Effect of anisotropy on the ground-state magnetic ordering of the spin-one quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ model on the square lattice

TL;DR

This study investigates the ground-state phases of a spin-1 $J_1^{XXZ}$--$J_2^{XXZ}$ model on a square lattice, revealing no intermediate disordered phase and characterizing the nature and boundaries of magnetic orderings across anisotropy parameters.

Contribution

The paper provides the first high-order coupled cluster analysis of the spin-1 $J_1^{XXZ}$--$J_2^{XXZ}$ model, showing the absence of intermediate disordered phases and accurately determining phase boundaries.

Findings

No intermediate disordered phase between Ne9el and stripe phases.

First-order quantum phase transitions for all $J_2/J_1$ and $\Delta$ values.

Phase boundary $J_2^c(\Delta)$ deviates from classical at $\Delta=1$, with $J_2^c(1)=0.55 \\pm 0.01$.

Abstract

We study the zero-temperature phase diagram of the $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ Heisenberg model for spin-1 particles on an infinite square lattice interacting via nearest-neighbour ($J_1 \equiv 1$) and next-nearest-neighbour ($J_2 > 0$) bonds. Both bonds have the same $XXZ$-type anisotropy in spin space. The effects on the quasiclassical N\'{e}el-ordered and collinear stripe-ordered states of varying the anisotropy parameter $\Delta$ is investigated using the coupled cluster method carried out to high orders. By contrast with the spin-1/2 case studied previously, we predict no intermediate disordered phase between the N\'{e}el and collinear stripe phases, for any value of the frustration $J_2/J_1$, for either the $z$-aligned ($\Delta > 1$) or $xy$-planar-aligned ($0 \leq \Delta < 1$) states. The quantum phase transition is determined to be first-order for all values of $J_2/J_1$ and $\Delta$. The position of the phase boundary $J_{2}^{c}(\Delta)$ is determined accurately. It is observed to deviate most from its classical position $J_2^c = {1/2}$ (for all values of $\Delta > 0$) at the Heisenberg isotropic point ($\Delta = 1$), where $J_{2}^{c}(1) = 0.55 \pm 0.01$. By contrast, at the XY isotropic point ($\Delta = 0$), we find $J_{2}^{c}(0) = 0.50 \pm 0.01$. In the Ising limit ($\Delta \to \infty$) $J_2^c \to 0.5$ as expected.