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 $\mathbb{Z}[t^±]$ 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 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 $\mathbb{Z}_{p^k} [t^±]$, 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.