Is the Luttinger liquid a limit of the Kitaev chain model?

Question

From what I understand, they are both models of electrons on a nanowire, but the Hamiltonian is different, just in the pairing term. When I look up the connection between them, there's surprisingly little literature on the topic. So I wonder if it's because it's too trivial? Is the Luttinger liquid model a limit of the Kitaev chain model?

Answer 1

The Luttinger liquid Hamiltonian for free electrons

$$ H_{LL}=\sum_{k=k_F-\Lambda}^{k=k_F+\Lambda} v_F k ( c_R^\dagger c_R + c_L^\dagger c_L) $$

describes a low-energy linear approximation of the real Hamiltonian. The point of this model is that for a linear spectrum we can perform bosonization to describe the system as free bosons, even if we add electron-electron interactions. This appears to be a universal low-energy model for 1D systems of interacting electrons.

The Kitaev chain Hamiltonian $$ H_{KC}=-\mu\sum_n c_n^\dagger c_n-t\sum_n (c_{n+1}^\dagger c_n+\textrm{h.c.}) + \Delta\sum_n (c_n c_{n+1}+\textrm{h.c.})\,. $$ describes a simple 1D superconducting lattice with BCS pairing between the electrons. This Hamiltonian can also be "bosonized" to model the system as free Cooper pairs. It does not account for general interactions, but captures a mean-field interaction causing BCS superconductivity.

The interactions that the LL Hamiltonian can account for does not include condensation into a superconducting phase, i.e. the LL Hamiltonian never predicts superconductivity (as far as I know). Therefore it is a poor model of a 1D superconductor. On the other hand, the LL model is an accurate approximation of e.g. the fractional quantum Hall edge state, where strong electron interactions lead to a very specific low-energy behavior.

In the no-pairing $\Delta \to 0$ limit, the KC Hamiltonian can indeed be approximated with a simple LL Hamiltonian. But the purpose of the KC Hamiltonian is to describe superconductivity without further electron interactions, and the purpose of the LL Hamiltonian is to describe strong electron interactions without superconductivity.

Comment question: Why don't we expect the same behavior if we're describing the same physical object (a nanowire)?

A nanowire is a particular realization of a one-dimensional system. Its behavior depends on what material it is made of, as well as the surrounding substrate, fields, etc. In the same way that different materials might be superconducting or ferromagnetic or whatever under different conditions, so too with nanowires of different materials.

Furthermore, the Luttinger Liquid model extends beyond nanowires to other one-dimensional systems such as edge states.

The physical shape of a condensed matter system is not enough to describe all its properties. Different models apply to different materials and conditions. So, depending on the nanowire, we might see very different behaviors under otherwise similar conditions, and so we need different models for different circumstances. As discussed above, the KC model might be useful for a superconducting nanowire, the LL model might be useful for a nanowire with strong electron interaction.