Here then I will turn to Majorana bound states, which are a condensed matter equivalent of the theorised Majorana fermions. What are Majorana fermions, and what exactly is a “condensed matter equivalent”? By “condensed matter equivalent” I am referring to a particle which exists as an emergent property of the behaviour of the ions and electrons in a particular solid material, rather than a fundamental particle like an electron or neutrino, those are the preserve of a different type of physicist. These condensed matter Majoranas are not quite like the fundamental version as we shall see. But they do have some analogous properties, and some novel properties, which make them very interesting in their own right.
To understand what a Majorana fermion is it is useful to take a (very) short tour through the history of quantum mechanics. The first relatively complete descriptions of quantum mechanics are the matrix mechanics of Heisenberg and the wave mechanics of Schrödinger. These are actually equivalent and the formulation of Schrödinger remains widely used (with a few additions). However they do not describe everything, and in particular they are not compatible with Einstein’s special theory of relativity. A version which was, and which can be used to describe electrons, was found by Dirac. However this lead to its own issue. Dirac’s equation predicted infinitely many negative energy states! To avoid the problems this would cause he suggested that these states are all filled up. However if one of these filled states is removed, a hole is left which will look much like an electron but with opposite charge, this is the electron’s anti-particle, called a positron. Such a particle was experimentally confirmed in the 1930s and particle-anti-particle pairs are now a standard feature of the standard model of particle physics. A particle and anti-particle destroy each other when they come into contact, leaving behind only light and energy. It was Ettore Majorana who, in 1937 one year before he disappeared mysteriously, found a solution to Dirac’s equation which was its own antiparticle! It is currently still unknown if these exist in nature. All the particles that make up matter that we know of come in particle and antiparticle pairs.
So, how can one get these strange particles in a solid in a laboratory? First we need an equivalent of particles and anti-particles. In solid materials the particles that we are normally looking at are the free electrons moving around which give rise to many of the conducting and thermal properties. These move in a background made by the atomic lattice. The exact make-up of this lattice, its interactions with the electrons, and how the electrons interact with each other, lead to the wide variety of phenomena which can be seen in solid materials. If we go back to our (very) simple picture of the electrons in ‘bands’, see the figure below, we can imagine we start with the lower band completely full of electrons. (The fact that only one electron can be in each state is the famous Pauli exclusion principle which is a fundamental property of electrons.) Now if we give some energy to one electron to raise it up into the upper band we have one electron which is free to move around the upper band, and one ‘hole’ left in the lower band. Shuffling the electrons around the lower band is equivalent to moving the hole around. So we can pretend that this hole is a particle which will have opposite charge to the electrons. Holes are the anti-particles of condensed matter, and we can write a formal transformation between electrons and holes mulch like the particle-antiparticle transformation. So what we need is a particle which is an equal combination of a hole and an electron, how to find one?
It so turns out that superconductors are systems which have a natural combination of electrons and holes. Due to the properties of the transformation between electrons and holes (our particles and anti-particles) if we want a particle which is both it has to have an energy of zero. If we have the states at the edges of a wire which is topological, as discussed last time, then in fact this has to have zero energy. So we need a topological superconductor, rather than a topological insulator. How to make such a topological superconductor was proposed in several seminal papers some 10 years ago (see for example here or here). Although no conclusive signature of these particles has been found several very promising experiments have been performed on a variety of possible platforms for these exotic particles, see the references here for a partial list.
These emergent Majoranas are interesting for several reasons. One is that they are not in fact fermions. They have the technical name of ‘non-abelian anyons’, which means that if I take two equivalent Majoranas, and swap their positions, I end up with something different than what I started with. If one tries this with bosonic particles the end result is the same as what I started with. If I try it with fermions (like electrons) the result is almost the same, except the function which is used to describe the fermions mathematically is multiplied by a minus sign. This leads to the Pauli exclusion provincials and also makes some other small changes which it is possible to find evidence of, but mostly everything looks the same. This corresponds to what one may expect, swap two things which start the same and nothing should change. This strange property of the Majoranas is behind one push to make a stable quantum computer out of them, which for example Microsoft is currently putting a large amount of effort and funding into.
