Multiple quadrupolar or nematic phases driven by the Heisenberg interactions in a spin-1 dimer system forming a bilayer

We explore several classes of quadrupolar ordering in a system of antiferromagnetically coupled quantum spin-1 dimers, which are stacked in the triangular lattice geometry forming a bilayer. Low-energy properties of this model are described by an $\mathcal{S}=1$ hard-core bosonic degrees of freedom defined on each dimer bond, where the singlet and triplet states of the dimerized spins are interpreted as the vacuum and the occupancy of boson, respectively. The number of bosons per dimer and the magnetic and density fluctuations of bosons are controlled by the interdimer Heisenberg interactions. In a solid phase where each dimer hosts one boson and the interdimer interaction is weak, a conventional spin nematic phase is realized by the pair fluctuation of bosons. Larger interdimer interaction favors Bose-Einstein condensates (BEC) carrying quadrupolar moments. Among them, we find one exotic phase where the quadrupoles develop a spatially modulated structure on the top of a uniform BEC, interpreted in the original dimerized spin-1 model as coexistent $p$-type nematic and ${120}^{\circ }$ magnetic correlations. This may explain an intriguing nonmagnetic phase found in ${\mathrm{Ba}}_3{\mathrm{ZnRu}}_2{\mathrm{O}}_9$.

Figure 1

(a) Schematic illustration of the spin-1 dimer model on a triangular lattice. (b) Intradimer and interdimer interactions, where $J>0, J^{\prime }, J^{\prime \prime }$ are the Heisenberg exchanges, and $B$ is the biquadratic exchange. (c) Eigenenergy levels of ${\mathcal{H}}_{\text{intra}}$ of an isolated dimer. $s, t$, and $q$ denote the singlet, triplet, and quintet states of the dimer, respectively. (d) Eigenenergy levels of ${\mathcal{H}}_{\text{intra}}$ of two isolated dimers without the interdimer $J^{\prime }$ and $J^{\prime \prime }$. (e), (f) Examples of the second-order perturbation processes. Processes in (e) return to the original state $|s,t_0\rangle$. Processes in (f) exchange the ${\mathcal{S}}^z=+1$ and $-1$ triplets, which are the origins of the biquadratic interactions between two triplets ${\left({\mathcal{S}}_i·{\mathcal{S}}_j\right)}^2$.

Figure 2

Ground-state phase diagram of the spin-1 dimer triangular lattice at (a) $B/J=0.2$ and (b) 0.4, obtained by the numerical diagonalization of the effective model with $N=12$. Filled and open circles represent the first- and second-order phase transitions, where the transition between FM-BEC and FQ-$p$-BEC is weakly first order. FM, AFM, FQ represent the ferromagnetic, antiferromagnetic, and ferroquadrupolar phases, and $\langle n_t\rangle \approx 1$ and $\lesssim 0.9$ are the solid and BEC states of bosons. Colors in the phase diagram are the density plot of the triplet number $\langle n_t\rangle$. The small $J^{\prime }, J^{\prime \prime }$ region marked with red square in (b) encloses the spin nematic (SN) phase [see Fig. 8]. (c) Phase diagram on the plane of $J^{\prime }/J$ and $B/J$, whose fixed $B/J$ lines correspond to the $J^{\prime }=J^{\prime \prime }$ line of the phase diagrams in (a) and (b).

Figure 3

$J^{\prime }/J$ dependencies of the physical quantities at $B/J=0.2$ when (a)–(c) $J^{\prime \prime }/J=-0.2$ and (d)–(f) $J^{\prime \prime }/J=+0.1$. (a), (d)  Total energies $e_{\text{all}}$ and the contributions from major terms in the effective Hamiltonian $e_t,e_P,e_{\mathcal{J}}$, and $e_{\mathcal{B}}$. (b), (e) Triplet densities $\langle n_t\rangle$. (c), (f) Spin [Eq. (6)] and quadrupole [Eq. (7)] structure factors at $\mathrm{\Gamma }, \mathrm{K}, \mathrm{M}$ points in the reciprocal space. Quadrupole structure factors denoted as $\overline{\mathcal{Q}}\left(\mathit{k}\right)$ are normalized to be compared with the spin structure factors $\mathcal{S}\left(\mathit{k}\right)$ (see Sec. 3a3 for details).

Figure 4

Spatial correlation functions for $(J^{\prime }/J,J^{\prime \prime }/J)=(-0.2,+0.2)$ in FQ-BEC and $(+0.2,-0.2)$ in FQ-$p$-BEC phases at $B/J=0.2$: the quadrupolar correlations in (a), (b) $\langle {\mathcal{Q}}_1·{\mathcal{Q}}_j\rangle$ and the boson-boson correlation in (c), (d) $\langle n_1{n}_j\rangle$. Areas of the circles are proportional to the amplitude of correlations $\left|\langle {\mathcal{Q}}_1·{\mathcal{Q}}_j\rangle \right|$ or $\left|\langle n_1{n}_j\rangle \right|$. Red and blue circles in (a) and (b) correspond to the signs of $\langle {\mathcal{Q}}_1·{\mathcal{Q}}_j\rangle$, positive and negative, respectively.

Figure 5

$J^{\prime }/J$ dependence of the physical quantities at $B/J=0.4$ when (a)–(c) $J^{\prime \prime }/J=-0.1$ and (d)–(f) $J^{\prime \prime }/J=+0.2$. (a), (d) Total energies $e_{\text{all}}$ and the contributions from major terms in the effective Hamiltonian $e_t,e_P,e_{\mathcal{J}}$, and $e_{\mathcal{B}}$. (b), (e) Triplet densities $\langle n_t\rangle$. (c), (f) Spin [Eq. (6)] and quadrupole [Eq. (7)] structure factors at $\mathrm{\Gamma }, \mathrm{K}, \mathrm{M}$ points in the reciprocal space. Quadrupole structure factors denoted as $\overline{\mathcal{Q}}\left(\mathit{k}\right)$ are normalized to be compared with the spin structure factors $\mathcal{S}\left(\mathit{k}\right)$ (See Sec. 3a3 for details).

