Excitation-detector principle and the algebraic theory of planon-only abelian fracton orders

We study abelian planon-only fracton orders: a class of three-dimensional (3d) gapped quantum phases in which all fractional excitations are abelian particles restricted to move in planes with a common normal direction. In such systems, the mathematical data encoding fusion and statistics comprises a finitely generated module over a Laurent polynomial ring $\mathbb{Z}[t^\pm]$ equipped with a quadratic form giving the topological spin. The principle of remote detectability requires that every planon braids nontrivially with another planon. While this is a necessary condition for physical realizability, we observe - via a simple example - that it is not sufficient. This leads us to propose the $\textit{excitation-detector principle}$ as a general feature of gapped quantum matter. For planon-only fracton orders, the principle requires that every detector - defined as a string of planons extending infinitely in the normal direction - braids nontrivially with some finite excitation. We prove this additional constraint is satisfied precisely by perfect theories of excitations - those whose quadratic form induces a perfect Hermitian form. To justify the excitation-detector principle, we consider the 2d abelian anyon theory obtained by spatially compactifying a planon-only fracton order in a transverse direction. We prove the compactified 2d theory is modular if and only if the original 3d theory is perfect, showing that the excitation-detector principle gives a necessary condition for physical realizability that we conjecture is also sufficient. A key ingredient is a structure theorem for finitely generated torsion-free modules over $\mathbb{Z}_{p^k} [t^\pm]$, where $p$ is prime and $k$ a natural number. Finally, as a first step towards classifying perfect theories of excitations, we prove that every theory of prime fusion order is equivalent to decoupled layers of 2d abelian anyon theories.