(\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 \(\tfrac{1}{2}\ln(3)\) 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.