Multipolar spin liquid in an exactly solvable model for $j_{\mathrm{eff}}$ = $\frac{3}{2}$ moments

We study an exactly solvable model with bond-directional quadrupolar and octupolar interactions between spin-orbital entangled $j_{\mathrm{eff}}=\frac{3}{2}$ moments on the honeycomb lattice. We show that this model features a multipolar spin liquid phase with gapless fermionic excitations. In the presence of perturbations that break time-reversal and rotation symmetries, we find Abelian and non-Abelian topological phases in which the Chern number evaluates to 0, $\pm 1$, and $\pm 2$. We also investigate quantum phase transitions out of the multipolar spin liquid using a parton mean-field approach and orbital wave theory. In the regime of strong integrability-breaking interactions, the multipolar spin liquid gives way to ferroquadrupolar-vortex and antiferro-octupolar ordered phases that harbor a hidden spin-$\frac{1}{2}$ Kitaev spin liquid. Our work unveils mechanisms for unusual multipolar orders and quantum spin liquids in Mott insulators with strong spin-orbit coupling.

Figure 1

Schematic representation of fractionalized $j_{\mathrm{eff}}=\frac{3}{2}$ moments on the honeycomb lattice. Here $x, y$, and $z$ bonds are represented by red, green, and blue solid lines, respectively. The two sublattices, A and B, are represented by up-pointing and down-pointing triangles. The inset in the top right corner shows the matter (${\theta }^{\gamma }$) and gauge (${\eta }^{\gamma }$) Majorana fermions. Each plaquette is associated with the conserved quantity ${\widehat{W}}_p=-e^{i\pi (J_1^x+J_2^y+J_3^z+J_4^x+J_5^y+J_6^z)}$.

Figure 2

Dispersion relation for Majorana fermion excitations in the exactly solvable MSL model. Here we set $K_o=0.4K_q$. (a) For the unperturbed model in Eq. (9), the system exhibits a gapless spectrum with Dirac nodes at the $\mathbf{K}$ point. (b) In the presence of strain fields ${\varepsilon }_2$ and ${\varepsilon }_3$ and time-reversal-symmetry-breaking field $h$ [see Eq. (17)], the excitation spectrum becomes gapped. Here we set ${\varepsilon }_2={\varepsilon }_3=0.4K_q$ and $h=0.1K_q$.

Figure 3

Topological phase diagram of the MSL as a function of the symmetry-breaking fields ${\varepsilon }_2$ and $h$. Here we set $K_o=0.2K_q$ and ${\varepsilon }_2={\varepsilon }_3$. The purple, yellow, and blue regions correspond to Chern numbers 0, 1, and 2, respectively.

Figure 4

Edge states of the MSL for a honeycomb lattice with zigzag edges. Here we use a cylinder with $L=50$ unit cells along the open direction. We fix the parameters as $K_o=h={\varepsilon }_2={\varepsilon }_3=0.2K_q$ in the regime of ${\mathcal{N}}_{\text{Ch}}=\pm 2$ (see Fig. 3).

Figure 5

Schematic representation of the ferroquadrupolar-vortex state on the honeycomb lattice. The arrows represent the direction of the vector defined by $(\langle O_2\rangle ,\langle O_3\rangle )=2(\langle {\tau }^x\rangle ,\langle {\tau }^z\rangle )\propto (cos{\varphi }_n,-sin{\varphi }_n)$, where ${\varphi }_n=(n-1)\pi /3$, with $n=1,\cdots ,6$ being the sublattice index.

Figure 6

Ground-state phase diagram obtained from the parton mean-field theory setting $K_o=0.2K_q$. Here AO and FQV refer to the antiferro-octupolar and ferroquadrupolar-vortex phases, respectively. In addition, the system exhibits a phase with coexisting AO and FQV orders.

Figure 7

Energy dispersion of the mean-field Hamiltonian for $K_o=0.2K_q, K_q^{\prime }=1.3K_q$, and three different values of $K_o^{\prime }$ in the broken-symmetry phases: (a) $K_o^{\prime }=0.65K_q$ in the AO phase, (b) $K_o^{\prime }=0.10K_q$ in the FQV phase, and (c) $K_o^{\prime }=0.40K_q$ in the coexistence region. Here the Brillouin zone is defined with respect to the unit cell shown in Fig. 1.

Figure 8

Quantum correction to the ground-state energy of FQV and AO states [see Eqs. (28) and (32)]. The dispersion relation of pseudo-orbital waves in the FQV state becomes complex for $K_o^{\prime }>0.746K_q^{\prime }$, signaling an instability. Similarly, the AO state becomes unstable for $K_o^{\prime }<0.5K_q^{\prime }$.

Figure 9

(a) Honeycomb lattice for a translationally invariant $\pi$-gauge flux configuration $W_p=-1$. The dashed blue and filled red bonds refer to the bond variables $u_{{\langle jl\rangle }_{\gamma }}=+1$ and $u_{{\langle jl\rangle }_{\gamma }}=-1$, respectively. In terms of this configuration, the physical unit cell contains four sublattices, which are indicated by numbers and also by different regular polygons. (b) Behavior of the MSL GS energies $E_{\mathrm{GS}}^{+}\equiv E_{\mathrm{GS}}\left(\{W_p=+1\}\right)$ and $E_{\mathrm{GS}}^{-}\equiv E_{\mathrm{GS}}\left(\{W_p=-1\}\right)$ for the zero- and $\pi$-gauge flux configurations, respectively. Note that $E_{\mathrm{GS}}^{+}$ is always lower than $E_{\mathrm{GS}}^{-}$, which is consistent with Lieb's theorem [69].