time, however, it has been difficult to realize Anderson's original vision in such solvable models, according to which the ground state has an unbroken SU(2) spin rotational symmetry and is dominated by fluctuation of singlet bonds. The kagome lattice is the simplest lattice geometry for which parent Hamiltonians stabilizing a prototypical spin-1$/$2 short-ranged RVB wave function has been constructed and strong evidence has been given that this state belongs to a topological phase. The uniqueness of the desired RVB-type ground states has, however, not been rigorously proven for the simplest possible such Hamiltonian, which acts on 12 spins at a time. Rather, this uniqueness has been demonstrated for a longer ranged (19-site) variant of this Hamiltonian by Schuch et al., via making contact with powerful results for projected entangled-pair states. In this paper, we extend this result to the 12-site Hamiltonian. Our result is based on numerical studies on finite clusters, for which we demonstrate a ``ground state intersection property'' with implications for arbitrary system size. We also review the relations between various constructions schemes for RVB parent Hamiltonians found in the literature.
The one-dimensional Heisenberg model in the presence of orbital degeneracy is studied at the
SU(4) symmetric point by means of Bethe ansatz. Following Sutherland's previous work on an equivalent model, we discuss the ground state and the low-lying excitations more extensively in connection to the spin systems with orbital degeneracy. We show explicitly that the ground state is a
SU(4) singlet. We study the degeneracies of the elementary excitations and the spectra of the generalized magnons consisting of these excitations. We also discuss the complex 2-strings in the context of the Bethe ansatz solutions.
Based on the generalized valence bond picture, a Schwinger boson mean-field theory is applied to the symmetric SU(4) spin-orbital systems. For a two-leg SU(4) ladder, the ground state is a spin-orbital liquid with a finite energy gap, in good agreement with recent numerical calculations. In two-dimensional square and triangle lattices, the SU(4) Schwinger bosons condense at $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ and $(\ensuremath{\pi}/3,\ensuremath{\pi}/3),$ respectively. Spin, orbital, and coupled spin-orbital static susceptibilities become singular at the wave vectors; the Bose condensation arises at twice this value. It is also demonstrated that there are spin, orbital, and coupled spin-orbital long-range orderings in the ground state.