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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.

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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.

  1. There's the toy model of a perfectly clean, uniform, 1D wire with quantized conductance.
  2. 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.

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  • $\begingroup$ Thank you for the reply. I agree with what what you say in that with systems with weak interactions can be roughly said to be Fermi gases. However, if we reduce electron density then interactions become much more important. And yet when the reasons behind various novel phenomena are discussed, such as fractional plateaus, the filling mechanisms of the quasi 1d Fermi gas bands (ie bands due to confinement in the 2nd dimension) are used. Why, when clearly in a non Fermi gas regime, do we still discuss in terms of these bands? $\endgroup$ Oct 18 '20 at 8:46
  • $\begingroup$ I've added an addendum to my answer. $\endgroup$
    – lnmaurer
    Oct 18 '20 at 21:12

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