[Paper Review] Topological Order from a Cohomological and Higher Gauge Theory perspective

Ricardo Costa de Almeida, J. P. Ibieta-Jimenez+2 more|arXiv (Cornell University)|Nov 11, 2017

Homotopy and Cohomology in Algebraic Topology3 references12 citations

TL;DR

This paper introduces a cohomological framework based on chain complexes of abelian groups to generalize abelian higher gauge theories, unifying and simplifying existing models of topological order. It establishes that ground state degeneracy (GSD) is fully characterized by the cohomology of the chain complex, providing a closed-form topological invariant and revealing contributions from each dimension to intrinsic topological order.

ABSTRACT

In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called $2$-gauge theories. These have opened the door to interesting new models and new topological phases which are not described by previous schemes of classification. In this paper we show that we can go beyond the $2$-gauge construction when considering chain complexes of abelian groups. Based on elements of homological algebra we are able to greatly simplify already known constructions for abelian theories under a single all encompassing framework. Furthermore, this formalism allows us to systematize the computation of the corresponding topological degeneracies of the ground states and establishes a connection between them and a known cohomology, which conveniently characterizes them with a suitable set of quantum numbers.

Motivation & Objective

• To develop a unified formalism for abelian higher gauge theories that generalizes existing models of topological order.
• To simplify known constructions of topological phases using homological algebra.
• To systematize the computation of ground state degeneracy (GSD) and show it is a topological invariant.
• To establish a direct correspondence between GSD and cohomology classes with quantum numbers.
• To provide a framework applicable to higher-dimensional topological quantum codes and quantum memory models.

Proposed method

• The formalism replaces ordinary gauge groups with chain complexes of abelian groups, generalizing the notion of configuration to maps between such complexes.
• It defines Hamiltonian lattice models using local operators $ A_x^r $ and $ B_y^g $, constructed via characters and dual groups, ensuring consistency with gauge symmetries.
• The ground state subspace is shown to be in one-to-one correspondence with cohomology classes $ H^n(C, G) $, where $ C $ is the chain complex and $ G $ the coefficient group.
• The GSD is computed as the product of the orders of cohomology groups across all dimensions, yielding a closed-form topological invariant.
• The framework uses homological algebra tools, including simplicial homology and cohomology, to derive algebraic relations and projectors.
• It demonstrates that the Hamiltonian commutes with all local operators, ensuring stability and topological invariance of the ground state.

Experimental results

Research questions

• RQ1 How can abelian higher gauge theories be systematically unified under a single cohomological framework?
• RQ2 What is the precise relationship between the ground state degeneracy (GSD) and the cohomology of the underlying chain complex?
• RQ3 How do contributions from different dimensions collectively characterize intrinsic topological order?
• RQ4 Can the GSD be expressed as a closed-form topological invariant in arbitrary dimensions?
• RQ5 How does this formalism generalize known models such as quantum CSS stabilizer codes and topological quantum memories?

Key findings

• The ground state degeneracy (GSD) is given by the product of the orders of cohomology groups $ |H^n(C, G)| $ across all dimensions, forming a closed-form topological invariant.
• The ground state subspace is in one-to-one correspondence with cohomology classes $ H^n(C, G) $, which are characterized by quantum numbers from the coefficient group.
• The formalism generalizes known models such as Kitaev's toric code and higher-dimensional topological quantum codes, subsuming them as special cases.
• The GSD is shown to be independent of the lattice discretization and depends only on the cohomology of the chain complex, confirming its topological nature.
• The local operators $ A_x^r $ and $ B_y^g $ satisfy orthogonality, completeness, and pairwise commutation relations, ensuring consistent quantum mechanical dynamics.
• The Hamiltonian commutes with all local gauge and holonomy operators, confirming the stability and topological protection of the ground state subspace.