分形子、磁通子和自由子的新模型
A New Model for Fractons, Fluxons, and Freeons
摘要：
我们提出了一个晶格自旋模型的立方晶格，共享的三维复曲面代码和X-立方体分形模型的许多属性。该模型由Z_3自由度的链接，有顶点，立方体，和plaquette条款。作为一种稳定子代码，基态被精确求解。在只有顶点项和立方项的情况下，我们证明了该体系的基态简并度为3^（L3+3L-1），其中L是立方晶格的线性维数.除了分形子之外，还有自由顶点激发，我们称之为自由子。随着元胞体项的加入，GSD大大减少到3^3，包括分形子、通量子和自由子激发，其中只有自由子被解除限制。如果只存在顶点项（A_v）和立方体项（B_c），则称之为AB模型;如果还包括元格项（C_p），则称之为ABC模型。由顶点项和元胞项组成的AC模型是Z_3三维复曲面程序。AB模型的广义GSD源于连接不同基态的局域和非局域逻辑算子的存在。后面的操作符与Z_3 X-cube模型的逻辑操作符相同。分形子激发是不动的，并伴随着通量子的产生-具有非零通量的斑块。在ABC模型中，这种通量子的产生会消耗能量，最终会限制分形子。与过去的分形子模型不同，顶点激发可以自由地向任何方向移动，并在通过通量子或分形子簇时获得非平凡的统计相位。
英文摘要：
We propose a lattice spin model on a cubic lattice that shares many of the properties of the 3D toric code and the X-cube fracton model. The model, made of Z_3 degrees of freedom at the links, has the vertex, the cube, and the plaquette terms. Being a stabilizer code the ground states are exactly solved. With only the vertex and the cube terms present, we show that the ground state degeneracy (GSD) is 3^(L3+3L-1) where L is the linear dimension of the cubic lattice. In addition to fractons, there are free vertex excitations we call the freeons. With the addition of the plaquette terms, GSD is vastly reduced to 3^3, with fracton, fluxon, and freeon excitations, among which only the freeons are deconfined. The model is called the AB model if only the vertex (A_v) and the cube (B_c) terms are present, and the ABC model if in addition the plaquette terms (C_p) are included. The AC model consisting of vertex and plaquette terms is the Z_3 3D toric code. The extensive GSD of the AB model derives from the existence of both local and non-local logical operators that connect different ground states. The latter operators are identical to the logical operators of the Z_3 X-cube model. Fracton excitations are immobile and accompanied by the creation of fluxons - plaquettes having nonzero flux. In the ABC model, such fluxon creation costs energy and ends up confining the fractons. Unlike past models of fractons, vertex excitations are free to move in any direction and pick up a non-trivial statistical phase when passing through a fluxon or a fracton cluster.
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