SU(4)-symmetric spin-orbital liquids on the hyperhoneycomb lattice

We study the effective spin-orbital model that describes the magnetism of $4d^1$ or $5d^1$ Mott insulators in ideal tricoordinated lattices. In the limit of vanishing Hund's coupling, the model has an emergent SU(4) symmetry, which is made explicit by means of a Klein transformation on pseudospin degrees of freedom. Taking the hyperhoneycomb lattice as an example, we employ parton constructions with fermionic representations of the pseudospin operators to investigate possible quantum spin-orbital liquid states. We then use variational Monte Carlo (VMC) methods to compute the energies of the projected wave functions. Our numerical results show that the lowest-energy quantum liquid corresponds to a zero-flux state with a Fermi surface of four-color fermionic partons. In spite of the Fermi surface, we demonstrate that this state is stable against tetramerization. A combination of linear flavor wave theory and VMC applied to the complete microscopic model also shows that this liquid state is stable against the formation of collinear long-range order.

Figure 1

Hyperhoneycomb lattice as a base-centered orthorhombic lattice with an eight-point basis. The disks and triangles indicate that the lattice is bipartite, while the different colors represent the different sublattices $r=1,\cdots ,8$. The primitive lattice vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$ (see Appendix pp1) are also shown.

Figure 2

Representation of the (a) zero-flux and (b) $\pi$-flux states. Each vertex corresponds to a basis point of the hyperhoneycomb lattice as labeled in Fig. 1. Solid (dashed) lines represent bonds with ${\chi }_{ij}=+1$ (${\chi }_{ij}=-1$).

Figure 3

Mean-field dispersion of fermions in the (a) zero-flux and (b) $\pi$-flux states. The dashed line marks the Fermi level at quarter filling. The high symmetry points of the Brillouin zone are specified in Fig. 4. The energy scale in this plot is set by $|{\chi }_{ij}|=1$.

Figure 4

(a) Brillouin zone of the base-centered orthorhombic lattice. (b) Fermi surface of the zero-flux state inside the Brillouin zone. Different colors represent different bands in Eq. (32).

Figure 5

Variational Monte Carlo ground-state energy, per site and in units of $J$, for the different mean-field states as a function of the inverse of the particle number. The dashed lines are linear extrapolations to the data.

Figure 6

Graphical representation of a covering of tetramers on the hyperhoneycomb lattice. Following Eq. (47), the larger magenta disks indicate the sites in which the chemical potential is modified. Likewise, the hopping amplitudes are modified on the magenta bonds and leads to two types of nonequivalent bonds that are labeled $a$ and $b$.

Figure 7

Variational Monte Carlo ground-state energy for the zero-flux complex fermions Ansatz as a function of the tetramerization order parameter $r$ for two different system sizes.

Figure 8

Stripy phase with ordering in $M^z=\pm 3/2$. This is the only collinear ordered state stable at the linear flavor wave theory level among the two- and four-color ordered states.

Figure 9

Linear flavor-wave dispersion of the $M^z=\pm 3/2$ stripy phase for (a) $\eta =J_H/U=0$ and (b) $\eta =J_H/U=0.1$. (a) shows the spectra of eight bands, 16 degenerate bands with zero energy, and Goldstone modes at the $\mathrm{\Gamma }$ point. (b) shows the dispersion of the 24 bands after the inclusion of Hund's coupling induced perturbations. The lack of Goldstone modes is due to the absence of continuous symmetry on the underlying Hamiltonian.

Figure 10

(a) Energy per site of three different states as a function of the ratio $\eta =J_H/U$. The energy of the stripy phase was estimated using linear flavor-wave theory, whereas the energy of the QSOLs based on complex fermions was evaluated with VMC. (b) VMC ground-state energy for zero-flux complex fermions in the presence of a staggered potential favoring stripy order. The four curves correspond to different values of the local moment $m$.

Figure 11

Nodal line (red) of the zero flux state. The thin black lines represent the edges of the first Brillouin zone of the base-centered orthorhombic lattice.