Gate-voltage-driven quantum phase transition at $0.7 (2e^2/h)$ in quantum point contacts

We investigate a quantum phase transition (QPT) in quantum point contacts by analyzing the gate-voltage-dependent quasiparticle energy at the Fermi level at zero temperature. This energy is computed using the local density of states at the site of the localized spin, which is extracted from the replicated gate-voltage-dependent differential conductance shaped by entangled-state tunneling. The QPT occurs between symmetric ($G \geq 0.7 G_0$) and asymmetric ($G < 0.7 G_0$) Kondo coupling states, where $G_0 = 2e^2/h$, and is driven by the migration of a localized spin in response to the side-gate voltage. The asymmetric state exhibits two distinct Kondo temperatures, while the symmetric state has only one. The existence of two Kondo temperatures in the $G < 0.7 G_0$ regime accounts for both the anomalous gate-voltage dependence of the zero-bias anomaly width and the inability to define a Kondo temperature in the $G < 0.7 G_0$ region.