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

Abstract: The density matrix renormalisation group (DMRG) is one of the most powerful numerical methods for strongly correlated condensed matter systems. We extend DMRG to the case with the $\mathrm{SU}(N_c)$ symmetry with $N_c > 2$, including two-dimensional systems. As a killer application, we simulate the ground state of the $\mathrm{SU}(4)$ Heisenberg model on the honeycomb lattice, which can potentially be realised in cold atomic systems and solid state systems like $α$-ZrCl$_3$. We keep up to 12800 $\mathrm{SU}(4)$ states equivalent to more than a million $\mathrm{U}(1)$ states. This supermassive DMRG simulation reveals the quantum spin-orbital liquid ground state, which has been conjectured for more than a decade. The methodology developed here can be extended to any classical Lie groups, paving the way to a next-generation DMRG with a full symmetry implementation.