Pauli stabilizer formalism for topological quantum field theories and generalized statistics

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 $p$.

Figures (2)

• Figure 1: Simplicial complexes used in different spacetime dimensions (adapted from Ref. kobayashi2024generalized). In $(1{+}1)$D we use a segment subdivided by an interior vertex; in $(2{+}1)$D a triangle with a central vertex; and in $(3{+}1)$D a tetrahedron subdivided by a central vertex. Each complex is understood as embedded into a larger spatial manifold (indicated by $\cdots$). For convenience in computations, we often compactify the ambient manifold by wrapping the segment into a circle $S^ = \partial \Delta^2$, the triangle into a 2-sphere $S^2 = \partial \Delta^3$, and the tetrahedron into a 3-sphere $S^3 = \partial \Delta^4$. We assume that for any simplex $\Delta$ there exists a finite-depth unitary $U_\Delta$ that creates an invertible excitation on its boundary $\partial\Delta$. For example, the string operator $U_{01}$ creates a particle at vertex $1$ and an anti-particle at vertex $0$. Likewise, on the 2-simplex $\Delta_{012}$, the membrane operator $U_{012}$ creates a loop excitation along the boundary edges $\partial\Delta_{012}$.
• Figure 2: The 24-step process for detecting the statistics of loop excitations with ${\mathbb Z}_2$ fusion in three spatial dimensions and higher (adapted from Ref. kobayashi2024generalized). The loop excitations are supported on the edges of the centered tetrahedron $\partial\langle 01234\rangle$. For ${\mathbb Z}_2$ loops, reversing the orientation does not change the physical configuration, so the initial and final states, related by an orientation flip, represent the same loop.

Theorems & Definitions (22)

• Theorem 2.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Definition 1
• Definition 2
• Definition 3
• Definition 4