Dirac spin liquids represent a class of highly-entangled quantum phases in two dimensional Mott insulators, featuring exotic properties such as critical correlation functions and absence of well-defined low energy quasi-particles. Existing numerical works suggest that the spin-orbital
SU(4) symmetric Kugel-Khomskii model of Mott insulators on the honeycomb lattice realizes a Dirac spin-orbital liquid, described at low energy by
(2+1)d quantum electrodynamics (QED
3) with
Nf=8 Dirac fermions. We generalize methods previously developed for
SU(2) spin systems to analyze the symmetry properties and stability of the Dirac spin-orbital liquid. We conclude that the standard Dirac state in the
SU(4) honeycomb system, based on a simple parton mean-field ansatz, is intrinsically unstable at low energy due to the existence of a monopole perturbation that is allowed by physical symmetries and relevant under renormalization group flow. We propose two plausible alternative scenarios compatible with existing numerics. In the first scenario, we propose an alternative
U(1) Dirac spin-orbital liquid, which is similar to the standard one except for its monopole symmetry quantum numbers. This alternative
U(1) state represents a stable gapless phase. In the second scenario, we start from the standard
U(1) Dirac liquid and Higgs the
U(1) gauge symmetry down to
Z4. The resulting
Z4 Dirac spin-orbital liquid is stable. We also discuss the continuous quantum phase transitions from the
Z4 Dirac liquids to conventional symmetry-breaking orders, described by the QED
3 theory with
Nf=8 supplemented with a critical charge-
4 Higgs field. We discuss possible ways to distinguish these scenarios in numerics. We also extend previous calculations of the quantum anomalies of QED
3 and match with generalized lattice Lieb-Schultz-Mattis constraints.
While the enhancement of spin-space symmetry from the usual SU(2) to SU(N) is promising for finding nontrivial quantum spin liquids, its realization in magnetic materials remains challenging. Here, we propose a new mechanism by which SU(4) symmetry emerges in the strong spin-orbit coupling limit. In d^{1} 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 J_{eff}=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an SU(4)-symmetric Hubbard model. In the strong repulsion limit at quarter filling, as realized 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. By generalizing this model to other three-dimensional lattices, we also propose crystalline spin-orbital liquids protected by this emergent SU(4) symmetry and space group symmetries.
An emergent SU(4) symmetry discovered in the microscopic model for d1 honeycomb materials [M. G. Yamada, M. Oshikawa, and G. Jackeli, ] has enabled us to tailor exotic SU(4) models in real materials. In the honeycomb structure, the emergent SU(4) Heisenberg model would potentially have a quantum spin-orbital liquid ground state due to the , and we can also expect similar spin-orbital liquids in three-dimensional versions of the honeycomb lattice. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and entangled together 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 excitations at low temperature.
We compute the momentum-resolved dynamical spin structure factor $S(k,\ensuremath{\omega})$ of the SU(4) Heisenberg model on the honeycomb lattice assuming the $\ensuremath{\pi}$-flux Dirac spin liquid ground state by using two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the $\ensuremath{\pi}$-flux Fermi sea, and (ii) in the noninteracting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the
1/distance4 decay of the spin correlation function, and the local correlations show ${S}_{\text{MF}}^{33}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{3}$ behavior.
We compute the momentum resolved dynamical spin structure factor
S(k,ω) of the SU(4) Heisenberg model on the honeycomb lattice assuming the
π-flux Dirac spin liquid ground state by two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the
π-flux Fermi sea and (ii) in the non-interacting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the
1/distance4 decay of the spin correlation function, and the local correlations show
SMF33(ω)∝ω3 behavior.
We perform large-scale density matrix renormalization group simulations of the SU(4) Heisenberg model on the honeycomb lattice to address the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find reliable numerical evidence that the ground state is a gapped spin-orbital liquid, presumably with a Z_{4} topological order, characterized by a finite topological entanglement entropy close to ln(4), the absence of both SU(4) and lattice symmetry breaking, and a variationally optimized ground-state energy well below the previously proposed π-flux variational state. By exploiting full SU(4) symmetry and keeping up to 12 800 SU(4) multiplets, corresponding to more than 1×10^{6} U(1) states, we achieve unprecedented accuracy for two-dimensional SU(4) quantum magnets. Finite-size scaling of energies and entanglement entropies supports a robust gapped phase in the two-dimensional limit, while a gapless critical state on narrow cylinders is identified as a proximate remnant of a Dirac spin-orbital liquid. Our results find the SU(4) honeycomb Heisenberg model a realization of a gapped topological spin-orbital liquid and provide convincing numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.