First- and Second-Order Phase Transition in Parallel Triple Quantum Dots: The Roles of Symmetric and Asymmetric Hopping

Abstract

By means of the numerical renormalization group method, I study the quantum phase transition (QPT) and the electronic transport in parallel triple quantum dot system with symmetric and/or asymmetric hopping. For symmetric hopping $$t_{1} = t_{2}$$ and zero magnetic field $$B = 0$$ , I find a first order transition between spin quadruplet and doublet as $$t_{1}$$ ( $$t_{2}$$ ) increases. With increasing $$B$$ , a second order QPT between $$S_{z} = 1/2$$ of the doublet and $$S_{z} = 3/2$$ of the quadruplet is observed. For asymmetric hopping $$t_{1} \ne t_{2}$$ , the QPT depends closely on the other hopping. For fixed $$t_{1} < \varGamma $$ , where $$\varGamma $$ is the hybridization function between the dots and the leads, a first order transition is observed as $$t_{2}$$ increases, while for $$t_{1} \ge \varGamma $$ , a crossover occurs. In the presence of $$B$$ , the transition between $$S_{z} = 1/2$$ and $$S_{z} = 3/2$$ is a first order QPT for $$t_{1} < \varGamma $$ , while a second order for $$t_{1} \ge \varGamma $$ .