The Fate of Crystalline Topological Phenomena in the Continuum

The continuum limit is a widely used theoretical construct for obtaining continuum field-theories of crystalline systems. We study the formulation of a continuum limit for gapped bosonic phases and ask whether their topological properties survive passage to the continuum. Using methods in algebraic topology and category theory, we give a rigorous formulation of the continuum limit and construct a surjective global map relating crystalline topological phases across all finite point-group symmetries to continuum invertible topological phases. Consequently, we find that some lattice phases admit no continuum limit, while distinct lattice phases that do admit a continuum limit can share the same continuum image, hence implying that some lattice topological data can collapse. Conversely, we find that every continuum invertible phase admits a faithful crystalline realization. In addition to the general framework, we apply it to several examples, including studies of a 2D rotation-symmetric phase, higher-order topological phases, and the mixed spin-lattice anomaly of the deconfined quantum critical point between an antiferromagnet and valence bond solid.