Symplectic Deformations, Non Commutative Scalar Fields and Fractional Quantum Hall Effect

TL;DR

This paper explores how symplectic structure deformations lead to non-commutative quantum field theories, with applications to fractional quantum Hall states, revealing new effects on edge excitations and dispersion relations.

Contribution

It introduces a novel approach linking symplectic deformations to non-commutative fields and applies it to fractional quantum Hall effect, deriving generalized filling factors and velocity shifts.

Findings

Derived a generalized fractional filling factor matching Jain states

Showed noncommutativity induces velocity shifts in edge modes

Obtained a nonlinear dispersion relation consistent with recent studies

Abstract

We clearly show that the symplectic structures deformations lead, upon quantization, to quantum theories of non commutative fields. Two variants of deformations are considered. The quantization is performed and the modes expansions of the quantum fields are derived. The Hamiltonians are given and the degeneracies lifting induced by the deformation is also discussed. As illustration, we consider the noncommutative chiral boson fields in the context of fractional quantum Hall effect. A generalized fractional filling factor is derived and shown to reproduce the Jain Hall states. We also show that the coupling of left and right edge excitations of a quantum Hall sample, gives rise a noncommutative chiral boson theory. The coupling or the non-commutativity induces a shift of the chiral components velocities. A non linear dispersion relation is obtained corroborating some recent analytical and numerical analysis.