一个可能存在Chern-Simons相的磁模型
A magnetic model with a possible Chern-Simons phase
摘要：
描述了$2-$维自旋$={1/2}$粒子量子力学系统的一个基本的定域哈密顿量族$H_{\c ,\ell}, \ell = 1,2,3, ldots$。在环面上，基态空间$G_{\circ,\ell}$是$(\log)$广泛简并的，但在$ł$微扰下应该坍缩为一个具有完整数学描述的任意子系统：在$q= e^{2 πi/\ell +2}$的$SO(3)-$ Chern-Simons模函子的量子二重，我们称之为$DE \ell$。汉密尔顿$H_{\circ,\ell}$定义了\underline{quantum} \underline{loop} \underline{gas}。我们认为，$\ell = 1$和2，$G_{\circ,\ell}$是不稳定的，真正发生的$G_{ε, \ell} \cong DE\ell$的崩溃扰动。对于$\ell \geq 3$，$G_{\circ,\ell}$是稳定的，在这种情况下，查找$G_{ε,\ell} \cong DE \ell$必须需要$ε> ε_\ell > 0$、有限系统尺寸的帮助、表面粗糙化（见第3节）或其他一些技巧，因此最初使用引号$ł\quad$“。一个假设的相图包括在介绍。
英文摘要：
An elementary family of local Hamiltonians $H_{\c ,\ell}, \ell = 1,2,3, ldots$, is described for a $2-$dimensional quantum mechanical system of spin $={1/2}$ particles. On the torus, the ground state space $G_{\circ,\ell}$ is $(\log)$ extensively degenerate but should collapse under $ł$perturbation" to an anyonic system with a complete mathematical description: the quantum double of the $SO(3)-$Chern-Simons modular functor at $q= e^{2 πi/\ell +2}$ which we call $DE \ell$. The Hamiltonian $H_{\circ,\ell}$ defines a \underline{quantum} \underline{loop}\underline{gas}. We argue that for $\ell = 1$ and 2, $G_{\circ,\ell}$ is unstable and the collapse to $G_{ε, \ell} \cong DE\ell$ can occur truly by perturbation. For $\ell \geq 3$, $G_{\circ,\ell}$ is stable and in this case finding $G_{ε,\ell} \cong DE \ell$ must require either $ε> ε_\ell > 0$, help from finite system size, surface roughening (see section 3), or some other trick, hence the initial use of quotes $ł\quad$". A hypothetical phase diagram is included in the introduction.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
