从双Chern-Simons-Maxwell格点规范理论到Toric码的推广
From Doubled Chern-Simons-Maxwell Lattice Gauge Theory to Extensions of the Toric Code
摘要：
我们正则化紧凑和非紧凑的阿贝尔陈-西蒙斯-麦克斯韦理论的空间晶格使用的哈密顿公式。本文考虑格点上规范场的二重化理论及其对偶格点。该理论的希尔伯特空间是局部希尔伯特空间的乘积，每个局部希尔伯特空间都与一个链环和相应的对偶链环相关联。与链路对相关联的两个电场算子不对易。在规范群$\mathbb{R}$的非紧情况下，每个局部希尔伯特空间都类似于一个带电“粒子”在链对群空间$\mathbb{R}^2$中在恒定的“磁”背景场中运动。在紧凑的情况下，链路对群空间是由量子化的“磁”通量的$k$单元穿过的环面$U(1)^2$，其中$k$是陈-西蒙斯理论的水平。环面$U(1)^2$的完整性产生了两个自伴扩展参数，它们形成了两个非动力学背景格点规范场，明确地打破了从$U(1)$到$\mathbb{Z}(k)$的明显规范对称性。一个链接对的局部希尔伯特空间然后分解成一个磁平移群的表示。在大“光子”质量的纯陈-西蒙斯极限中，这导致了Kitaev复曲面码的$\mathbb{Z}(k)$对称变体，由两个非动力学背景格点规范场自伴扩展。原始晶格和对偶晶格上的电荷相互服从任意统计，统计角为$\frac{2 π}{k}$。由自伴延拓参数引起的非阿贝尔$U(k)$ Berry规范场在量子信息处理中可能是有趣的。
英文摘要：
We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group $\mathbb{R}$, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space $\mathbb{R}^2$ in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus $U(1)^2$ threaded by $k$ units of quantized "magnetic" flux, with $k$ being the level of the Chern-Simons theory. The holonomies of the torus $U(1)^2$ give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from $U(1)$ to $\mathbb{Z}(k)$. The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a $\mathbb{Z}(k)$-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle $\frac{2 π}{k}$. Non-Abelian $U(k)$ Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.
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