Variational Method for Interacting Surfaces with Higher-Form Global Symmetries

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 $\hat{\phi}[C_p^{}]$ charged under a $p$-form global symmetry. Applying the variational principle, we derive a functional Schr\"{o}dinger 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 $U(\psi^*\psi)$ 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.