We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) methods to study the phases and transitions displayed by a class of sign-free designer Hamiltonians for $\mathrm{SU}\left(N\right)$ analogs of spin $S=1$ quantum antiferromagnets on the square lattice. The $\mathrm{SU}\left(N\right)$ spins are generators of the single-row two-column representation (complex conjugate of single-row two-column representation) on $A$ ($B$) sublattices of the square lattice, and the Hamiltonian is designed to explore the competition between the nearest-neighbor antiferromagnetic exchange couplings $J$ and four-spin interactions $Q$ that favor a plaquette-ordered valence bond solid (p-VBS) ground state. We find that this state is indeed established at large $Q/J$ for all $N>3$. For $3<N\le 9$, the ground state exhibits a direct first-order quantum phase transition from a small-$Q/J$ Néel ordered antiferromagnetic state to this large-$Q/J$ p-VBS state. The ground state at $Q/J=0$ for $N\ge 10$ has been previously reported to be a valence bond nematic state, dubbed the Haldane nematic in recent literature. For small nonzero $Q/J$ and $N\ge 10$, we additionally find an unusual intermediate state in which the four-site plaquette energy has a Bragg peak at wave vector $(\pi ,\pi )$ with no accompanying Bragg peaks at wave vectors $(\pi ,0)$ and $(0,\pi )$. This $(\pi ,\pi )$ state also appears to be metastable at $Q/J=0$, as evidenced by low-temperature histograms of the Haldane nematic and $(\pi ,\pi )$ order parameters. Deep in the p-VBS phase of the ground state phase diagram, we find the temperature-driven melting of the p-VBS order is in the Ashkin-Teller universality class. In this regime, we identify an interesting signature of Ashkin-Teller criticality in four-site plaquette singlet correlations at wave vector $(\pi ,\pi )$; this is in addition to the expected critical fluctuations of the conventional p-VBS order parameter at wave vectors $(\pi ,0)$ and $(0,\pi )$.

中文翻译：

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S=1 方晶格反铁磁体的 SU(N) 类似物中尼尔相、霍尔丹向列相、板状价键固相和 (π,π) 价键固相之间的竞争

我们使用随机级数展开 (SSE) 量子蒙特卡罗 (QMC) 方法来研究一类无符号设计哈密顿量所显示的相位和转变：$苏（氮）$自旋的类似物$S=1$方形晶格上的量子反铁磁体。这$苏（氮）$自旋是单行两列表示的生成器（单行两列表示的复共轭）$A$($乙$) 方晶格的子晶格，哈密顿量旨在探索最近邻反铁磁交换耦合之间的竞争$J$和四自旋相互作用$问$有利于血小板有序价键固体（p-VBS）基态。我们发现这种状态确实是大范围建立的$问/J$对全部$氮>3$。为了$3<氮\le 9$，基态表现出从小到大的直接一阶量子相变$问/J$尼尔命令反铁磁态达到这个大-$问/J$p-VBS状态。基态在$问/J=0$为了$氮\ge 10$此前曾报道过这是一种价键向列态，在最近的文献中被称为霍尔丹向列态。对于小的非零$问/J$和$氮\ge 10$，我们还发现了一种不寻常的中间态，其中四位点板能量在波矢处具有布拉格峰$（\pi ,\pi ）$波矢量处没有伴随的布拉格峰$（\pi ,0）$和$（0,\pi ）$。这$（\pi ,\pi ）$状态似乎也处于亚稳态$问/J=0$，霍尔丹向列相的低温直方图证明了这一点$（\pi ,\pi ）$订单参数。在基态相图的 p-VBS 相深处，我们发现 p-VBS 级的温度驱动熔化属于 Ashkin-Teller 普适性类别。在这种情况下，我们在波矢量的四位点斑块单重态相关性中发现了 Ashkin-Teller 临界性的一个有趣特征$（\pi ,\pi ）$;这是波矢量处传统 p-VBS 阶次参数预期临界波动的补充$（\pi ,0）$和$（0,\pi ）$。