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

TL;DR

Extends DMRG to SU($N_c$) symmetry in two dimensions and applies it to the SU($4$) Heisenberg model on the honeycomb lattice, achieving bond dimensions up to $m=12800$ states ($\approx 10^6$ U(1) states). Demonstrates a gapped quantum spin-orbital liquid with topological order, evidenced by a topological entanglement entropy of $\gamma_{\text{top}} \approx 1.33$ (≈ $\ln 4$) and no symmetry breaking in the entanglement spectrum. Finite-size scaling yields a 2D ground-state energy per site of $E/N \approx -0.9210(6)$, lower than previous VMC estimates and inconsistent with a $\pi$-flux Dirac spin liquid. The results support a gapped $Z_4$ spin liquid generalizing Anderson's RVB to SU($4$), with potential realizations in cold-atom and solid-state systems and possible charge-4e superconductivity upon doping; the work also establishes a scalable, symmetry-aware DMRG framework for classical Lie groups.

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.

Figures (4)

• Figure 1: Overall features of the DMRG simulation.a, Overview of the zigzag-edge cylinder geometry, which is called ZC in the previous literature. Gong2013b, Phase diagram about $L_y$ (cylinder DMRG). The rung singlet phase appears when $L_y=4$, while a gapped spin liquid phase appears when $L_y \geq 8$. Transition occurs at $L_y=6$, which is gapless and belongs to the $\mathrm{SU}(4)$ level-1 Wess-Zumino-Witten universality class. c, Entanglement entropy $S$ observed by bipartition of the cylinder disconnected around the cross section at $x$. Approximate independence of $x$ shows the gapped nature of the phase. Even-odd effect only appears in the $L_y=10$ case.
• Figure 2: Comparison of entanglement spectra.a, Entanglement spectrum of the $\mathrm{SU}(3)$ Heisenberg model on the $12 \times 12$ square lattice with $m=6400$. For each value, an irrep is associated as shown in the legend. The lowest values for (0, 0, 0), (2, 1, 0), (3, 0, 0) and (3, 3, 0) irreps show a linear behaviour about the quadratic Casimir, as indicated by the black line. b, Entanglement spectrum of the $\mathrm{SU}(4)$ Heisenberg mode on the honeycomb lattice with $m=12800$. ZC6-32 cylinder is used. The entanglement spectrum shows a random behaviour, which is consistent with the spin liquid ground state.
• Figure 3: Expectation values of bond operators for the ZC6-12 cylinder. The fluctiation of expectation values of bond operators is displayed by the thickness of bonds. Blue bonds indicate a minus value and red bonds indicate a plus value with respect to the average, and the exact value is annotated on the bond.
• Figure 4: Finite size scaling of energy and entanglement entropy.a, Finite size scaling of energy about $L_y$. Blue dots are plotted for each $L_y$. A red line is for $L_y \mod 4 = 0$, and a purple line is for $L_y \mod 4 = 2$. A black line shows the VMC result from Ref. Corboz2012. b, Finite size scaling of entanglement entropy about $L_y$. Blue dots are plotted for each $L_y$, while the point for $L_y=6$ is missing because of its gapless nature. A red line is for an exact fit for $L_y>6$, and a purple line shows its linear component.