So, finally, what did we actually do? We looked at the topologically protected states at the edges not of wires, but of two dimensional flat crystals. Along their (one dimensional) edges there can also be these special states. In two dimensions instead of having a single state on an edge we find many states forming a “band” all of which exist on the edge. It is expected that if we look in the middle of this band at zero energy we have a Majorana. However, unexpectedly it turns out that this is not always the case and our paper was concerned with understanding when this does and does not happen, see the figure above. This can help us in the search for these strange particles.
We found that whether or not the Majoranas are there is tied closely to the symmetries of the material, and in particular the symmetry it has at the point in the band structure where the Majorana should exist. This point needs a higher symmetry than what surround spot or no Majorana can be present.
The next step is to look for experimental signatures of these states along the edges. Can we tell the difference between the case when too is there and when it is not?
]]>Topological insulators are materials which in the bulk are insulating, so that they do not conduct electricity. On the surface there are however conducting electrons which, much like the states on the edges of the quantum Hall effect we have been looking at, are assured of being there regardless of the properties of the edge (well almost, reality is typically messier than such simple statements can encompass). There is in fact a direct correspondence between a specific number, which is an integer called a topological index, and the amount of states on the edges of the topological insulator. (‘State’ is a name we give to any particular possible configuration of an electron or other particle in a specific set-up. So here I really mean the number of possible electrons.) As the details of the edge are unimportant these electrons find it easy to move around the surface, leading to the good conduction.
So, is there a general way of understanding what kind of insulators will have these states on their surface? The short answer is yes. It turns out that it is topology and symmetry which tells us the answer, hence the name ‘topological insulator’. This picture has been slowly put together over the last 40 years, the last 10 of which have really seen an explosion of interest in these materials. The 2016 Nobel prize in physics was awarded exactly for contributions to these ideas.
I will not focus very much on the symmetry point here, just to say that the material needs to satisfy certain symmetry requirements. That these insulating materials can be classified by their topology was the really groundbreaking new idea. Topology is popularly explained as the study of what types of objects can be deformed into each other without cutting, breaking, or joining them. The classic example is of a coffee cup and a ring doughnut, one can be deformed into the other because they each have just one hole. However you can not make a sphere into a coffee cup without cutting a hole in it. These belong then to different topological classes.
How does this apply then to an insulator? After all what is an insulator? For our purposes it is a crystal structure of some number of different elements, which does not conduct electricity. In fact insulators come in a variety of types, but we are interested only in one type here. This has a nice simple explanation in terms of what they electrons in the crystal are doing. In a crystal some of the electrons belonging to each atom are bound tightly to the atom. Others are either spread out over the whole crystal, shared by all the atoms; or confined to an atom but not necessarily the one they ‘belong’ to. (These are covalent and ionic bonds.) The behaviour of these electrons moving around in the structure which is given by the (relatively static) atoms or ions is what we are interested in. We want to know if they can move, and we therefore have a conductor, or if they are stuck, and we have an insulator.
We can make a nice simple cartoon of how much energy different electrons can have:
Each green or purple circle is a possible electron state in the material. However we have only enough electrons available to fill up the purple circles. There is then a big energy gap until there are any more states. Now, if we want to do anything to this material, like make a current go through it, we need to move the electrons. But as we can see the only way to do this is by putting in more energy than the big gap, to move an electron into an empty gap. Hence our material is insulating. Note if we put more electrons in the material so the upper green circles are partially filled, then it becomes easy to move the electrons and we will have a metal. A real insulator looks much more complicated, but this contains the basic idea.
So, what on earth does this have to do with topology? Rather than deforming a shape, we deform the properties of this lower filled brand of electrons. Depending on how it behaves different types can have different indices, much like counting the holes in an object for the topology of 3D shapes. No band with a particular index can be changed into one with a different index, unless we join it to the another band, when it will cease to be insulating. That is the topology of a band insulator. That materials can be classified in this way has brought a whole brand new way of thinking to condensed matter physics. The example from last time possesses exactly this relation between its topology, and the robust conducting states on its edge and it is possible to prove this direct relation between the two.
