Self-Dual $\nu=1$ Bosonic Quantum Hall State in Mixed Dimensional QED

TL;DR

This paper investigates a self-dual bosonic quantum Hall state in a mixed-dimensional QED setup, deriving exact relations between transport coefficients at the self-dual point and connecting bosonic and fermionic descriptions through dualities.

Contribution

It establishes exact transport relations at the self-dual point and demonstrates their consistency across bosonic and fermionic dual descriptions, highlighting the role of dualities and cancellations.

Findings

Exact relations between transport coefficients at the self-dual point.

Mapping of bosonic quantum Hall state to a fermionic Fermi sea.

Success of RPA in reproducing exact relations due to parity term cancellation.

Abstract

We consider a (2+1) dimensional Wilson-Fisher boson coupled to a (3+1) dimensional U(1) gauge field. This theory possesses a strong-weak duality in terms of the coupling constant e, and is self-dual at a particular value of $e$. We derive exact relations between transport coefficients for a $\nu=1$ quantum Hall state at the self-dual point. Using boson-fermion duality, we map the $\nu=1$ bosonic quantum Hall state to a Fermi sea of the dual fermion, and observed that the exact relationships between transport coefficients at the bosonic self-dual point are reproduced by a simple random phase approximation, coupled with a Drude formula, in the fermionic theory. We explain this success of the RPA by pointing out a cancellation of a parity-breaking term in the fermion theory which occurs only at the self-dual point, resulting in the fermion self-dual theory explored previously. In addition, we argue n that the equivalence of self-dual structure can be understood in terms of electromagnetic duality or modular invariance, and these features are not inherited by the non-relativistic cousins of the present model.