The algebraic spin liquid in the SU(6) Heisenberg model on the kagome lattice

TL;DR

The paper argues that the Dirac spin liquid (DSL) is the ground state of the SU($6$) Heisenberg model on the kagome lattice in the fundamental representation, supported by mean-field theory and variational Monte Carlo (VMC) for a 12-site quadrupled unit cell. The authors show the DSL remains energetically favorable among SU($6$) singlet ansätze and remains locally stable under real perturbations and globally stable against David-star type instabilities in substantial regions of parameter space, though time-reversal breaking perturbations can destabilize it in certain regimes. They characterize the DSL via static and dynamical structure factors, revealing triangular-shaped plateaus around extended K points and a gapless continuum of flavoron excitations, with VMC results closely mirroring mean-field predictions for low-energy behavior. The work also lays out a detailed phase diagram including chiral flux states and provides methodological tools for evaluating three-site ring exchanges and projector-based correlations, with implications for realizing and detecting DSL physics in ultracold SU($6$) systems such as $^{173}$Yb in optical lattices.

Abstract

We explore the Dirac spin liquid (DSL) as a candidate for the ground state of the Mott insulating phase of fermions with six flavors on the Kagome lattice, particularly focusing on realizations using $^{173}$Yb atoms in optical lattices. Using mean-field theory and variational Monte Carlo simulations, we demonstrate that the Dirac spin liquid (DSL) has the lowest variational energy among SU(6) symmetry-preserving trial wave functions with a periodicity of a 12-site unit cell, as well as uniform chiral states with larger unit cells. It remains a local minimum even when small second-nearest neighbor and ring exchange interactions are introduced. To characterize the DSL, we calculate the static and dynamic structure factor of the Gutzwiller projected wavefunction and compare it with mean-field calculations. The static structure factor shows triangular-shaped plateaus around the $\mathrm{K}$ points in the extended Brillouin zone, with small peaks at the corners of these plateaus. The dynamical structure factor consists of a gapless continuum of fractionalized excitations. Our study also presents several complementary results, including bounds for the ground state energy, methods for calculating three-site ring exchange expectations in the projective mean field, the boundary of ferromagnetic states, and the non-topological nature of flat bands in the DSL band structure.

Figures (24)

• Figure 1: The 12-site quadrupled unit cell of the mean-field Hamiltonian Eq. (\ref{['eq:mean_field_Hamiltonian']}) in real space and Eq. (\ref{['eq:mean-field_Hamiltonian_in_k_space']}) in reciprocal space, describing the Dirac spin liquid ansatz. The black solid bonds represent $t^{\text{DSL}}_{i,j} = - 1$ and the white (inverted) $t^{\text{DSL}}_{i,j} = 1$. The product of the hoppings around every elementary triangular and hexagonal plaquette is $-1$, corresponding to a $\pi$ flux threading through each plaquette. The $2 \mathbf{a}_1$ and $2 \mathbf{a}_2$ are the primitive vectors of the quadrupled unit cell, where $\mathbf{a_1}= \left(1,0 \right)$ and $\mathbf{a}_2=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$ are the primitive vectors of the unit cell of the kagome lattice. The red dashed line shows the antiperiodic boundary condition along the edge of the cluster parallel to $\mathbf{a}_2$, which we impose to make the mean-field ground state non-degenerate.
• Figure 2: (a) The Brillouin zones of the kagome lattice (black hexagon) and of the quadrupled unit cell having 12 sites (blue shaded hexagon). The blue points at $\Gamma$ and $\mathrm{M}$ momenta denote the Fermi points of the DSL (the $\mathrm{M}$'s are also $\Gamma_{\text{MF}}$). (b) The bands of the DSL mean-field Hamiltonian $\mathcal{H}_{\text{DSL}}(\mathbf{k})$ (\ref{['eq:mean-field_Hamiltonian_in_k_space']}) along the path $\mathrm{M}_{\text{MF}}-\Gamma_{\text{MF}}-\mathrm{K}_{\text{MF}}-\mathrm{M}_{\text{MF}}$ in the Brillouin zone of the 12-site unit cell. Partons describing SU(2) spins occupy the lowest three bands up to the Dirac point at the $\Gamma$ point, while for the SU(6), they occupy only the lowest band (drawn by the thick line). Each dispersive band is two-fold degenerate, while the flat band is four-fold degenerate.
• Figure 3: The plot of the $(v_1,v_2,v_3)$ values that minimize the symmetry invariant form of the energy Eq. (\ref{['eq:c2c3c4']}) of a three-dimensional irreducible representation ($T_1$ or $T_2$).
• Figure 4: (a) the chiral $A_{1g}$ ansatz and (b) the staggered $A_{1u}$ ansatz. Since $t_{i,j} = t_{j,i}^{*}$, the direction matters for the complex hopping amplitudes. The white arrows pointing from $j$ to $i$ denote $t_{i,j} = e^{i \varphi}$, while the black arrows correspond to $t_{i,j} = -e^{i \varphi}$. The fluxes of upward pointing triangles are $\phi = \pi + 3 \varphi$ in both of them since the product of the hoppings around every triangle in the clockwise direction is $\propto -e^{i 3 \varphi}$. However, the fluxes in the downward-pointing triangles and the hexagons differ in the two cases.
• Figure 5: In the first row, we show the hopping structure of the real perturbations of the Dirac spin liquid having a single free parameter $\delta$. Different shades represent different absolute values of the hoppings. The white bonds show positive hoppings (each ansatz has a $\pi_{\hexagon} \pi_{\triangle} \pi_{\bigtriangledown}$ flux structure, just as the DSL). The black bonds have amplitude 1, the dark red hoppings have amplitude $1 + \delta$, and the light reds $1 - \delta$. In the midle row, the red points show the $\Delta \langle \mathcal{P}_{\triangle} + \mathcal{P}^{-1}_{\triangle} \rangle$, the blue points $\Delta \langle \mathcal{P}_{\text{1st}} \rangle$, and the green points $\Delta \langle \mathcal{P}_{\text{2nd}} \rangle$ defined in Eq. (\ref{['eq:expectation_values_relative_to_the_DSL']}), while the solid lines are the fitted parabolas. The bottom row shows the local stability of these ansätze (as explained in sec. \ref{['sec:local_stability_against_real_perturbations']}), as a function of $K$ and $J_2$, fixing $J_1 = 1$. The DSL is the lowest energy state in the red region, and the perturbation wins in the blue region.