Pauli stabilizer formalism for topological quantum field theories and generalized statistics

TL;DR

This paper develops a stabilizer formalism for topological quantum field theories, constructing lattice models with extended excitations and generalized statistics across multiple dimensions, advancing understanding of topological phases and quantum error correction.

Contribution

It introduces new lattice gauge theories as Pauli stabilizer models for various TQFTs, including fermionic-loop toric codes and higher-form gauge theories, with a framework for analyzing extended excitation statistics.

Findings

Realized all twisted 2-form gauge theories in (4+1)D as stabilizer models.

Discovered fermionic loop statistics with a phase of -1 in (4+1)D.

Constructed new topological orders like fermionic-membrane and volume toric codes.

Abstract

Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the $e^2m^2$-loop in the (4+1)D $\mathbb{Z}_4$ toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of $-1$. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by $H^5(B^2G,U(1))$. Beyond (4+1)D, the fermionic-loop toric codes form a family of $\mathbb{Z}_2$ topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using $\mathbb{Z}_4$ qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of $\mathbb{Z}_2$ topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that $p$-dimensional excitations in $2p+2$ spatial dimensions can support anyonic $p$-brane statistics for only even $p$.