The $\mathrm{SU}\left(4\right)$ Heisenberg model on a honeycomb lattice is expected to host a quantum spin-orbital liquid at low temperature with a surprising candidate material, $\alpha \text{−}\mathrm{Zr}{\mathrm{Cl}}_3$. We employed the canonical thermal pure quantum state method to investigate the finite-temperature phase of this model. Exploiting the full symmetry of $\mathrm{SU}\left(4\right)$, calculation up to a 24-site cluster, which is equivalent to 48 sites in the spin-$\frac{1}{2}$ language, is possible. This state-of-the-art computation with large-scale parallelization enables us to capture the thermodynamic properties of the $\mathrm{SU}\left(4\right)$ Heisenberg model on a honeycomb lattice. In particular, the specific heat shows a characteristic peak-and-shoulder structure, which should be related to the nature of the low-temperature quantum spin-orbital liquid phase. We also discuss what can be concluded from the assumption that the ground state is gapped and symmetric in view of the generalized Lieb-Schultz-Mattis theorem.

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• Quantum spin liquid
• Specific heat
• Spin liquid
• SU(N) symmetries