]]>The bulk-boundary correspondence is a relation between the bulk properties of a material, by which I mean how a large chunk of it behaves overall, and the possible existence of special particles that on the surface. In fact it says under certain conditions they must exist on the surface, no matter what you do to the surface. To understand how this might work let us consider a simpler system than the one we will turn to shortly. The system we will consider is known to exhibit what is called the quantum Hall effect. Let us imagine we have a two dimensional material in which some of the electrons are free to move around. If we apply a magnetic field perpendicular to the material, then the electrons want to form ‘cyclotron orbits’ around the direction of the field. This is just part of the strange way that charged particles and magnetic fields interact, see figure 1. The size of the radius of the orbit depends on the strength of the magnetic field, and in a material it will also depend on the material properties. Going back to our two dimensional material we have not one, but many electrons moving around in the magnetic field. They will try and set up motion that we can think of as being like in figure 2, with all of them circling around in little orbits. The actual size of the orbits will depend on the strength of the magnetic field, the material parameters, and so on.
The Hall effect, named after Edwin Hall who discovered it in 1879, is the phenomenon where a applying a voltage drop across such a sample causes a current in a perpendicular direction, see figure 2. As the electrons are now moving preferentially along the direction fo the dashed blue arrow, as opposed to the opposite direction, they tend to be turned more to their left, than to their right. This causes us a current across the sample. The quantum Hall effect, is the quantum analogue of this and in typical quantum style the current becomes quantised into discrete plateaus. In fact there are many other Hall effects too, but we shall leave this here for now, awaiting another time.
Now we can ask what happens at the edges of such a set/up, see figure 3. In this case as the electrons cannot complete the circular orbits they are forced into moving along the edge as shown by the green patch in figure 3. This is what is known as “the bulk boundary correspondence”, i.e. the properties of the bulk of the material tell us something very specific about what will happen at the edge. Not only that but it doesn’t depend at all on the details of the edge! The edge can be as disordered and as messy as possible and the path of the electron will still follow it. This is not confined to the type of system we show here, but happens in several wide classes of materials. The two we are interested in are called topological insulators and topological superconductors. Let’s leave that for next time though…
First there are several simplifications in this picture which I think it is worth mentioning. Note that in a real system the electrons are not localised at particular points and going round these orbits exactly as pictured. Rather they are expended throughout the material, but can be thought of as something like a combination of many of the little orbits we show here. This is related to the fact that for an electron in a crystal the electron will not be confined to a particular place somewhere, but will rather be spread out throughout the entire crystal. It will however have a specific momentum, this is of course related to the fact that no particle can simultaneously have a well defined position and momentum in quantum mechanics. One of the consequences of which is the famous Heisenberg uncertainty principle. This means that looking at pictures in of how things stand in real physical space is not always the best way, and in fact in this field one uses what is called reciprocal space. Instead of asking where everything is, we ask instead what its momentum is. This can take some getting used to, but once one has got used to it then it becomes a very powerful aid in tackling many problems. The reason why it is so useful is also fascinating, but alas this will also have to wait! In most topological insulators and superconductors it is not really a magnetic field which is causing such orbits either, rather they arise out of more complicated physics, but this picture serves as a very useful introduction to these ideas.
]]>One of the short chapters is on the four dimensional universe. Now I would guess that most people with a passing interest in physics would be familiar with some of the various ideas that go by the name of string theory. In particular that all string theories only ‘work’ in many more than four dimensions. In fact it is yet to be seen if anything much can be made of string theory even in ten or twenty-six dimensions, but so far more than four has always been a necessary ingredient. For those of you wondering, the four dimensions that might be expected of our universe are the three space dimensions and time. String theory has been one attempt since the early 1980s to combine the two big 20th century theories, the first being quantum mechanics and the standard model of particle physics, and the second being Einstein’s general theory of relativity. The first of these describes how things behave on a very small scale, and very accurately too. The latter is all about gravity and the large scale universe. These do not agree and so far all attempts to bring them together have not got very far. To do so is one of the outstanding great problems of modern physics.
In the book Matt Brown writes that in fact the world has more than four dimensions, unlike what you think you know, fitting into his overall theme. Now I have several problems with what is written there. Firstly it is far from clear that the claims for high dimensional universes made by the various forms of string theory are true. There is not a single piece of experimental evidence for it, and nor is there anything approaching a coherent theory which could be argued in its favour. More than four dimensions is just a requirement of these particular ideas. But these ideas are far from anything like a solid theory. As they do not yet predict anything, even any of the things we already know, they can hardly be used in favour of even more exotic ideas. They could turn out to be true, and they could not. The best we can say at the moment is that it looks like the universe has four dimensions, but it could perhaps be more.
