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 $t_{1} \ge \varGamma$.