This paper proposes concrete quantum-dot setups that would let experimentalists realize and tune two one-dimensional quantum models whose critical properties change smoothly along a line. The authors show how to build the quantum Ashkin–Teller model and the XYZ or “eight-vertex” model using currently developed quantum-dot Kitaev-chain hardware. One design uses microwave-like resonator modes to mediate interactions. A second design uses direct Coulomb (electrostatic) coupling and can reach stronger interaction strengths.

At a technical level the authors start from two transverse-field Ising chains and map them to two Kitaev chains. Kitaev chains are one-dimensional chains of spinless fermions with superconducting pairing; they are convenient to implement with spin-polarized quantum dots coupled through hybrid superconducting links. In the resonator-based proposal two kinds of bosonic modes are attached to each dot: a site mode that shifts the local chemical potential, and a link mode that modulates hopping and pairing between neighboring dots. In the limit where the resonator modes are much faster than the electronic dynamics (the “anti-adiabatic” limit) the resonators can be removed perturbatively. This produces effective four-fermion (four-Majorana) terms that realize the Ashkin–Teller interaction. The effective interaction strengths depend on the resonator couplings and frequencies, so changing those controls the critical behavior.

For the XYZ/eight-vertex line the authors outline a Coulomb-based implementation where direct electrostatic interactions between quantum dots generate the required couplings. The two critical lines — Ashkin–Teller and eight-vertex — are related by a non-local duality. That duality means physical order in one description appears as a topological, string-like order in the other. The duality also explains why the two models show different patterns of continuously varying exponents.