What is a Tomonaga-Luttinger Liquid?

I've written in the past (say here and here) about how we think about the electrons in a conventional metals as forming a Fermi Liquid.    (If the electrons didn't interact at all, then colloquially we call the system a Fermi gas.  The word "liquid" is shorthand for saying that the interactions between the particles that make up the liquid are important.  You can picture a classical liquid as a bunch of molecules bopping around, experiencing some kind of short-ranged repulsion so that they can't overlap, but with some attraction that favors the molecules to be bumping up against each other - the typical interparticle separation is comparable to the particle size in that classical case.)  People like Lev Landau and others had the insight that essential features of the Fermi gas (the Pauli principle being hugely important, for example) tend to remain robust even if one thinks about "dialing up" interactions between the electrons.  

A consequence of this is that in a typical metal, while the details may change, the lowest energy excitations of the Fermi liquid (the electronic quasiparticles) should be very much like the excitations of the Fermi gas - free electrons.  Fermi liquid quasiparticles each carry the electronic amount of charge, and they each carry "spin", angular momentum that, together with their charge, makes them act like tiny little magnets.  These quasiparticles move at a typical speed called the Fermi velocity.  This all works even though the like-charge electrons repel each other.

For electrons confined strictly in one dimension, though, the situation is different, and the interactions have a big effect on what takes place.  Tomonaga (shared the Nobel prize with Feynman and Schwinger for quantum electrodynamics, the quantum theory of how charges interact with the electromagnetic field) and later Luttinger worked out this case, now called a Tomonaga-Luttinger Liquid (TLL).  In one dimension, the electrons literally cannot get out of each other's way - the only kind of excitation you can have is analogous to a (longitudinal) sound wave, where there are regions of enhanced or decreased density of the electrons.  One surprising result from this is that charge in 1d propagates at one speed, tuned by the electron-electron interactions, while spin propagates at a different speed (close to the Fermi velocity).  This shows how interactions and restricted dimensionality can give collective properties that are surprising, seemingly separating the motion of spin and charge when the two are tied together for free electrons.

These unusual TLL properties show up when you have electrons confined to truly one dimension, as in some semiconductor nanowires and in single-walled carbon nanotubes.  Directly probing this physics is actually quite challenging.  It's tricky to look at charge and spin responses separately (though some experiments can do that, as here and here) and some signatures of TLL response can be subtle (e.g., power law responses in tunneling with voltage and temperature where the accessible experimentally reasonable ranges can be limited).   

The cold atom community can create cold atomic Fermi gases confined to one-dimensional potential channels.  In those systems the density of atoms plays the role of charge, and while some internal (hyperfine) state of the atoms plays the role of spin, and the experimentalists can tune the effective interactions.  This tunability plus the ability to image the atoms can enable very clean tests of the TLL predictions that aren't readily done with electrons.

So why care about TLLs?  They are an example of non-Fermi liquids, and there are other important systems in which interactions seem to lead to surprising, important changes in properties.  In the copper oxide high temperature superconductors, for example, the "normal" state out of which superconductivity emerges often seems to be a "strange metal", in which the Fermi Liquid description breaks down.  Studying the TLL case can give insights into these other important, outstanding problems.