变分法用于具有高阶全局对称性的相互作用表面系统
Variational Method for Interacting Surfaces with Higher-Form Global Symmetries
AI总结：
本文提出了一种变分法用于具有高阶全局对称性的相互作用表面系统，推导了功能性薛定谔方程，并展示了低能涨落的拓扑特性。
摘要：
我们开发了一种变分法，用于具有高阶全局对称性的相互作用表面系统。作为传统相互作用玻色子二次量子化哈密顿量的自然扩展，我们明确构建了一个以闭合表面算符$\hatϕ[C_p^{}]$为形式的二次量子化哈密顿量，该算符受$p$-形式全局对称性所带电。应用变分原理，我们推导出一个类似于常规玻色子系统中格罗斯-皮塔耶夫斯基方程的功能性薛定谔方程。在没有外部力的情况下，变分方程允许一个均匀解，该解由微观相互作用势$U(ψ^*ψ)$和化学势唯一确定。这个均匀解描述了一个均匀的玻色子表面气体。利用获得的能量泛函，我们证明当$p$-形式全局对称性为$\mathrm{U}(1)$时，低能涨落包含一个无质量的$p$-形式场$A_p^{}$，而当对称性为离散对称性时，$p$-形式场变得有质量，其低能极限由$\mathrm{BF}$型拓扑场论描述。因此，系统表现出阿贝尔拓扑序，具有任意子表面激发。然而，在存在外部力的情况下，解决功能性方程在一般情况下仍然具有挑战性。我们论证，然而，当外部力分别作用于质心和相对运动时，问题可以归结为解决常规格罗斯-皮塔耶夫斯基方程。此外，我们还给出了拓扑缺陷的解析解，作为常规玻色子系统中涡旋和域壁解的类比。最后，作为具体的微观模型，我们研究了一个$\mathbb{Z}_N^{}$晶格规范理论，并将我们的变分方法应用于该系统。
英文摘要：
We develop a variational method for interacting surface systems with higher-form global symmetries. As a natural extension of the conventional second-quantized Hamiltonian of interacting bosons, we explicitly construct a second-quantized Hamiltonian formulated in terms of a closed surface operator $\hatϕ[C_p^{}]$ charged under a $p$-form global symmetry. Applying the variational principle, we derive a functional Schrödinger equation analogous to the Gross-Pitaevskii equation in conventional bosonic systems. In the absence of external forces, the variational equation admits a uniform solution that is uniquely determined by a microscopic interaction potential $U(ψ^*ψ)$ and the chemical potential. This uniform solution describes a uniform gas of bosonic surfaces. Using the obtained energy functional, we show that low-energy fluctuations contain a gapless $p$-form field $A_p^{}$ when the $p$-form global symmetry is $\mathrm{U}(1)$, whereas the $p$-form field becomes massive for discrete symmetries, whose low-energy limit is described by a $\mathrm{BF}$-type topological field theory. As a consequence, the system exhibits abelian topological order with anyonic surface excitations. In the presence of external forces, however, solving the functional equation in full generality remains challenging. We argue, however, that the problem reduces to solving the conventional Gross-Pitaevskii equation when external forces act separately on the center-of-mass and relative motions. In addition, we present analytic solutions for topological defects as analogs of vortex and domain-wall solutions in conventional bosonic systems. Finally, as a concrete microscopic model, we study a $\mathbb{Z}_N^{}$ lattice gauge theory and apply our variational method to this system.
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