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We consider electrons in a quantum wire interacting via a long-range Coulombpotential screened by a nearby gate. We focus on the quantum phase transitionfrom a strictly one-dimensional to a quasi-one-dimensional electron liquid,that is controlled by the dimensionless parameter $n x_0$, where $n$ is theelectron density and $x_0$ is the characteristic length of the transverseconfining potential. If this transition occurs in the low-density limit, it canbe understood as the deformation of the one-dimensional Wigner crystal to azigzag arrangement of the electrons described by an Ising order parameter. Thecritical properties are governed by the charge degrees of freedom and the spinsector remains essentially decoupled. At large densities, on the other hand,the transition is triggered by the filling of a second one-dimensional subbandof transverse quantization. Electrons at the bottom of the second subbandinteract strongly due to the diverging density of states and becomeimpenetrable. We argue that this stabilizes the electron liquid as itsuppresses pair-tunneling processes between the subbands that would otherwiselead to an instability. However, the impenetrable electrons in the second bandare screened by the excitations of the first subband, so that the transition isidentified as a Lifshitz transition of impenetrable polarons. We discuss theresulting phase diagram as a function of $n x_0$.

Quantum phase transition in quantum wires controlled by an external gate