Conductance of an interacting quasi one dimensional wire using the method for a 1D Fermi gas? Assuming the electrons are non interacting and spin degenerate, the conductance of a quasi one dimensional quantum wire is quantised in units of $2\frac{e^2}{h}$. For small voltages, we simply count how many bands have their bottoms below the chemical potential and multiply this by $2\frac{e^2}{h}$. This is due to the electron velocity and 1D density of states cancelling for all energies, when we do the integral over occupied energies for each occupied band.
Now we add electron-electron interactions. We don't have a fermi gas or even liquid now as we are in 1D. I naively thought that as the fermi gas/liquid no longer applies we couldn't rely on the above picture. But it appears that we roughly can. My question is this essentially; why?
I can sort of appreciate that it's a contact resistance so will be determined by the fermi liquid leads. Our wire itself is, as before, meant to be perfectly conducting, so the resistance should be determined by what goes on in the leads. (I'm still a little sketchy on this.) This aside, I still can't get my head around why we still talk about 1D bands and whether they have started to be filled. We don't have a fermi gas anymore and more importantly we don't have a fermi liquid. As we are in 1D we have something that resembles a Luttinger Liquid. So why do we still use 1D bands to explain the conductance?
As a final remark, there are many cases in the literature, for example the 0.7 structure, where the fact the electrons interact does have some bearing. The possible explanations still talk about the 1D bands and mechanisms describing how they're filled. But we don't have a fermi gas or fermi liquid; why talk about 1D bands? Furthermore, the conductance is now not determined just by the fermi liquid leads, the interactions within the wire must now be taken into account.
 A: Let's flip the question: if, in theory, we should have a Luttinger liquid in quasi-1D wires, then why is that so hard to observe in practice? (The Fermi-gas model works very well for many real-world systems.) I think that you are right that, in theory, you don't have a Fermi gas or liquid and the simple theory of 1D transport doesn't apply. However, in practice, electron-electron interactions are often so weak that they're ignored in transport problems (outside of calculating the band structure). You'd probably need a system that's quite long, pure, and cold before electron-electron interactions matter. If the interaction is too weak to matter, you're basically back to a Fermi gas.
ADDENDUM:
Let's separate two things.

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*There's the toy model of a perfectly clean, uniform, 1D wire with quantized conductance.

*There are actual nanostructures like quantum point contacts (and their associated 0.7 anomaly).

The first is a toy model. Toy models are meant to be simple. There's not much more to say.
The second basically requires some sort of numerical simulation of the transport problem.
Transport problems are often set up as follows: you two have "big" reservoirs (with simple physics) connected to a "small" scattering region where the interesting stuff happens. If you have a Fermi gas in the reservoirs, that means you'll inject single-electron plane waves into the scattering region. That's all that's going on here. In your scattering region, all kinds of things can happen. You can have variations in the potential due to gates or impurities. You can have Hartree-like electron-electron interactions. You can add more complicated interactions. If you include "enough" physics in your model for the scattering region, you'll include any effects of electron-electron interactions.
Sometimes the interactions "matter". Sometimes they don't. You can go thru all this work modeling a QPC with electron-electron interactions and find out that, lo and behold, the conductance is quantized almost exactly like the toy model predicts! (At least for some range of gate voltages, biases, etc.) It turns out that, even when you include electron-electron interactions, you don't get a Luttinger liquid when you attach a short channel between two reservoirs and shoot plane waves thru it.
That said, there are certainly situations where the big Fermi gas/small scattering region don't apply.
A: I would like to add a few points to the answer by @Inmauer (which I agree with).
Quantum point contact
One has to distinguish quantum point contacts and 1D electron gas. Quantum point contact can be any small opening between two conductors, which can be either two-dimensional or three-dimensional Fermi liquids. The quantization arises from the fact that the transversal modes are quantized in the contact area, but the structure is not one-dimensional.
This is also the setting for Landauer-Bütiker formalism, usually employed to derive the conductance quantization for non-interacting electrons. The formalism relies on the scattering matrix for electrons, but makes no any restrictive assumption about the dimensionality.
Nevertheless, if discussing non-interacting electrons, the multidimensional scattering problem problem can be reduced to one-dimensional one along the axis connecting Fermi seas, by projecting onto a suitable set of stationary eigenfunctions in transverse directions. But one-dimensional nature of the problem here is mathematical, not physical.
Quantum wires
Luttinger liquid is a theoretical concept - it is what we get instead of Fermi liquid for interacting electrons in one dimension. To study conductance in such a liquid, it has to be connected to reservoirs, which raises a question of whether it can be still considered one-dimensional. A possible widely studied geometry is a long quantum wire, where electrons can tunnel in and out from nearby gates parallel to the wire, this testing conductance in it. However, if we try to realize something resembling an extended quantum point contact, the Luttinger liquid theory doesn't quite hold.
0.7 anomaly
Anomalous quantization plateau around $0.7G_0=0.7\frac{2e^2}{h}$ is ubiquitously observed in quantum point contacts - whether short or long. Naturally, there have been attempts to tackle the problem from both ends, with perhaps a dozen (or a few dozen theories) published. While many theories successfully explain the appearance and the position of the conductance plateau, 0.7 anomaly exhibits a number of other features that are less easily captures - notably an activation dependence on temperature (i.e., the anomaly becomes more pronounced rather than washed out with higher temperature.)
The most notable short contact theory is that by Meir, Hirose and Wingreen, which suggests Kondo-like physics for the phenomenon, whereas the most influential long-contact theory is that by Matveev, who is also an expert in Luttinger liquids. There have been some controversy in deciding "who is right" with various experiments seeming to favor one or the other theory. Rejec and Meir have carried rigorous density functional modeling to demonstrate the possibility of Kondo physics in narrow constrictions, wheras Matveev's Winger crystal theory seemed to develop in the direction suggesting a cross-over with Kondo in short contacts. The more recent publications seem rather conciliatory, regarding the two approaches as the two limiting cases of the same phenomenon, see, e.g., this experiment.
