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Statistics of Abelian topological excitations

Hanyu Xue

TL;DR

The paper develops a self-contained microscopic theory for Abelian topological excitation statistics in arbitrary dimensions, deriving statistics from two basic quantum-mechanical axioms applied to excitation patterns and their realizations. It links the resulting invariants to cohomology via $H^{d+2}(K(G,d-p),\mathbb{R}/\mathbb{Z})$, while providing a computational route (Smith decomposition) to compute torsion statistics on finite lattices. The framework emphasizes locality, localization, and a dual expression view to capture phase data and operator-independence, with connections across dimensions through Eilenberg–MacLane spaces and potential applications to higher-form orders and condensation. It also develops tools like quantum cellular automata and condensation to relate local and global statistics, and outlines clear future directions toward non-Abelian generalizations and triangulation-independence proofs. Overall, the work offers a rigorous, kinematics-level bridge between microscopic lattice models and the cohomological classification of Abelian topological excitations, enabling practical computation and deeper theoretical insight.

Abstract

In this paper, we develop a novel theory that generalizes the concept of anyon statistics to Abelian topological excitations of any dimension. We axiomatize excitations as a selected collection of states and operators satisfying the configuration axiom and the locality axiom, purely based on many-body quantum mechanics. Upon these axioms, we define a rigorous and self-contained theory of statistics using only basic algebra and can be implemented on a computer. While our theory is developed independently, the results align with existing physical theories.

Statistics of Abelian topological excitations

TL;DR

The paper develops a self-contained microscopic theory for Abelian topological excitation statistics in arbitrary dimensions, deriving statistics from two basic quantum-mechanical axioms applied to excitation patterns and their realizations. It links the resulting invariants to cohomology via , while providing a computational route (Smith decomposition) to compute torsion statistics on finite lattices. The framework emphasizes locality, localization, and a dual expression view to capture phase data and operator-independence, with connections across dimensions through Eilenberg–MacLane spaces and potential applications to higher-form orders and condensation. It also develops tools like quantum cellular automata and condensation to relate local and global statistics, and outlines clear future directions toward non-Abelian generalizations and triangulation-independence proofs. Overall, the work offers a rigorous, kinematics-level bridge between microscopic lattice models and the cohomological classification of Abelian topological excitations, enabling practical computation and deeper theoretical insight.

Abstract

In this paper, we develop a novel theory that generalizes the concept of anyon statistics to Abelian topological excitations of any dimension. We axiomatize excitations as a selected collection of states and operators satisfying the configuration axiom and the locality axiom, purely based on many-body quantum mechanics. Upon these axioms, we define a rigorous and self-contained theory of statistics using only basic algebra and can be implemented on a computer. While our theory is developed independently, the results align with existing physical theories.

Paper Structure

This paper contains 26 sections, 32 theorems, 211 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background of Statistics
  3. Framework for Statistics
  4. Excitation patterns and their realizations
  5. Phase data, localization, and statistics
  6. Expressions: a dual point of view
  7. Classification of statistical expressions
  8. Example: hidden permutation symmetry and properties of the T-junction process
  9. Symmetry restoration in the T-junction process
  10. Initial state independence
  11. Finite order of $[e]$
  12. Computational Methods and Conjectures
  13. Computing $T(m)$
  14. Some discussion
  15. Constructing statistical processes
  16. ...and 11 more sections

Key Result

Theorem 3.1

Figures (8)

  • Figure 1: (Eq. (4) in Previous) The T-junction process exchange two anyons at $\overline{1}$ and $\overline{2}$. Although the two anyons are colored differently, they must be identical, so the initial state and the final state are the same.
  • Figure 2: (Fig. 4 of Previous) The 24-step process for detecting the statistics of loops with $G = \mathbb{Z}_2$ fusion in (3+1)D, which is an optimization of the $36$-step process found in FHH21. For $\mathbb{Z}_2$ loops, different orientations correspond to the same configuration state, indicating that the initial and final configurations are reversed and illustrating the loop-flipping process.
  • Figure 3: When $X$ is a triangle, square, or other polygons, computations suggest $T_0(X,G)\simeq H_3(G,{\mathbb Z})$.
  • Figure 4: We have tried several regular cell complex of $S^2$, and computations suggest $T_0(X,G)\simeq H_4(K(G,2),{\mathbb Z})$ and $T_1(X,G)\simeq H_4(G,{\mathbb Z})$. Note that $H_4(K(G,2),{\mathbb Z})$ is the familiar anyon statistics, and the 1-skeleton of the first graph coincide with that of the T-junction process.
  • Figure 5: Non-planar graphs corresponds to particles in 3 dimensions. For $G=\oplus_i{\mathbb Z}_{N_i}$, we have $T=\oplus_i{\mathbb Z}_{(N_i,2)}$: exchange phase can only be $0$ (bosons) or $\pi$ (fermions) and braiding between different particles is trivial.
  • ...and 3 more figures

Theorems & Definitions (100)

  • Remark 2.1
  • Definition 3.1
  • Definition 3.2
  • Remark 3.1
  • Definition 3.3
  • Remark 3.2
  • Remark 3.3
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • ...and 90 more