Transition from a one-dimensional to a quasi-one-dimensional state in interacting quantum wires

At low density, all electrons in a quantum wire occupy the lowest state of transverse quantization, and it is natural to view the system as one-dimensional. As the density is increased, the electrons start to populate the second subband, resulting in a transition to a quasi-one-dimensional state. I will discuss this transition in the presence of electron-electron interactions in a model that neglects electron spins. Clearly, in the non-interacting case the transition is accompanied by the emergence of a second gapless excitation mode. On the other hand, at very strong interactions, the one-dimensional electrons form a Wigner crystal, and the transition corresponds to it splitting into two chains. Unlike the non-interacting electrons, this two-row (zigzag) crystal still has only one acoustic excitation branch. This raises the question of how the nature of the transition to a quasi-one-dimensional state changes with interaction strength. We can show that in the vicinity of the transition already arbitrarily weak interactions open a gap in the second excitation mode. We then argue that only one gapless mode exists near the transition at any interaction strength.