Figure 6

(a), (b) Phase diagrams of the effective model up to the first order in $J^{\prime }/J$ and $J^{\prime \prime }/J$ at (a) $B/J=0.2$ and (b) $B/J=0.4$, obtained by the analysis of the results of the numerical diagonalization on the $N=12$ cluster.

Figure 7

Low-energy excited states as a function of $S(S+1)$ for (a) FQ-BEC, (b) FQ-$p$-BEC, (c) AFM-solid phases for $N=12$ cluster. Filled circles and triangles are the $\mathrm{\Gamma }$ and $\mathrm{K}$ points of the Brillouin zone and open circles are from the other $\mathit{k}$ points. Those following the solid line are tower of states indicating the long-range order. (d) Triplet density (left) and structure factors at $\mathrm{\Gamma }, \mathrm{K}$ points (right), including $\mathcal{N}\left(\mathit{k}\right)$ and $\mathcal{C}\left(\mathit{k}\right)$ which gives the information on the details of the magnetic properties of the dimerized spin ${\mathit{S}}_{i_{\gamma }}$. $J^{\prime }/J$ is varied from FQ-BEC ($J^{\prime }/J<0.15$), FQ-$p$-BEC ($0.15<J^{\prime }/J<0.25$) to AFM-BEC phases.

Figure 8

(a) Low-energy excited states as a function of $S(S+1)$ at $J^{\prime }/J=0.01$ and $J^{\prime \prime }/J=-0.01, B/J=0.4$. (b) Phase diagram at small $J^{\prime }/J$ and $J^{\prime \prime }/J$ and $B/J=0.4$ determined by the Anderson tower analysis, which is the magnification of the region indicated by red square at the center part of Fig. 2. SN phase is found for $J^{\prime }\sim -J^{\prime }$, where the lowest excited state is $S=2$. The boundaries are given both from the crossing of $S=1,2$ levels (filled circle) and from the structural factor (open circle). Physical quantities along the fixed $J^{\prime \prime }/J=0.01$ line in the phase diagram are given in Fig. 12.

Figure 9

Energy bands of the eigenstates of the quadratic Hamiltonian [Eq. (20)] for (a) $J^{\prime }-J^{\prime \prime }<0$ and (b) $J^{\prime }-J^{\prime \prime }>0$. We set $B/J=0.2$, and use the parameters $t, P$, and $\mu$ defined in Eq. (4), at the first order.

Figure 10

Three different types of fluctuations that contribute to the formation of the spin nematics. Single and double arrows represent the spin $\frac{1}{2}$ and spin 1, respectively. (a) In the spin-$\frac{1}{2}$ dimer system, ring exchange interaction that permutates spin $\frac{1}{2}$'s as $(1,2,3,4)\leftrightarrow (2,3,4,1)$ in the upper panel (see Refs. [29, 40]) and the second-order perturbation terms operated twice, ${(J^{\prime \prime }{\mathit{s}}_i·{\mathit{s}}_j)}^2$, with ${\mathit{s}}_i$ the spin-$\frac{1}{2}$ operator discussed in Refs. [41, 66], work in the similar manner. (b) Fluctuation between on-bond spin 1's that are equivalent to those of (a). (c) In our spin-1 dimer system, the pair-creation and annihilation term ($P$) plays a major role which originates from the first order in $J^{\prime }$ and $J^{\prime \prime }$.

Figure 11

(a), (b) Typical second-order perturbation processes over three-dimers. (a) Processes of “correlated hopping” and (b) pair creation of bosons. Ellipses mark the pair of sites to which the perturbation Hamiltonian ${\mathcal{H}}_{\text{inter}}$ operates. (c) Energy levels of ${\mathcal{H}}_{\text{intra}}$ of the three spin-1 dimer states. (d)–(g) $J^{\prime }/J$ dependencies of the ground-state energies of the effective Hamiltonian ${\mathcal{H}}_{\text{eff}}$ with and without ${\mathcal{H}}_{\text{3}\text{body}}$ and the original spin Hamiltonian $\mathcal{H}$ on the nine-dimer triangular lattice for $(B/J,(J^{\prime }+J^{\prime \prime })/J)=$ (d) $(0.2,+0.2)$, (e) $(0.2,-0.2)$, (f) $(0.4,+0.2)$, and (g) $(0.4,-0.2)$.

Figure 12

(a) Low-energy excited states of $S=0,1,2$ sectors as a function of small $J^{\prime }/J$ with fixed $J^{\prime \prime }/J=0.01$, where we find a crossing of the low-lying $S=1$ and 2 levels shown more clearly in the inset (extracted data from the main panel). The region near $J^{\prime }\sim 0$ where $S=2$ is the lowest excited state is the typical spin nematic phase found in the $S=1$ BLBQ models. (b) Spin [Eq. (6)], quadrupole [Eq. (7)], and staggered spin [Eq. (18)] structure factors at $\mathrm{\Gamma }, \mathrm{K}, \mathrm{M}$ points. (c) The contributions from major terms in the effective Hamiltonian $e_t,e_P,e_{\mathcal{J}}$, and $e_{\mathcal{B}}$. (d) Triplet densities $\langle n_t\rangle$.