Topological Defect Network Representations of Fracton Stabilizer Codes

Abstract

A topological defect network (TDN) is formed by a network of topological defects embedded within a topological quantum field theory (TQFT). TDNs were introduced recently for the purpose of describing fracton topological phases of matter using the framework of defect TQFT. Their effectiveness has been demonstrated through numerous examples, yet a systematic construction was lacking. Here we solve this problem by formulating a method to construct TDNs for a wide range of lattice Hamiltonians. Our method takes a lattice Hamiltonian as input, applies an ungauging procedure, then creates a refined lattice within each unit cell, followed by regauging the system to produce a TDN as output. For topological Calderbank-Shor-Steane (CSS) Pauli stabilizer models, this procedure is guaranteed to produce a phase-equivalent TDN. This provides TDN representations of canonical fracton models for which no such construction was previously known, including Haah’s cubic code and Yoshida’s infinite family of fractal spin liquid models. We demonstrate the applicability of our method beyond CSS stabilizer models by constructing TDNs for non-CSS models including Chamon’s model and the semionic X-cube model.