Bumping Around in One Dimension – Part One

In the first blog proper I thought I would discuss a collection of papers on one dimensional systems, i.e. wires. In condensed matter physics when we speak of one dimension, we really only mean it approximately. It is like a very narrow corridor in which one can only move forwards and backwards, if you come up to someone else there is no way round. However, given enough energy you could clamber over each other. This is pretty much what we mean by one dimensional. The wire (1) is narrow enough that the particles inside can only pass around each other with a lot of energy. So much that we can generally ignore this and pretend it just never happens. (A lot of physics’ power happens to lie in just pretending things don’t happen, and when they are rare enough this really works well!)

Now, what particles are we talking about in these wires? Electrons – just like one thinks of flowing through the wiring at home. When they are stuck inside these narrow wires and they bump into each other there is no way around, and hence how they interact become very important. Those of you who know some physics may be confused by why this should be interesting, after all electrons are negatively charge, and two charged things will interact via the Coulomb force. It turns out that he force between electrons in many simple materials is one of those things we can pretty much ignore, and pretend it doesn’t happen (2). This in itself is a fascinating fact which maybe I will come back to in a future blog. (The cases where we can’t pretend lead to some strange phenomena too.)

So what do we have? A collection of electrons, stuck in a narrow wire, interacting. In principle we can write down some equations which describe this very accurately. But we can’t solve them. Even this relatively simple set up is impossible to solve. We can get a lot of information from this with some reasonable approximations. It turns out that if the set up is not too hot, and room temperature will often count as cold here, that we have an example of emergence. The wire full of interacting electrons can be described by a much simpler type of particle. The nice thing here is that we can sometimes even derive the emergent behaviour, which is often not the case. However even when we can’t there are arguments as to why the simpler behaviour should still be a good description, as we shall see.

My own work comes in when trying to get a better grip on how widely we can do this. What can we do to the wire and still see the same thing happening. But before we try and understand that, we should understand the simplest case first.

Imagine that we have many electrons all in the wire jostling up next to each other. They can’t get around each other so they are all pretty much stuck. So how does anything move in such a case? The answer is much like sound waves do in a solid. One part bumps into the next which bumps into the next and so on and a wave made of all these bumping events travels through the material. A similar thing occurs for the electrons in the wire. We can get a very good description of what is happening by just treating these sound waves (3). The reasons why this is so helpful is that, unlike for the electrons, different sound waves do not interact with each other!  Thus we can treat them one by one which is considerably easier. That this will work even for electrons which interact with each other is the insight that Sin-Itiro Tomonaga had in his original paper which started this field back in 1950.

Such behaviour of particles in the wire has several interesting consequences. One I would like to discuss is the phenomenon called spin-charge separation. The electrons which are moving around inside the wire have several intrinsic properties. They have a mass, they have a charge, and they have what is called spin. Assuming that mass and charge will probably mean something to most people already, let me explain what spin is. Spin is a quantum mechanical property of particles, meaning that it does not show up in any classical theory (like the theory which describes how a ball flies through the air). Spin can be thought of as an intrinsic form of angular momentum. For reasons which are simultaneously both fascinating and too far from our topic to explain here, an electron’s spin can be thought of as a having one of two values: plus or minus one half. It is often pictured as a little arrow pointing either up or down.

Now as both spin and charge are carried by the same electrons, once you move the electrons around then both spin and charge must be moved. It turns out however that for a collection of electrons this is no longer true! The spin can move one way and the charge in a different direction, even though the electrons themselves still have both spin and charge. This property becomes manifest explicitly in the description of our so-called sound waves. (Note that sometimes I talk about waves, and some times about particles, the mixing up of these descriptions is characteristic of quantum mechanics.) For an example of how this can occur see Fig. 1.

Fig. 1. (a) A particular configuration of the spins (shown as arrows) of the electrons. Each purple circle represents the charge of an electron (-e). (b) Let us now take an electron out, the gap is given by the black circle. (c) This gap allows electrons to move, or equivalently of the gap to move. (d) We can also swap electrons around. If we keep on doing this then we see that on the one hand the black hole, which can be thought of as carrying around a lack of charge +e, and the two red arrows, which carry around excess spin move independently. This is spin-charge separation!

It was an insight of recent Nobel Laureate Duncan Haldane in a 1981 paper that in fact this argument can be applied to a wide variety of wire-like systems with different types of particles and different types of interactions. The differences between all the different types of particles and interactions ends up encoded in a few constants for the simplified theory. The simpler theory of non-interacting particles, which describes the ‘sound waves’ of the electrons, has a higher symmetry. This is an example of how a low energy approximate theory can have a much higher symmetry than the full theory. This is counter to what one often finds when looking for ‘grand unified theories’ which attempt to unify different physical laws. In that case it is though that the low energy theory (which describes the work we see) has fewer symmetries than the unified theory which stands behind it. It seems that both of these occur in nature. Symmetry plays a very important role in modern physics, and we will hopefully return to that theme soon.

  1.  or carbon nanotube, or…
  2.  We don’t really pretend it doesn’t happen at all, it does make some changes but they can be taken into account by just talking about electrons which happen to have more, or less, mass and by other similar strategies. Mass being of course the physicists more careful term for what is colloquially known as weight. The difference is a perennial favourite of school physics teachers, so I will not go on about it any more here.
  3.  A quick aside for those who know a little about the behaviour of electrons in solids. The actual description tends to rely on a picture of what happens when we describe the electrons not by where they are, but by how much momentum they have. This is the natural description in a regular material, but takes us too far from our topic here.

Further (Much More Technical) Reading:

Remarks on Bloch’s Method of Sound Waves applied to Many-Fermion Problems – Sin-Itiro Tomonaga

Effective Harmonic-Fluid Approach to Low-Energy Properties of One-Dimensional Quantum Fluids – Duncan Haldane

A brief introduction to Luttinger liquids – Johannes Voit

Luttinger Liquids: The Basic Concepts – Kurt Schonhammer

One-dimensional quantum wires: A pedestrian approach to bosonization – Sebastian Eggert

Some of My Own Work:

Conductance in Inhomogeneous Quantum Wires: Luttinger Liquid Predictions and Quantum Monte Carlo Results – D. Morath, N. Sedlmayr, J. Sirker, and S. Eggert

Conducting Fixed Points for Inhomogeneous Quantum Wires: A Conformally Invariant Boundary Theory – N. Sedlmayr, D. Morath, J. Sirker, S. Eggert, and I. Affleck

Theory of the Conductance of Interacting Quantum Wires with Good Contacts and Applications to Carbon Nanotubes – N. Sedlmayr, P. Adam, and J. Sirker

Transport and Scattering in Inhomogeneous Quantum Wires – N. Sedlmayr, J. Ohst, I. Affleck, J. Sirker, and S. Eggert

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