Ground-state phase diagram and magnetic properties of a tetramerized spin-$1∕2$ $J_1\text{−}{J}_2$ model: BEC of bound magnons and absence of the transverse magnetization

We study the ground state and the magnetization process of a spin-$1∕2$ $J_1\text{−}{J}_2$ model with a plaquette structure by using various methods. For small interplaquette interaction, this model is expected to have a spin gap and we computed the first and the second excitation energies. If the gap of the lowest excitation closes, the corresponding particle condenses to form magnetic orders. By analyzing the quintet gap and magnetic interactions among the quintet excitations, we find a spin-nematic phase around $J_1∕J_2\sim -2$ due to the strong frustration and the quantum effect. When high magnetic moment is applied, not the spin-1 excitations but the spin-2 ones soften and dictate the magnetization process. We apply a mean-field approximation to the effective Hamiltonian to find three different types of phases (a conventional BEC phase, “striped” supersolid phases, and a $1∕2$ plateau). Unlike the BEC in spin-dimer systems, this BEC phase is not accompanied by transverse magnetization. Possible connection to the recently discovered spin-gap compound $\left(\mathrm{Cu}\mathrm{Cl}\right)\mathrm{La}{\mathrm{Nb}}_2{\mathrm{O}}_7$ is discussed.

Figure 1

Two-dimensional square lattice with a plaquette structure to be considered in this paper. Filled circles denote spin-$1∕2\mathrm{s}$ connected by the usual exchange interactions. Thin lines (both solid and broken) imply that the interactions are multiplied by the distortion parameter $\lambda$ on these bonds.

Figure 2

Single plaquette. Dots represent spin $1∕2\mathrm{s}$, and the solid and the dashed lines respectively represent Heisenberg interaction with the couplings $J_1$ and $J_2$ between $S=1∕2$ spins.

Figure 3

(Color online) The energy of the triplets ∣1,0;1⟩, ∣0,1;1⟩ and the quintet ∣1,1;2⟩. We take the units of energy as $J_2$, and the energy is plotted as a function of $J_1∕J_2$.

Figure 4

Interplaquette interactions associated with the plaquette $n$ (shown by a thick line).

Figure 5

(Color online) The dispersion relation of the excitation energy of the triplet $p^{\prime }$ in Eq. (20), which has the lower energy of the two triplets, for the parameters $\lambda =0.3$, $J_1=-0.8$, $J_2=1$.

Figure 6

(Color online) The dispersion relation of the excitation energy $E_t^{-}\left(\mathbf{k}\right)$ of the triplet at $\lambda =0.3$, $J_1=-0.8$, $J_2=1$.

Figure 7

(Color online) The dispersion relation of the excitation energy $E_q\left(\mathbf{k}\right)$ of quintet at $\lambda =0.3,J_1=-0.8,J_2=1$.

Figure 8

(Color online) The value of $\lambda$ at which the energy gaps ${\Delta }_t$ and ${\Delta }_q$ close. The energy gap of the triplets ${\Delta }_t$ is given in Eq. (D5), which is not reliable near $J_1∕J_2=-2$ because of the pole there, and that of the quintet ${\Delta }_q$ is in Eq. (27), which is not reliable near $J_1∕J_2=0$.

Figure 9

(Color online) Plot of the value of $\lambda$ when the energy gaps are equal to 0. We use ${\Delta }_{t,\mathrm{mod}}$ [Eq. (D6)] for the triplets, which is reliable even in the vicinity of $J_1∕J_2=-2$. The curve for the quintet is the same as in Fig. 8. We only plot the region $-2<J_1∕J_2<-1.5$.

Figure 10

(Color online) The schematic phase diagram of the ground state determined by the particle whose excitation gap closes first. In the green region, the quintet and the singlet condense, in the red the triplet and the singlet, and in the blue the singlet. In the region marked by blue, the energy gap exists. The phase shown by red may be considered as CAF state. The nature of the green phase, where the quintet condensation occurs, is closely investigated using an effective Hamiltonian $H_{\mathrm{qu}}$ [Eq. (32)].

Figure 11

Clusters involved in the three-points interaction in Eq. (32). The plaquette corresponding to $i^{″}$ is always located on the center of the clusters. We identify all clusters obtained from a given one by rotation by $\pi ∕2,\pi ,3\pi ∕2$, and reflection.

Figure 12

(Color online) The schematic phase diagram obtained in a similar manner to that in Fig. 10. We zoom up the region around $J_1∕J_2=-2$ in Fig. 10. In the two regions on the left [green $(\mathrm{q}+\mathrm{s},1)$ and yellow $(\mathrm{q}+\mathrm{s},2)$], the quintet and the singlet condense and we determined the resulting magnetic orders by a mean-field approximation to the magnetic Hamiltonian $H_{\mathrm{qu}}$. In the green region, the quintet and the singlet condense, and the spin-nematic phase appears. In the red $(\mathrm{t}+\mathrm{s})$, on the other hand, conventional ferromagnetic order is stabilized. The red and the blue (s) region are the same as Fig. 10.

Figure 13

(Color online) The energy of eigenstates of a single plaquette as a function of magnetic field $h$.

Figure 14

Two-sublattice structures assumed in the calculation: (i) striped (left) and (ii) checkerboard (right) cases. Circles (whether filled or open) denote the strongly coupled plaquettes shown by thick lines in Fig. 1.

Figure 15

(Color online) Schematic classification of the magnetization curve. (i) In the green region, the curve is smooth and the system is always in the normal BEC phase. (ii) In the red region, the magnetization curve has a $1∕2$ plateau. Except at the plateau, the system is in the normal BEC phase. (iii) The region where we have additional supersolid phases around the $1∕2$ plateau is highlighted in blue. (iv) In the region colored by yellow, magnetization jumps to saturation and the magnetization process is steplike. The concrete expression of curves is shown in Fig. 16. The phase boundary is only schematic.

Figure 16

(Color online) Magnetization curve for various values of the distortion parameter $\lambda$. The frustration parameter is fixed to $J_1∕J_2=-1.4$. The colors of the curves correspond to those used in Fig. 15 (except for yellow). The blue curve has supersolid phase around the $1∕2$ plateau and the phase transition between the normal BEC and the supersolid phase is of second order. All curves in BEC and supersolid phase are convex down because of the three point interaction $\gamma$ in Eq. (39).

Figure 17

Magnetization curve obtained from Eq. (39) by using the parameter set in Eq. (46).