Gate-voltage-driven quantum phase transition at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.7</mml:mn> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>e</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <m

We investigate a quantum phase transition in quantum point contacts by analyzing the gate-voltage-dependent zero-temperature quasiparticle energy at the Fermi level. This energy is computed from the local density of states at the site of the localized spin obtained from replicated gate-voltage-dependent differential conductance line shapes generated by entangled-state tunneling. The transition occurs between states with left-right symmetric Kondo couplings (G⩾0.7G0withG0=2e2/h) and left-right asymmetric Kondo couplings (G<0.7G0) and is triggered by the migration of a localized spin in response to the side-gate voltage. The asymmetric phase hosts two distinct Kondo temperatures, whereas the symmetric phase has one. The coexistence of two Kondo temperatures explains both the anomalous gate-voltage dependence of the zero-bias-anomaly width and the indeterminate Kondo temperature in theG<0.7G0regime.