Topological $Z_4$ spin-orbital liquid on the honeycomb lattice

We perform large-scale density matrix renormalization group simulations of the $\mathrm{SU}(4)$ Heisenberg model on the honeycomb lattice and resolve the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find compelling numerical evidence that the ground state is a gapped $Z_4$ spin-orbital liquid, characterized by a finite topological entanglement entropy close to $\ln(4)$, the absence of both $\mathrm{SU}(4)$ and lattice symmetry breaking, and a variationally optimized ground-state energy well below competing Dirac spin liquid states. By exploiting full $\mathrm{SU}(4)$ symmetry and keeping up to 12,800 $\mathrm{SU}(4)$ multiplets, corresponding to more than one million $\mathrm{U}(1)$ states, we achieve unprecedented accuracy for two-dimensional $\mathrm{SU}(4)$ 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 establish the $\mathrm{SU}(4)$ honeycomb Heisenberg model as a concrete realization of a gapped $Z_4$ spin-orbital liquid and provide robust numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.