Supersolid phase and magnetization plateaus observed in the anisotropic spin-$\frac{3}{2}$ Heisenberg model on bipartite lattices

We study the spin-$3/2$ Heisenberg model including easy-plane and exchange anisotropies in one and two dimensions. In the Ising limit, when the off-diagonal exchange interaction $J$ is zero, the phase diagram in magnetic field is characterized by magnetization plateaus that are either translationally invariant or have a two-sublattice order, with phase boundaries that are macroscopically degenerate. Using a site-factorized variational wave function and perturbational expansion around the Ising limit, we find that superfluid and supersolid phases emerge between the plateaus for small finite values of $J$. The variational approach is complemented by a density matrix renormalization group study of a one-dimensional chain and exact diagonalization calculations on small clusters of a square lattice. The studied model may serve as a minimal model for the layered Ba${}_2$CoGe${}_2$O${}_7$ material compound, and we believe that the vicinity of the uniform 1/3 plateau in the model parameter space can be observed as an anomaly in the measured magnetization curve.

Figure 1

Phase diagram in the Ising limit as a function of the anisotropy and magnetic field. The spin configurations on the A and B sublattices are shown, as well as the degeneracies of the ground-state manifolds on the phase boundaries (the dashed line is a first-order phase boundary). Long arrows represent the $S^z=\pm 3/2$ spin states, while the short ones the $S^z=\pm 1/2$ spin states. The coordination number $\zeta =2$ for the chain and $\zeta =4$ for the square. $F1$ and $F3$ are uniform phases, while the others break the translational invariance and are twofold degenerate.

Figure 2

(a) Instabilities (thick lines) of the gapped phases in the square lattice as obtained from the perturbation theory for $J/J_z=0.2$. We also show the $J=0$ phase boundaries of Fig. 1 with thin lines for comparison. (b) Variational phase diagram as a function of $\Lambda /\zeta J$ and $h_z/\zeta J$ for large exchange anisotropy $J/J_z=0.2$ and a bipartite lattice with $\zeta$ neighbors. Dashed lines stand for first-order, while solid lines denote second-order phase transitions. The solid dot ending the first-order transition represent a tricritical point.

Figure 3

(a) Schematic figure for the first-order hopping process that occurs during the instability of uniform phases $F1$ and $F3$, where the dispersion is $\propto 4{\gamma }_{\mathbf{k}}$. (b) Schematic representation of the second-neighbor correlated hopping that gives the dispersion $\propto 16{\gamma }_{\mathbf{k}}^2$. There are 8 neighboring places where the magnon can hop through a virtual state on the B site.

Figure 4

Variational phase diagram for $h=0$ as a function of $\Lambda /J_z$ and $J/J_z$. Solid lines stand for continuous (second-order) phase boundaries, while the dashed line denotes the first-order phase boundary of the phase $A3$.

Figure 5

Order parameters as a function of $\Lambda /J_z$ for different values of $J/J_z$. The axial antiferromagnets have finite staggered magnetization, and zero $U\left(1\right)$ order parameter. In the planar superfluid phases $m_z^{\mathrm{st}}=0$ but the expectation value of $O_{U\left(1\right)}$ is finite, while the conical antiferromagnet exhibits both types of order.

Figure 6

Phase diagram as a function of $\Lambda /J$ and $h_z/J$ ($J_z=J$).

Figure 7

Order parameters as a function of magnetic field for different values of $\Lambda$ parameter. The fully and partially polarized ferromagnets $F3$ and $F1$ exhibit finite magnetization $m_z$, while in the conical ferromagnetic phase ${\mathrm{SF}}_{\mathrm{F}}$ the expectation values of both $m_z$ and $O_{U\left(1\right)}$ are finite.

