Exact diagonalization and cluster mean-field study of triangular-lattice XXZ antiferromagnets near saturation

Quantum magnetic phases near the magnetic saturation of triangular-lattice antiferromagnets with XXZ anisotropy have been attracting renewed interest since it has been suggested that a nontrivial coplanar phase, called the $\pi$-coplanar or $\mathrm{\Psi }$ phase, could be stabilized by quantum effects in a certain range of anisotropy parameter $J/J_z$ besides the well-known 0-coplanar (known also as $V)$ and umbrella phases. Recently, Sellmann et al. [Phys. Rev. B 91, 081104(R) (2015)] claimed that the $\pi$-coplanar phase is absent for $S=1/2$ from an exact-diagonalization analysis in the sector of the Hilbert space with only three down-spins (three magnons). We first reconsider and improve this analysis by taking into account several low-lying eigenvalues and the associated eigenstates as a function of $J/J_z$ and by sensibly increasing the system sizes (up to 1296 spins). A careful identification analysis shows that the lowest eigenstate is a chirally antisymmetric combination of finite-size umbrella states for $J/J_z\gtrsim 2.218$ while it corresponds to a coplanar phase for $J/J_z\lesssim 2.218$. However, we demonstrate that the distinction between 0-coplanar and $\pi$-coplanar phases in the latter region is fundamentally impossible from the symmetry-preserving finite-size calculations with fixed magnon number. Therefore, we also perform a cluster mean-field plus scaling analysis for small spins $S\le 3/2$. The obtained results, together with the previous large-$S$ analysis, indicate that the $\pi$-coplanar phase exists for any $S$ except for the classical limit $\left(S\to \infty \right)$ and the existence range in $J/J_z$ is largest in the most quantum case of $S=1/2$.

Figure 1

Three magnetic phases predicted for the quantum triangular XXZ model in the presence of strong magnetic fields. (a) Spin moments on three sublattices A, B, and C in each phase are illustrated. (b) Schematic representation of the suggested phase diagram [29, 30] in the vicinity of the saturation field $H_s=3(J+2J_z)S$.

Figure 2

Two series of clusters that are considered in the present ED analysis.

Figure 3

The low-lying eigenenergies of the three-magnon sector for $N=36$ as a function of $J/J_z$. We plot the eigenenergies of the ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_1},{\mathrm{\Psi }}_{{\mathrm{\Gamma }}_2}$, and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ states measured from that of ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$. The lower panel is the enlarged view showing the energy difference between ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$. In the shaded region marked by the double-headed arrow, the ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ state has the lowest energy.

Figure 4

Same as in Fig. 3, but for $N=324$. There is no $J/J_z$ range where the ${\mathrm{\Gamma }}_3$ state has the lowest energy.

Figure 5

Lowest eigenstate diagram of $1/N$ versus $J/J_z$. The eigenenergy-crossing points between the ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_{1,2}}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$ states (marked by the circles), between the ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_{1,2}}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ states (diamonds), and between the ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$ states (inverted triangles) are shown. The boundary lines are just a guide for the eye. The left (right) panel shows the data for the series of $N=36,81,...,1296$ $\left(N=27,48,...,1200\right)$.

Figure 6

Overlaps between ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_m}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_m}^{*}$ as a function of $1/N$ at $J/J_z=1$. The top and bottom panels show the data for the series $N=36,81,...,1296$ and $N=27,48,...,1200$, respectively. All data seem to approach 1 as $N\to \infty$.

Figure 7

The energy-crossing point around $J/J_z\sim 1$ between the eigenstates ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$ as a function of $1/N$. The circles and inverted triangles show the data for the series $N=36,81,...,1296$ and $N=27,48,...,1200$, respectively.

Figure 8

(a) Log-log plots of the eigenenergy differences between ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_{1,2}}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$ (circles) and between ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$ (inverted triangles) as a function of the system size $N$. The slope of each pair of the two neighboring data points are shown in (b) for the former and (c) for the latter as a function of $1/N$. The right-hand panel of (a) is the schematic diagram of the energy levels in the range $0<J/J_z\lesssim 1.03$. We show the data at $J/J_z=0.5$ for the cluster series $N=27,48,...,1200$ as an example.

Figure 9

Log-log plots of the eigenenergies of ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ (circles) and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$ (diamonds) measured from that of the lowest eigenstate ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_{1,2}}$ as a function of the system size $N$ at $J/J_z=0.5$. We show the data for different magnon numbers $(n=3$, 5, and 7). The lines for $n=3$ $\left(n=5\right)$ are linear fitting functions of the $N=27,48,75$ $\left(N=48,75,108\right)$ data for each data series. The vertical dashed lines mark the point at which the magnon density $n/N=7/108\approx 0.065$.

Figure 10

Log-log plots of the eigenenergies of ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_3}$ (circles) and ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_4}$ (diamonds) measured from that of the lowest eigenstate ${\mathrm{\Psi }}_{{\mathrm{\Gamma }}_{1,2}}$ as a function of the system size $N$ at the fixed magnon density $n/N=0.065$. We show the data for different anisotropy parameters $(J/J_z=0.5$, 1, and 2). The lines are a linear fitting function for each data series.

Figure 11

Series of the clusters used in the CMF+S analysis. The parameter $\lambda$ is defined by $\lambda \equiv N_B/\left(3N_C\right)$. The bottom illustrations show examples of the independent clusters that have to be considered under the three-sublattice ansatz.

Figure 12

Cluster-size scaling of the CMF+S method for the phase boundaries just below the saturation field for (a) $S=1$ and (b) $S=3/2$. The transition points ${(J/J_z)}_{c1}$ and ${(J/J_z)}_{c2}$ correspond to the transitions between the 0-coplanar and $\pi$-coplanar phases and between the $\pi$-coplanar and umbrella phases, respectively. The symbols $(\times )$ mark the semianalytical values ${(J/J_z)}_{c{2}^{*}}$ from the dilute Bose-gas expansion [34].

Figure 13

Phase boundaries among the two coplanar and umbrella states just below the saturation field. We show the current CMF+S results for $S=1$ and $S=3/2$ together with the previous $S=1/2$ values [29] for ${(J/J_z)}_{\mathrm{c}1}$ (inverted triangles) and ${(J/J_z)}_{\mathrm{c}2}$ (diamonds). The dilute Bose-gas results for arbitrary $S$ (circles) [34] and within the large-$S$ approximation (dashed and dash-dotted curves) [30] are also plotted for comparison. Note that the arbitrary-$S$ dilute Bose-gas analysis provides a quantitatively precise value for the coplanar-umbrella transition point ${(J/J_z)}_{\mathrm{c}{2}^{*}}$ just below the saturation, but which coplanar state (0-coplanar or $\pi$-coplanar) appears for $J/J_z\le {(J/J_z)}_{\mathrm{c}{2}^{*}}$ is unspecified [32, 34]. The boundary lines are just a guide for the eye.