The enhancement of the spin-space symmetry from the usual
SU(2) to
SU(N) with
N>2 is promising for finding nontrivial quantum spin liquids, but the realization of
SU(N) spin systems in real materials is still challenging. Although there is a proposal in cold atomic systems, in magnetic materials with a spin-orbital degree of freedom it is difficult to achieve the
SU(N) symmetry by fine tuning. Here we propose a new mechanism by which the
SU(4) symmetry emerges in the strong spin-orbit coupling limit. In
d1 transition metal compounds with edge-sharing anion octahedra, the spin-orbit coupling gives rise to strongly bond-dependent and apparently
SU(4)-breaking hopping between the
Jeff=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an
SU(4)-symmetric Hubbard model, which means that the system has a hidden symmetry in spite of its large spin-orbit coupling. In the strong repulsion limit at quarter filling, as expected in
α-ZrCl
3, the low-energy effective model is the
SU(4) Heisenberg model on the honeycomb lattice, which cannot have a trivial gapped ground state and is expected to host a gapless spin-orbital liquid. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and correlated together at low temperature due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic "orbitalon" excitations at low temperature, which we have named here in distinction from orbitons in the symmetry-broken Jahn-Teller phases.
The SU(4) Heisenberg model can serve as a low-energy model of the Mott insulating state in materials where the spins and orbitals are highly symmetric or in systems of alkaline-earth atoms on an optical lattice. Recently, it has been argued that on the honeycomb lattice, the model exhibits a unique spin-orbital liquid phase with an algebraic decay of correlations [P. Corboz et al., Phys. Rev. X 2, 041013 (2012)]. Here we study the instability of the algebraic spin-orbital liquid toward the spontaneous formation of SU(4) singlet plaquettes (tetramerization). Using a variational Monte Carlo approach to evaluate the projected wave function of fermions with $\ensuremath{\pi}$-flux state, we find that the algebraic liquid is robust and that a small finite value of the next-nearest exchange is needed to induce tetramerization. We also studied the phase diagram of a model which interpolates between the nearest-neighbor Heisenberg model and a Hamiltonian for which the singlet-plaquette-product state is an exact ground state.
Previous investigations have suggested that the simplest spin-orbital model on the simplest frustrated lattice can host a nematic quantum spin-orbital liquid state. Namely, the orbital degeneracy of the SU(4) Kugel-Khomskii model tends to enhance quantum fluctuations and stabilize a quantum spin-orbital liquid exhibiting stripy features on the triangular lattice, as revealed by the state-of-the-art method of the density matrix renormalization group boosted by Gutzwiller projected wave functions. In this work, using the variational quantum Monte Carlo method, we study several spin-orbital liquid states, including a uniform $\ensuremath{\pi}$-flux state, three stripy states, and a plaquette state, on the $L\ifmmode\times\else\texttimes\fi{}L$ torus up to
L=24. It turns out that one of these stripy states, called the ``stripe-II'' state, is energetically favored. This ground state breaks the
C6 symmetry of the lattice, resulting in a reduced
C2 symmetry and doubled unit cells while preserving the SU(4) spin-orbital rotation symmetry. Such a nematic quantum spin-orbital liquid state can be characterized by a parton Fermi surface (FS) consisting of open orbits in the Brillouin zone, in contrast to the circular FS of the uniform $\ensuremath{\pi}$-flux state.
We study the
S=1/2 Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor interactions
J1 and
J2 with the density matrix renormalization group, on long open cylinders with widths up to nine lattice spacings. In an intermediate
J2 region $0.06\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.17$, we find evidence for a spin liquid (SL) state with short range spin-spin, bond-bond, and chiral correlation lengths, bordered by a classical ${120}^{\ensuremath{\circ}}$ N\'eel ordered state at small
J2 and by a two sublattice collinear magnetically ordered state at larger
J2. Focusing on
J2/J1=0.1, we find a number of signatures of a gapped SL phase.
Previous investigations have suggested that the simplest spin-orbital model on the simplest frustrated lattice can host a nematic quantum spin-orbital liquid state. Namely, the orbital degeneracy of the SU(4) Kugel-Khomskii (KK) model tends to enhance quantum fluctuations and stabilize a quantum spin-orbital liquid exhibiting stripy features on the triangular lattice, as revealed by the state-of-the-art method of the density matrix renormalization group boosted by Gutzwiller projected wave functions. In this work, using the variational quantum Monte Carlo method, we have studied several spin-orbital liquid states, including a uniform
π flux state, three stripy states, and a plaquette state, on the
L×L torus up to
L=24. It turns out that one of these stripy states, called the "stripe-II" state, is energetically favored. This ground state breaks the
C6 symmetry of the lattice, resulting in a reduced
C2 symmetry and doubled unit cells, while preserving the SU(4) spin-orbital rotation symmetry. Such a nematic quantum spin-orbital liquid state can be characterized by a parton Fermi surface (FS) consisting of open orbits in the Brillouin zone, in contrast to the circular FS of the uniform
π-flux state.
The SU(4)-symmetric spin-orbital model on the honeycomb lattice was recently studied in connection to correlated insulators such as the
eg Mott insulator
Ba3CuSb2O9 and the insulating phase of magic-angle twisted bilayer graphene at quarter filling. Here we provide a unified discussion of these systems by investigating an extended model that includes the effects of Hund's coupling and anisotropic, orbital-dependent exchange interactions. Using a combination of mean-field theory, linear flavor-wave theory, and variational Monte Carlo, we show that this model harbors a quantum spin-orbital liquid over a wide parameter regime around the SU(4)-symmetric point. For large Hund's coupling, a ferromagnetic antiferro-orbital ordered state appears, while a valence-bond crystal combined with a vortex orbital state is stabilized by dominant orbital-dependent exchange interactions.