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.

💡 Research Summary

In this work the authors address the long‑standing question of the ground‑state nature of the SU(4) Heisenberg model on the honeycomb lattice by means of large‑scale density‑matrix renormalization group (DMRG) simulations that fully exploit the non‑Abelian SU(4) symmetry. Using a GPU‑accelerated implementation they keep up to 12 800 SU(4) multiplets (equivalent to more than one million U(1) states) and study cylindrical geometries with circumferences Ly = 4, 6, 8, 10, 12 and lengths up to Lx = 32 (36 for Ly = 6). The Hamiltonian is H = ∑⟨ij⟩ Pij, where Pij swaps the fundamental SU(4) representations on nearest‑neighbour sites, i.e. it can be written as a product of spin‑½ and orbital‑½ operators.

The key findings are threefold. First, the extrapolated ground‑state energy per site in the two‑dimensional limit is E/N = −0.9210(6), obtained from a power‑law scaling in 1/Ly with exponent p = 3. This value is significantly lower than the best variational Monte‑Carlo estimate (−0.894) for a π‑flux Dirac spin liquid, thereby ruling out that state as the true ground state. Second, extensive analysis of the entanglement spectrum shows no Anderson tower of states; instead the spectrum is random and composed of many SU(4) irreducible representations, indicating that neither SU(4) nor lattice translational symmetry is broken. Real‑space bond expectation values ⟨Pij⟩ are uniform up to small fluctuations that decay with increasing Ly, and spin‑spin correlations decay exponentially with a correlation length of only 2–4 lattice spacings for Ly ≥ 8, confirming a gapped phase.

Third, the topological entanglement entropy γtop extracted from the scaling of the bipartite entanglement entropy SEE with Ly (using data for Ly = 8, 10, 12) yields γtop = 1.33(3), which is remarkably close to ln 4 ≈ 1.386. This value is the hallmark of a Z4 topological order, implying a sixteen‑fold ground‑state degeneracy on a torus. The authors also identify a special Ly = 6 cylinder that remains gapless; its entanglement spectrum and scaling are consistent with an SU(4) level‑1 Wess‑Zumino‑Witten conformal field theory, providing a quasi‑one‑dimensional remnant of the Dirac spin‑orbital liquid proposed in earlier field‑theoretical work. As Ly increases beyond 6, the system crosses over to the gapped Z4 phase.

Methodologically, the study leverages Young‑tableau based labeling of SU(4) multiplets and the Schur‑Weyl duality to pre‑compute Clebsch‑Gordan and subduction coefficients, which are stored in hash tables for rapid access during DMRG sweeps. Truncation errors are kept below 10⁻⁵ for the widest cylinders, and extrapolations to infinite bond dimension are performed linearly for energies and quadratically for entanglement entropies, providing controlled error estimates.

Overall, the combination of (i) a variationally exact low energy, (ii) the absence of any symmetry breaking in both real‑space observables and the entanglement spectrum, and (iii) a topological entanglement entropy consistent with ln 4, constitutes compelling evidence that the SU(4) honeycomb Heisenberg model realizes a gapped Z4 spin‑orbital liquid. This establishes a concrete, highly symmetric platform for Z4 topological order in two dimensions, with direct relevance to cold‑atom systems possessing emergent SU(N) symmetry and to solid‑state candidates such as α‑ZrCl₃. The work also opens avenues for exploring doped versions of the model, where exotic charge‑4e superconductivity has been theoretically predicted.