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

TL;DR

This paper introduces the excitation-detector principle for abelian planon-only fracton orders, linking algebraic structures to physical realizability and providing a classification framework for these quantum phases.

Contribution

It proposes the excitation-detector principle as a necessary condition for physical realizability of planon-only fracton orders and establishes a structure theorem for related algebraic modules.

Findings

The excitation-detector principle is satisfied by perfect theories of excitations.

Compactified 2d theories are modular if and only if the original 3d theory is perfect.

Every prime fusion order theory is equivalent to layered 2d abelian anyon theories.

Abstract

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.