Figure 8

Expectation value of order parameters per site as a function of $h/J_z$ for different values of $\Lambda /J_z$. In the axial antiferromagnetic phases $A1$ and $A3$ only the staggered magnetization has finite expectation value. For $\Lambda =0$ there is a first-order phase transition from the completely polarized $A3$ phase to the superfluid phase. All the other field-induced transitions are second-order transitions. The superfluid phase exhibits finite magnetization $m_z$ and finite staggered in-plane magnetization $O_{U\left(1\right)}$. The ferromagnetically ordered phases $F3$ and $F1$ are characterized by finite magnetization $m_z$, while the plateaus ($P1$ and $P2$) have an additional finite staggered magnetization $m_z^{\mathrm{st}}$. In the supersolid phase all four order parameters have finite expectation values.

Figure 9

The phase diagram of the anisotropic $S=1$ model in the (a) Ising limit for a bipartite lattice with coordination number $\zeta$ and (b) for the square lattice ($\zeta =4$) when $J=0.2J_z$, obtained from the variational calculation.

Figure 10

Zero-field phase diagram for the infinite chain as a function of $\Lambda /J_z$ and $J/J_z$, as obtained from DMRG calculation with $\chi =25$. Panel (a) shows the staggered magnetization. Panel (b) shows the half-chain entanglement entropy, i.e., the von Neumann entropy of the reduced density matrix for a bipartition of the chain into two half chains.

Figure 11

Phase diagram as a function of $\Lambda /J_z$ and $h/J_z$ for (a)–(c) $J/J_z=0.1$ and (d)–(f) $J/J_z=0.2$. We show the uniform and the staggered magnetization along the $z$ axes, where the plateau phases can be identified. The large increase of the entanglement entropy indicates gapless phases. The phase diagram is obtained from DMRG calculation with $\chi =25$.

Figure 12

Estimate of the central charge from the entanglement entropy for four different points in the LL phase and one point in the supersolid phase (the $h/J_z=1.95$, $\Lambda /J_z=0.5$ point). The solid lines are fits based on the $S=\frac{c}{6}\ln \xi +\text{const.}$ formula, with $c$ set to 1. In all these points the gapless phases are characterized by $c=1$ central charge.

Figure 13

The magnetic field dependence of the order parameters as a function of the magnetic field for $J/J_z=0.2$ and $\Lambda /J_z=0.5$. The curves are the result of DMRG calculations with $\chi =40$ and with a small field $h_x/J_z={10}^{-4}$. In (a), the nonvanishing off-diagonal order parameter $m_x^{\text{st}}$ shows the extension of the gapless phases. The finite value of the $m_z^{\text{st}}$ and $m_x$ indicate a robust supersolid phase, as seen in (b).

Figure 14

The magnetization has a square root singularity at (a) lower $h_{c,1}/J_z=1.8579$, and (b) upper $h_{c,2}/J_z=2.0195$ critical field. The solid lines show the $m_z^2\approx 2.68(h-h_{c,1})/J_z+16.9{(h-h_{c,1})}^2/J_z^2$ and ${(1-m_z)}^2\approx 1.27(h_{c,2}-h)/J_z+13.2{(h_{c,2}-h)}^2/J_z^2$ fits to the magnetization curves.

Figure 15

The first few lowest lying energy levels of a 10-site cluster for (a) $S^z=0$ and (b) $S^z=1$ as a function of $\Lambda /J_z$. We set $J=0.2J_z$. The inset shows the available $\mathbf{k}$ points in the Brillouin zone.

Figure 16

The gap $\Delta =E_{(\pi ,\pi )}-E_{(0,0)}$ as a function of $\Lambda /J_z$ and $h/J_z$ for the 10-site cluster. The solid curves separate the different $S^z$ sectors.

Figure 17

Magnetization as a function of magnetic field, as obtained from variational calculation and exact diagonalization. Here $J=0.2J_z$, and $\Lambda /J_z=$0, 1.5, and 4.5. $h_{\text{sat}}$ is the saturation field [Eq. (21)].