A Tale of Two Plateaus: Competing Orders in Spin-1 and Spin-$\tfrac{3}{2}$ Pyrochlore Magnets

Abstract

We use large-scale density-matrix renormalization group simulations with bond dimensions up to $20\ 000$ to determine the magnetization curves of spin-1 and spin-$\tfrac{3}{2}$ pyrochlore Heisenberg antiferromagnets. Both models exhibit a robust half-magnetization plateau, and we find that the same 16-site state (quadrupled unit cell) is selected in both cases on the largest 64-site cubic cluster we consider for the plateau state. This contrasts sharply with the effective quantum dimer model prediction which favors the ``R'' state, and demonstrates the breakdown of the perturbative mechanism at the Heisenberg point. These results provide a nonperturbative characterization of field-induced phases in pyrochlore magnets and predictive guidance for spin-1 and spin-$\tfrac{3}{2}$ materials.

Figures (5)

• Figure 1: On-site magnetization patterns on the $m/m_{\rm sat}=\tfrac{1}{2}$ plateau for the cubic $N=64$ cluster. Only one cubic unit cell is shown. Red spheres mark minority-spin sites; sphere radii are proportional to $|\langle S_i^z\rangle|$. For the "S" state we find $\langle S_i^z\rangle \sim 0.88$ (majority) and $\sim -0.64$ (minority) for $S=1$, and $\langle S_i^z\rangle \sim 1.36$ (majority) and $\sim -1.09$ (minority) for $S=\tfrac{3}{2}$ at bond dimension $\chi=12\,000$. The corresponding values for the "R" state are essentially the same.
• Figure 2: Normalized magnetization curves of pyrochlore clusters in a magnetic field. Upper panel: $m/m_{\rm sat}$ versus $h/h_{\rm sat}$ for the cubic $N=32$ cluster and spin lengths $S=\tfrac{1}{2},1,\tfrac{3}{2}$, where $m=\langle\psi_0|S^z_{\rm tot}|\psi_0\rangle$, $m_{\rm sat}=NS$, and $h_{\rm sat}=8JS$. Lower panel: magnetization for the $S=1$ model on clusters up to $N=108$; for $N=48$ and $N=64$ only the upper half of the curve is reliably accessible, while for $N=108$ only the high-field regime is shown. Inset: finite-size extrapolation of the lower and upper critical fields of the $m/m_{\rm sat}=\tfrac{1}{2}$ plateau, yielding $h_- = 4.22(1)J$ and $h_+ = 4.70(1)J$ in the thermodynamic limit; the fit is restricted to cubic clusters and includes the primitive 16-site cubic unit cell.
• Figure 3: Spin structure factor of the spin-1 model on different clusters at the half-magnetization plateau, computed at bond dimension $\chi=10\,000$. The upper (lower) panels show cuts in the $[hhl]$ ($[hl0]$) plane. The appearance of line-like features signals dimensional reduction and the breaking of threefold rotational symmetry, whose form depends on the cluster geometry.
• Figure 4: Variance-based extrapolation of the energy per site for different plateau candidates. The upper panel shows results for the spin-1 model on the cubic 64-site cluster, while the lower panel presents data for the spin-$\tfrac{3}{2}$ model on the 32- and 64-site clusters. For clarity, variance values for the 32-site cluster are rescaled by a factor of 10. Error bars are defined as one quarter of the difference between the lowest-variance data point and the extrapolated value. Numbers next to the symbols indicate the bond dimension $\chi$. For spin-$\tfrac{3}{2}$, data for the $\boldsymbol{q}=0$ state are omitted for clarity, as its energies lie above those of the other candidates.
• Figure 5: Spin structure factor of the spin-$\tfrac{3}{2}$ model at the half-magnetization plateau for different cluster sizes, computed at bond dimension $\chi=10\,000$. The upper (lower) panels show cuts in the $[hhl]$ ($[hl0]$) plane. While the "R" state breaks threefold rotational symmetry, its structure-factor signature on the 32-site cluster appears nearly symmetric in the $[hhl]$ plane, in contrast to the pronounced line features observed for the "S" state.