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In this work, we introduce an anyon condensation web that interconnects abroad class of 2D fracton gauge theories with multipolar conservation laws at amicroscopic level. We find that condensation of anyons triggers the emergenceof additional spatially modulated symmetries, which has the general effect ofincreasing the number of super-selection anyon sectors, as well as the groundstate degeneracy for systems with periodic boundary conditions. As explicitexamples, we start with the rank-2 toric code model and implement various anyoncondensation protocols, resulting in a range of 2D fracton-like theories, eachwith a distinct gauge structure. We also expand the scope of anyon condensationby introducing lattice defects into spatially modulated gauge theories anddemonstrate that these geometric defects can be viewed as effective anyoncondensations along the branch cut. Furthermore, we introduce the MultipartiteEntanglement Mutual Information measure as a diagnostic tool to differentiateamong various distinct multipole conserving phases. A captivating observationis the UV sensitivity of the mutual information sourced from multipartiteentanglement in such spatially modulated gauge theories, which depends on thegeometric cut and the system size, and exhibits periodic oscillations at largedistances.

Anyon Condensation Web and Multipartite Entanglement in 2D Fracton-like Theories