Geometric Presentations of Braid Groups for Particles on a Graph

Byung Hee An,Tomasz Maciazek

doi:10.1007/s00220-021-04095-x

Byung Hee An, Tomasz Maciazek

Open Access

https://doi.org/10.1007/s00220-021-04095-x

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Abstract

We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for 3-connected planar graphs such a quotient reconstructs the well-known planar braid group. For 2-connected graphs this approach leads to generalisations of the Yang–Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.

Highlights

The study of non-abelian quantum statistics is currently at the forefront of research concerning quantum computers [1], the fractional quantum Hall effect [2] and superconductivity [3]
If the considered particles are constrained to move in R2, this means that such a quantum system features a unitary representation of the braid group
Unitary representations of the planar braid group Bn(R2) give rise to non-abelian anyons relating to the fractional quantum Hall effect, tensor categories utilised in quantum computing [8] and the field-theoretic description of quantum statistics [3,9]

Summary

Introduction

The study of non-abelian quantum statistics is currently at the forefront of research concerning quantum computers [1], the fractional quantum Hall effect [2] and superconductivity [3]. Unitary representations of the planar braid group Bn(R2) give rise to non-abelian anyons relating to the fractional quantum Hall effect, tensor categories utilised in quantum computing [8] and the field-theoretic description of quantum statistics [3,9]. There exists no explicit description of a universal set of generators and relations for graph braid groups in terms of geometric moves of particles on a given graph. Other authors proposed effective discrete hopping hamiltonians for interacting anyons that are defined on discrete configuration spaces of graphs Dn( ) ⊂ Cn( ) [15] For abelian anyons, this led to full classification of abelian quantum statistics on graphs [16] in terms of the first homology group H1(Cn( ), Z). Homology groups of graph configuration spaces as well as presentations of graph braid groups are subjects of independent interest in mathematics, see [19,20,21]

Preliminaries

Presentations of Graph Braid Groups

Y -exchanges

Planar Triconnected Graphs

A Generalisation of the Yang–Baxter Equation