Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exchange and exclusion in the non-abelian anyon gas

Douglas Lundholm, Viktor Qvarfordt

Abstract

We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on $N$ strands. Allowing for arbitrary rank (dependent on $N$) and non-abelian representations, and letting $N \to \infty$, this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincaré and Hardy inequalities) and a local exclusion principle (also implying a Lieb-Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.

Exchange and exclusion in the non-abelian anyon gas

Abstract

We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on strands. Allowing for arbitrary rank (dependent on ) and non-abelian representations, and letting , this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincaré and Hardy inequalities) and a local exclusion principle (also implying a Lieb-Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.

Paper Structure

This paper contains 49 sections, 41 theorems, 408 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Anyons --- abelian vs. non-abelian
  3. Some mathematically rigorous results for abelian anyons
  4. Main new results
  5. Algebraic anyon models
  6. Fusion
  7. Fusion rules
  8. Fusion diagrams and spaces
  9. F-symbols: Associativity of fusion
  10. Braiding
  11. R-symbols: Commutativity of fusion
  12. B-symbols: Braiding of standard fusion states
  13. Pentagon and hexagon equations
  14. Vacuum, bosons and fermions
  15. Abelian anyons
  16. ...and 34 more sections

Key Result

Theorem 1.1

For any sequence of abelian $N$-anyon models $\rho_N\colon B_N \to \textup{U}(1)$, $\rho_N(\sigma_j) = e^{i\alpha\pi}$, with statistics parameters $\alpha = \alpha(N) \in [0,1]$ and n-anyon exchange parameters $\alpha_n = \alpha_n(N) \in [0,1]$, we have the following uniform bounds for the ground-st where and Further, there exist universal constants $C_2 \ge C_1 > 0$ such that, if $\alpha$ is in

Figures (10)

  • Figure 1.1: One- respectively two-particle loops of abelian anyons in the plane, with their respective phases and braid diagrams obtained by projecting and ordering the particles on the horizontal axis and with time running upwards on the vertical axis. In each loop $p$ other particles are enclosed, and the total obtained phase is $\alpha\pi$ times the number of simple braids appearing in the diagram.
  • Figure 1.2: Left: The popcorn function $\alpha \mapsto \alpha_*$. Right: A numerical lower bound to the function $\alpha \mapsto f(j_{\alpha_*}'^2)$ from LarLun-16. The continuous orange curve indicates $\alpha \mapsto c(\alpha_2)$ for comparison (see text).
  • Figure 1.3: Anyon models.
  • Figure 2.1: Some typical fusion/splitting/braiding diagrams.
  • Figure 2.2: Diagram corresponding to the pentagon equation \ref{['eq:pentagon']}.
  • ...and 5 more figures

Theorems & Definitions (94)

  • Theorem 1.1: Uniform bounds for the ideal abelian anyon gas LunSei-17
  • Theorem 1.2: Uniform bounds for the ideal non-abelian anyon gas
  • Definition 2.1: F operator
  • Lemma 2.2
  • proof
  • Definition 2.3: R operator
  • Definition 2.4: B operator
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 84 more