# Effective hamiltonians for anyons on graphs via self-adjoint extensions of the Landau operator

Lead Research Organisation:
University of Bristol

Department Name: Mathematics

### Abstract

Anyons are (quasi)particles that obey so-called fractional quantum statistics - their statistical properties are neither bosonic nor fermionic. Such particles are known to exist in two- dimensional systems (2D lattices or thin metallic strips) and one-dimensional systems (quantum wires). Much less is known about the behaviour of anyons on networks formed from quantum wires, i.e. on quantum graphs. Formally, we consider a graph as a one-dimensional CW-complex. We form its configuration space, by considering the space of all un-ordered tuples of length n that consist of distinct point

Sn is the permutation group. Graph braid group on n strands is defined as the fundamental group.

Similarly, one can consider a configuration space of a topological space. Some important special cases are

R3 - in three-dimensional space there are only bosons or fermions (sometimes in disguise). The braid group Brn(R3) is simply the permutation group, Sn.

R2 - this setting leads to exotic statistics. The nontrivial topology of the configuration space of R2 supports anyons whose fractional statistics is realised in physical models as unitary representations of the planar braid group, Brn(R2). They appear in solid state physics in certain models of superconductors, in fault-tolerant quantum computing and in Chern-Simons theories.

Leinaas and Myrheim (1977) show that the dynamics of anyons can be studied by inserting magnetic fluxes in the "holes" of the configuration space. The corresponding hamiltonian is then found via the minimal coupling principle which says that the momentum of kth particle is given by pk + Ak, where Ak is the local magnetic potential. The corresponding hamiltonian, pk2, is called the Landau operator. To solve the time-independent Schrödinger equation, we first need to find the correct gluing conditions for corresponding to situations where i) a particle is on a junction of the graph and ii) two particles come close to each other. The mathematical theory that tells us how to find such gluing conditions is the theory of self-adjoint extensions of symmetric operators. The aim is to look at specific families of graphs, starting with the simplest T-junction and then proceeding to general star graphs, the lasso graph, wheel graphs, etc. The project is open-ended.

Sn is the permutation group. Graph braid group on n strands is defined as the fundamental group.

Similarly, one can consider a configuration space of a topological space. Some important special cases are

R3 - in three-dimensional space there are only bosons or fermions (sometimes in disguise). The braid group Brn(R3) is simply the permutation group, Sn.

R2 - this setting leads to exotic statistics. The nontrivial topology of the configuration space of R2 supports anyons whose fractional statistics is realised in physical models as unitary representations of the planar braid group, Brn(R2). They appear in solid state physics in certain models of superconductors, in fault-tolerant quantum computing and in Chern-Simons theories.

Leinaas and Myrheim (1977) show that the dynamics of anyons can be studied by inserting magnetic fluxes in the "holes" of the configuration space. The corresponding hamiltonian is then found via the minimal coupling principle which says that the momentum of kth particle is given by pk + Ak, where Ak is the local magnetic potential. The corresponding hamiltonian, pk2, is called the Landau operator. To solve the time-independent Schrödinger equation, we first need to find the correct gluing conditions for corresponding to situations where i) a particle is on a junction of the graph and ii) two particles come close to each other. The mathematical theory that tells us how to find such gluing conditions is the theory of self-adjoint extensions of symmetric operators. The aim is to look at specific families of graphs, starting with the simplest T-junction and then proceeding to general star graphs, the lasso graph, wheel graphs, etc. The project is open-ended.

### Organisations

## People |
## ORCID iD |

Tomasz Maciazek (Primary Supervisor) | |

Oliver Brown (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/W52413X/1 | 30/09/2021 | 29/09/2025 | |||

2765041 | Studentship | EP/W52413X/1 | 30/09/2022 | 29/09/2026 | Oliver Brown |