AI 中文总结
研究任意维度任意定向细胞复形上\({\mathbb Z}_k\) \(p\)形式规范场的哈密顿框架，以\(p = 2\)，\(k = 2\)为例，通过多种方法求解相关模型，经实验得出如通量面积损失比例等结果，置于高形式哈密顿量和编码框架。
AI 中文摘要
我为任意维度的任意定向细胞复形上的\({\mathbb Z}_k\) \(p\)形式规范场开发了一个哈密顿框架。规范量子比特由\(p\)元胞定义，带电边界量子比特由\((p - 1)\)元胞定义，高斯定律生成器由边界映射\(\partial_p\)定义，磁校验由\(\partial_{p + 1}\)定义。相同的细胞结构产生局部着装威尔逊算子，在\(k = 2\)时产生与量子纠错相关的卡尔德班克 - 肖尔 - 斯特恩校验复形。然后专门研究\(p = 2\)，\(k = 2\)的情况，此时磁 3 元胞项不存在且一形式高斯定律可精确求解。物理希尔伯特空间由面元电通量变量参数化，而链路配置被重建为演化通量域的动态边界。简化后的哈密顿量是一个伊辛型面元模型，其局部横向场项是边界着装威尔逊算子\(\sigma_p^z\prod_{\ell\in\partial p}\tau_\ell^z\)的物理图像。管帽淬火比较具有相同初始边界环的两个初始通量填充。在\(4×4\)、\(6×4\)和\(5×5\)环面上的精确对角化发现，帽损失了其占用通量面积的\(20\%\) - \(37\%\)，而管几乎保持固定。有限尺寸缩放确定了张力与密度比的动态交叉点，接近\((m/\varepsilon_E)_c\simeq1.89\)。未简化的面元加链路编码提供局部高斯定律校验和直接数字实现，而简化的仅面元哈密顿量提供精确基准。结果将特定的顶形式放电和宇宙学常数中和计算置于一般的高形式哈密顿量和编码框架内。
英文摘要
I develop a Hamiltonian framework for ${\mathbb Z}_k$ $p$-form gauge fields on arbitrary oriented cell complexes in arbitrary dimensions. Gauge qudits are defined by $p$-cells, charged boundary qudits by $(p-1)$-cells, Gauss-law generators by boundary map $\partial_p$, and magnetic checks by $\partial_{p+1}$. The same cellular structure produces local dressed Wilson operators, and at $k=2$ a Calderbank-Shor-Steane check complex relevant to quantum error correction. I then specialize to $p=2, k=2$, where the magnetic 3-cell term is absent and the one-form Gauss-law can be solved exactly. The physical Hilbert space is parameterized by plaquette electric-flux variables, while the link configuration is reconstructed as the dynamical boundary of the evolving flux domains. The reduced Hamiltonian is an Ising-type plaquette model, where its local transverse-field term is the physical image of the boundary-dressed Wilson operator $σ_p^z\prod_{\ell\in\partial p}τ_\ell^z$. A tube-cap quench compares two initial flux fillings with the same initial boundary loops. Exact diagonalization on $4\times4$, $6\times4$, and $5\times5$ tori finds that the cap loses $20$-$37\%$ of its occupied-flux area, while the tube remains nearly pinned. A finite-size scaling locates a dynamical crossover of tension-to-density ratio near $(m/\varepsilon_E)_c\simeq1.89$. The unreduced plaquette-plus-link encoding provides local Gauss-law checks and a direct digital implementation, while the reduced plaquette-only Hamiltonian supplies the exact benchmark. The result places the specific top-form discharge and the cosmological constant neutralization calculation inside a general higher-form Hamiltonian and coding framework.
Comments33 pages; 5 figures
