标题： “反交换”量子自旋液体$\mathbb{Z}_2$ 显示英文标题

标题： «Anticommuting» $\mathbb{Z}_2$ quantum spin liquids

摘要： 我们讨论了一类具有键依赖性伊辛耦合的格点量子哈密顿量$S=\frac{1}{2}$，其具有由众多局部守恒荷$\mathbb{Z}_2$组成的、且大小与系统规模成比例增长的相互反对易代数{\guillemotleft }{\guillemotright }，这一模型由 [arXiv:2407.06236] 提出，并包括了术语的命名。 这种相互反对易代数让人联想到自旋$\frac{1}{2}$的泡利矩阵代数，但其结构被编码在\emph{局域守恒荷}中。 这些模型在基态时具有有限的残余熵密度，其简化的但非平凡的简并计数以及伴随的量子自旋液体性质已在 [arXiv:2407.06236] 中得到证明。 自旋液体依赖于局部守恒的$\mathbb{Z}_2$充电在几何上具有位间链接的特性，在存在{\guillemotleft }反交换{\guillemotright }结构的情况下是非常自然的。 可以将其与六边形蜂窝状 Kitaev 自旋-$\frac{1}{2}$模型中六边形小平面的局部守恒$\mathbb{Z}_2$充电的键间链接特性进行对比，这会导致局部代数彼此交换。 在这项工作中，我们阐明了这种量子自旋液体流动性与一些带隙量子自旋液体中的多体拓扑序之间的差异，Kitaev toric 代码是其典型代表。 toric 代码属于更广泛的 Levin-Wen 或字符串网构造类，这些构造对于局部守恒的电荷具有彼此交换的代数。 我们将就{\guillemotleft }反交换{\guillemotright } $\mathbb{Z}_2$ 量子自旋液体类别中可能存在的多种多体现象做出若干精确陈述，并且这些量子自旋液体与广泛的剩余基态熵共存。 我们还将指出一个具有局域支撑的相互交换代数，它们在这些量子自旋液体的 Kitave 表示中自然以多线性马约拉纳形式表达。 显示英文摘要

摘要： We discuss a class of lattice $S=\frac{1}{2}$ quantum Hamiltonians with bond-dependent Ising couplings with a mutually {\guillemotleft}anticommuting{\guillemotright} algebra of extensively many local $\mathbb{Z}_2$ conserved charges that was introduced in [arXiv:2407.06236] including the nomenclature. This mutual algebra is reminiscent of the spin-$\frac{1}{2}$ Pauli matrix algebra but encoded in the structure of \emph{local conserved charges}. These models have finite residual entropy density in the ground state with a simple but non-trivial degeneracy counting and concomitant quantum spin liquidity as proved in [arXiv:2407.06236]. The spin liquidity relies on a geometrically site-interlinked character of the local conserved $\mathbb{Z}_2$ charges that is rather natural in presence of an {\guillemotleft}anticommuting{\guillemotright} structure. One may contrast this with say the bond-interlinked character of the local conserved $\mathbb{Z}_2$ charges on the hexagonal plaquettes of the Kitaev honeycomb spin-$\frac{1}{2}$ model which leads to a mutually commuting local algebra. In this work, we elucidate the differences of this kind of quantum spin liquidity in relation to many-body topological order found in some gapped quantum spin liquids whose canonical example is the Kitaev toric code. The toric code belongs to the more general class of Levin-Wen or string net constructions that possess mutually commuting algebras for the local conserved charges. We will make several exact statements on the kinds of many-body order that can be present within the class of {\guillemotleft}anticommuting{\guillemotright} $\mathbb{Z}_2$ quantum spin liquids co-existing with extensive residual ground state entropy. We will also point out a mutually commuting algebra with local support that are naturally expressed as multi-linear Majorana forms in the Kitaev representation of these quantum spin liquids.