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Fractal symmetries: Ungauging the cubic code

Dominic J. Williamson

TL;DR

This work develops a general gauging duality for submanifold and fractal symmetries in Pauli Hamiltonians, connecting pre- and post-gauged physics and enabling the construction of short-range entangled phases with fractal-like symmetries. By leveraging Haah's polynomial formalism, it derives a comprehensive gauging framework, analyzes the resulting gauge structure, and relates CSS stabilizer codes to their gauged counterparts. The authors demonstrate concrete dual pairs, including toric code–Ising and cubic code–fractal Ising models, and present self-dual cluster models that host fractal symmetries. These results broaden the toolkit for exploring fracton orders and fractal SET phases, with potential implications for topological quantum computation and transversal gate design.

Abstract

Gauging is a ubiquitous tool in many-body physics. It allows one to construct highly entangled topological phases of matter from relatively simple phases and to relate certain characteristics of the two. Here we develop a gauging procedure for general submanifold symmetries of Pauli Hamiltonians, including symmetries of fractal type. We show a relation between the pre- and post-gauging models and use this to construct short-range entangled phases with fractal-like symmetries, one of which is mapped to the cubic code by the gauging.

Fractal symmetries: Ungauging the cubic code

TL;DR

This work develops a general gauging duality for submanifold and fractal symmetries in Pauli Hamiltonians, connecting pre- and post-gauged physics and enabling the construction of short-range entangled phases with fractal-like symmetries. By leveraging Haah's polynomial formalism, it derives a comprehensive gauging framework, analyzes the resulting gauge structure, and relates CSS stabilizer codes to their gauged counterparts. The authors demonstrate concrete dual pairs, including toric code–Ising and cubic code–fractal Ising models, and present self-dual cluster models that host fractal symmetries. These results broaden the toolkit for exploring fracton orders and fractal SET phases, with potential implications for topological quantum computation and transversal gate design.

Abstract

Gauging is a ubiquitous tool in many-body physics. It allows one to construct highly entangled topological phases of matter from relatively simple phases and to relate certain characteristics of the two. Here we develop a gauging procedure for general submanifold symmetries of Pauli Hamiltonians, including symmetries of fractal type. We show a relation between the pre- and post-gauging models and use this to construct short-range entangled phases with fractal-like symmetries, one of which is mapped to the cubic code by the gauging.

Paper Structure

This paper contains 14 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Gauging
  4. Hamiltonian construction
  5. Gauging procedure
  6. Basic properties of the gauging procedure
  7. Properties of the gauging complex
  8. A construction from CSS stabilizer Hamiltonians
  9. Cluster state construction & gauging
  10. Fractal Symmetries
  11. 2D toric code - Ising model
  12. 3D cubic code - fractal symmetry Ising model
  13. Self dual cluster models
  14. Conclusions

Key Result

Proposition 1

The operator gauging map is invertible for symmetric operators $O$ in the following sense $\emph{$\text{Tr}$}_{({\underline{\mathbf{j}}},t)\in\Gamma} \left(\mathcal{G}[O] \cdot \bigotimes_{({\underline{\mathbf{j}}},t)\in\Gamma} \ket{0}\bra{0} \right)=O$.

Theorems & Definitions (12)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • ...and 2 more