We investigate the Heisenberg model on a decorated square (Fisher) lattice in the presence of first-neighbor $J_1$, second-neighbor $J_2$, and third-neighbor $J_3$ exchange couplings, with antiferromagnetic $J_1$. The classical ground-state phase diagram obtained within a Luttinger-Tisza framework is spanned by two antiferromagnetically ordered phases, and an infinitely degenerate antiferromagnetic chain phase. Employing classical Monte Carlo simulations we show that thermal fluctuations fail to lift the degeneracy of the antiferromagnetic chain phase. Interestingly, the spin-wave spectrum of the Néel state displays three Dirac nodal loops out of which two are symmetry protected while for the antiferromagnetic chain phase we find symmetry-protected Dirac lines. Furthermore, we investigate the spin $S=\frac{1}{2}$ limit employing a bond operator formalism which captures the singlet-triplet dynamics, and find a rich ground-state phase diagram host to a variety of valence bond solid orders in addition to antiferromagnetically ordered phases.

中文翻译：

---

Fisher格上沮丧的Heisenberg模型中的竞争订单

我们在存在第一个邻居的情况下研究装饰正方形（Fisher）晶格上的Heisenberg模型 ${Ĵ}_{\mathrm{1个}}$，第二邻居 ${Ĵ}_2$和第三邻居 ${Ĵ}_3$ 交换联轴器，带反铁磁 ${Ĵ}_{\mathrm{1个}}$。在Luttinger-Tisza框架内获得的经典基态相图由两个反铁磁有序相和一个无限简并的反铁磁链相构成。使用经典的蒙特卡洛模拟，我们证明了热涨落无法提升反铁磁链相的简并性。有趣的是，Néel态的自旋波谱显示了三个Dirac节点环，其中两个受到对称保护，而对于反铁磁链相，我们发现了对称保护的Dirac线。此外，我们调查了自旋$\mathrm{小号}=\frac{\mathrm{1个}}{2}$ 限制使用键操作符形式主义来捕获单重态-三重态动力学，并找到一个丰富的基态相图宿主，除了反铁磁有序相外，还具有多种价键固体阶。