A Family of Non-commutative geometries

TL;DR

This paper demonstrates that the non-commutative geometry in quantum Hall systems can be modified through self-adjoint extensions of the Hamiltonian, resulting in a family of geometries that may relate to fractional quantum Hall states.

Contribution

It introduces a family of non-commutative geometries parameterized by self-adjoint extension parameters in quantum Hall systems, providing a new way to understand geometric variations and fractional states.

Findings

Emergent non-commutative geometries form a one-parameter family.

Self-adjoint extension parameters influence the geometry and filling fractions.

Potential to model fractional quantum Hall effect through parameter tuning.

Abstract

It is shown that the non-commutativity in quantum Hall system may get modified. The self-adjoint extension of the corresponding Hamiltonian leads to a family of non-commutative geometries labeled by the self-adjoint extension parameters. We explicitly perform an exact calculation using a singular interaction and show that, when projected to a certain Landau level, the emergent non-commutative geometries of the projected coordinates belong to a one parameter family. There is a possibility of obtaining the filling fraction of fractional quantum Hall effect by suitably choosing the value of the self-adjoint extension parameter.