Now this perhaps one could just let slide, after all string theory is interesting so why not use this as the excuse to put a mention in the book. The book is after all clearly not meant to be a careful exposition of the current state of science! It is a fun book about some things you may or may not have realised about physics, biology, chemistry, and so on.
The second issue arises when he starts to explain what these dimensions are. This part of the chapter is just bizarre. The explanation of dimensions five and six seem to be a confusion with the idea of the many worlds interpretation of quantum mechanics. (Itself an incoherent idea in my opinion, but let’s save that for another time.) Dimension seven is then apparently the multiverse, or one version of it, i.e. many different universes with slightly different set-ups. The eighth dimension is another version of this. Now already there is a very simple mathematical problem here. Brown introduces two extra dimensions, but claims that they describe every possible permutation of the four fundamental forces. Clearly that is more than two dimensions! It would be as many dimensions as there were free parameters in the theories to change, which is something we do not in fact know. His final tenth dimension is supposed to be all of the previous ones together, which just suggests that he does not in fact know what a dimension is.
Now in fact in string theory the extra dimensions are nothing so exotic, nor could they be. What is described in the book is completely mathematically and physically incoherent. Instead the extra dimensions are assumed to be just like the usual three space dimensions but rolled up very small so one can not notice them, see the figure below. As an example imagine a two dimensional plane like a piece of paper. One can use the usual two Cartesian coordinates x and y to describe any point on the paper, and in that sense it is two dimensional. Now if you roll up the paper along one direction, it is still two dimensional. However you can roll it up so tight os that you can barely notice one direction and it will look essentially one dimensional. That is the usual idea behind the extra dimensions.
Or maybe I just missed a joke…
A two dimensional sheet rolled up to something a bit more like one dimension.
]]>The following formulation of the Berry paradox can be found on Wikipedia:
“The smallest positive integer not definable in under sixty letters.”
Like many paradoxes it does not so much uncover a deep philosophical problem, so much as a sloppy way with language (a prejudice of mine which would no doubt please Wittgenstein). The problem is that there is no consistent use of the word ‘definable’ in the paradox. If I am to take its formulation seriously then I have to assume that every single 59 letter string of the 26 latin letters of the English alphabet are used to refer to some number. In fact the above formulation does not stipulate the latin alphabet, but let’s assume that that is what is meant, it makes no difference to the argument. So perhaps we could have, with the letters on the left and the number represented on the right, (picking the integers represented at will):
a |
1 |
aa |
2 |
aaa |
3 |
raspberry pie |
31 |
strawberry pie |
314 |
We would also need to assign a number to such strings as
fhdjkfhdsjflskjfdsl |
1000 |
jhdfkjdslfjldsfmklds |
1001 |
and inevitably to such sentences as
The smallest positive integer not definable in under sixty letters |
30009 |
The smallest elephant under the bed |
3008 |
Note this would include such phrases as “twenty one”, “einundzwanzing”, “vingt et un”, and so on. Though none of these need to refer to the number 21. In general it does not matter which integer refers to which string of letters, though we might try and construct a simple rule to do this job. The important point is that once one has finished, the phrase “The smallest positive integer not definable in under sixty letters” already refers to a number, and it certainly does not have to refer to the number described by that string in English. Just as the German sentence does not refer to that number, or indeed the sentence translated into any other language which can be transcribed into the latin alphabet. Paradox over. The mistake is to assume that the strings of letters must refer to the number that they describe in English.
]]>It was the example given that confused me. I will try and summarize it here, hopefully not doing a disservice to it! Imagine you are sitting on a chair, without thinking about it one should say what you are feeling in your bottom. Most people will say that the chair is pushing up on them, and this is indeed the desired answer. At this point I thought the point was going to go onto Newton’s third law – that when one body exerts a force on another, there is an equal and opposite force being exerted in return. (The chair is pushing you up as you are pushing down on the chair.) I was quite spectacularly wrong with that guess! Instead Fay Dowker then asked what do we not feel? Well, we don’t actually consciously feel anything pulling us down. We don’t experience gravity in this situation, not as a force. This was proposed as an insight which tells us that there is something wrong with Newtonian physics which relies on forces, whereas in Einstein’s theory of gravitation no mention of forces is made, which we now know to be a more complete theory of gravity.
