$mathbb{Z}_3$ parafermion in the double charge-Kondo model

Quantum impurity models with frustrated Kondo interactions can support quantum critical points with fractionalized excitations.

Recent experiments [arXiv:2108.12691] on a circuit containing two coupled metal-semiconductor islands exhibit transport signatures of such a critical point.

Here we show using bosonization that the double charge-Kondo model describing the device can be mapped in the Toulouse limit to a sine-Gordon model.

Its Bethe-ansatz solution shows that a $mathbb{Z}_3$ parafermion emerges at the critical point, characterized by a fractional $ frac{1}{2}ln(3)$ residual entropy, and scattering fractional charges $e/3$. We also present full numerical renormalization group calculations for the model and show that the predicted behavior of conductance is consistent with experimental results.