Renormalization group analysis on emergence of higher rank symmetry and higher moment conservation

Higher rank symmetry and higher moment conservation have been drawn considerable attention from, e.g., subdiffusive transport to fracton topological order. In this paper, we perform a one-loop renormalization group (RG) analysis and show how these phenomena emerge at low energies. We consider a $d$-dimensional model of interacting bosons of d components. At higher-rank-symmetric points with conserved angular moments, the $a$-th bosons have kinetic energy only along the $x^a$ direction. Therefore, the symmetric points look highly anisotropic and fine-tuned. By studying RG in a wide vicinity of the symmetric points, we find that symmetry-disallowed kinetic terms tend to be irrelevant within the perturbative regime, which potentially leads to emergent higher-rank symmetry and higher-moment conservation at the deep infrared limit. While non-perturbative analysis is called for in the future, by regarding higher-rank symmetry as an emergent phenomenon, the RG analysis presented in this paper holds alternative promise for realizing higher-rank symmetry and higher-moment conservation in experimentally achievable systems.