I have a major problem with this line of argument though. It is true that normally one does not talk about forces in Einstein’s general theory of relativity (the name for his theory of gravity). Instead space and time themselves are described as distorting and curving around mass, and objects are just trying their best to follow straight paths in this space-time. This is where talk of rubber sheets and weights normally comes in, which I have always found a little unsatisfactory as the weight is only distorting the rubber sheet because of gravity. What kind of explanation of gravity relies on gravity! (I am only being half serious here…)
But what does the general theory of relativity have to say about the situation at hand, you sitting on a chair. Well, it has to agree with Newton’s theory of forces doesn’t it! If it didn’t it wouldn’t agree with everything we have learnt about gravity in the time since Newton that we see played out in the world around us. Differences between the theories, though they may be metaphysically quite distinct, only show up in observation and experiment at large scales or with very large masses. Not between the earth, a chair, and you (or me).
In fact it is not so difficult to start with Einstein’s theory and derive Newton’s for a simple set up. For example just you and the earth, at some distance from each other. (This is obviously a very specific meaning of “not so difficult” which assumes one has already spent half a year learning the maths needed to do general relativity, once you’ve got that bit done you are almost home and dry!) How does one go from distorted and curved space, to a force? Well, on the scale we are talking about most of this curvature just can not be seen, it is just too small, and we can safely ignore it. What we are left with is a quantity which varies in space which we call a ‘potential’, which depends on where you are, and where the earth is. The way in which this potential varies tells us what the force of gravity looks like, and of course this ends up agreeing with Newton’s law. If it didn’t Einstein’s theory would have been abandoned long ago, probably by Einstein himself!
So, if our experience does not agree with Newton on this thought experiment, then it doesn’t agree with Einstein either! It can’t possible favour one theory over the other, and it certainly isn’t telling us that forces aren’t real. So what is going on? Well, probably it is entirely down to how your brain works. Bearing in mind that we feel the effects of gravity all the time, it would be rather odd of the brain to be constantly telling us that we are being pulled down, but I am straying rather too far into conjecture now. As a last thought I don’t think it is so hard to find scenarios where we do feel the effects of gravity as a force… can you think of any?
]]>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.
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.
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
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:
]]>This blog will be mainly about physics, maybe a few other things that catch my fancy, but mainly physics. More than that it will be about the particular area of physics I work in, condensed matter physics. To be even more precise, it will mostly be about my own work on the topic as a theoretical physicist of some 10 years standing (already 14 if I include my PhD studies). Condensed matter physics, despite being one of the largest currently studied areas of physics [1], is rather neglected in popular science books. Is there a good reason for this? It of course does not have the instant glamour of black holes or the birth of the universe, and it is not in a hunt after the fundamental stuff of the world. Nor does it look at those very day questions which physics can try and answer, like why does time seem to flow only one way [2]? Nonetheless I think there is much which could be of interest to any intellectually curious person in this area of physics.
Condensed matter physics concerns itself with how solids behave. Most of it, I think, with the electronic and thermal properties of solids in all sorts of strange configurations and sizes. This is the are of physics which butts up against chemistry on one side, and is constantly pilfering useful ideas from high energy physics on the other. It encompasses superconductors, semiconductors, topological insulators, ferromagnets, quantum dots, graphene, nanowires, nano-‘just about anything’ in fact, and many other topics – some of which I plan to explore here.
The key idea which I think keeps me constantly fascinated by what I do, quite apart from the sheer joy of solving tricky problems, is the way in which condensed matter physics is built out of a few simple building blocks but leads to such a wonderful variety of phenomena. I think it is perhaps the perfect riposte to the, I suspect quite meaningless, charge that since is somehow reductionist. But more on this later…
[1] By numbers working on it rather than thing studied, I guess cosmology wins there.
[2] I said try. (Though this is perhaps unfair, as there are some very good ideas on this out there.)