Topology in Condensed Matter Physics

Geometry, the mathematical study of shapes, might be familiar to you from school. In geometry the representation of the shapes is very important. For example, an equilateral triangle is different from a right angled triangle. In topology, these two triangles and many other shapes are equivalent. In his introductory book on topology [1], Robert A. Conover writes “Topology is sometimes called ‘rubber sheet geometry’ because two objects are said to be topologically the same (topologically equivalent) if one can be stretched, shrunk, bent, or twisted to make it look like the other.” No tearing or sticking together is allowed. If two objects are topologically the same, we say that one can be continuously deformed into the other. For example, a triangle, a square and a circle are all topologically the same but they are all topologically different from a line segment. We would have to tear a triangle to make a line segment, and we would have to stick together a line segment to make a triangle.

Topology began as the study of curves, surfaces and other objects in two and three dimensional space [2] and it is defined simply as the mathematical study of the properties of an object that are preserved through continuous deformations of that object. In topology, objects are classified in a way that is independent of their representation in space. For example a doughnut (known in mathematics as a torus) and a coffee cup are topologically the same because both have one hole. Here, we have classified the objects by the number of holes they have, and this number is independent of the representation of the object in space. This leads to the idea of a topological invariant which, in this example, is the number of holes. As long as we deform the doughnut with no sticking or tearing, the number of holes will not change; a topological invariant is a property that stays the same as an object is continuously deformed.

Topological invariants turn out to be very useful in condensed matter physics because we use them to classify what are called topological phases of matter. These phases have properties that give topologically invariant results when measured. Topologically invariant for a quantum system means that the property depends on its global structure only. Any impurities or defects in the material will not affect the result of a measurement of the topologically invariant property. This is the case for a material exhibiting the quantum Hall effect—the conductance is a topological invariant.

The process of classifying topological phases of matter usually begins by asking whether the mathematics describing one phase of matter can be deformed to look like the mathematics describing another phase of matter, subject to some conditions. Deforming here means changing the parameters in the mathematical expressions. To use topology, we then find precise mathematical mappings that allow us to go from the mathematics which describes a certain phase of matter to a corresponding topological object.  

Physicists have begun to classify insulating materials using topology, and in the article on topology and the classification of matter, we explain what it means to deform the mathematics in the context of an insulator.

1. Robert Conover, A First Course in Topology: An Introduction to Mathematical Thinking
2. Wolfram MathWorld: http://mathworld.wolfram.com/Topology.html

Quantum Computing

The development of a new type of computation that takes advantage of quantum superpositions could lead to many scientific and technological breakthroughs. Here, we will briefly discuss the advantages of quantum computation, and how topology can help solve one of the biggest challenges we face in realising quantum computing.

There are certain problems that even the most powerful classical computer theoretically possible cannot solve. When both classical and quantum computers carry out calculations, they have to store information in their memory to use later on in the calculation: some problems would require so much memory on a classical computer that it would run out of space before getting to the end of the calculation. There is no way around this. Ultimately, classical computers perform computations in the same way you or I would, only much faster, and there are limits to what this type of computation can do. Quantum computers can solve many of the problems inaccessible to classical computers, and can also solve many problems that classical computers can much more quickly.

Simulating chemical bonding and chemical reactions, which are inherently quantum processes, is a challenge which classical computers can only scratch the surface of, but quantum computers can overcome—this is because it is much easier to simulate a system that is inherently quantum with a quantum system. The ability to simulate chemical processes could lead to the discovery of new medicines and materials. In general, the kind of calculations and simulations that physicists have to do will be made a great deal easier with quantum computers.

Despite the advantages provided by the use of quantum superpositions, they are also the biggest downfall of quantum computers. In typical quantum computers, the superposition states used are typically those of individual atomic ions (atoms with some electrons removed) or electrons. These superposition states are very sensitive and fragile, meaning that small perturbations and interactions with the environment can cause them to collapse: the superposition state is destroyed and instead we find the system in one of the states that formed the superposition—this process is known as decoherence. Errors therefore occur when the superpositions collapse during computations, due to interactions with the environment.

A new type of quantum computer, known as a topological quantum computer, has been proposed by theoretical physicists. Topological quantum computers are well protected from the collapse of superpositions because they use braided particle world lines to perform calculations. To understand this, we will discuss exactly what a world line is, how world lines can be braided and why they make quantum computers much more stable to perturbations from the environment. It will be useful to have read the article on topology in condensed matter physics before reading the section below.

A world line describes a particle’s path through space-time, which is really not as complicated as it sounds. Say we have an ant walking in a circle on a table, we can mark its position every second, for example, and plot this on a graph (note that the ant is constrained to move in two spatial dimensions, assuming that it cannot jump off the surface of the table). This graph will have to be in three dimensions because we need two numbers to specify where the ant is on the table (one number to tell us how far along the length of the table the ant is and another to tell us how far along the width of the table the ant is) and a third to tell us the time corresponding to each position. The animation below shows the graph: the red line is the path that the ant follows on the table and the blue line is what physicists would call the world line of the ant. Each set of three numbers telling us where the ant is at what time corresponds to a single point in our three dimensional visualization. Joining all these points together gives a curve which is the world line. Although the world line can be visualised as a curve in three dimensional space, which is what the animation shows, it’s not actually in three dimensional space because one of the axes corresponds to time. You may have noticed that we would have needed a four-dimensional animation if our ant were able to jump (i.e. move is a third spatial dimension). While we can’t produce such an animation, the world line is still defined in the same way.

The particles needed in a topological quantum computers are called anyons which are emergent particles that only arise in two dimensional systems. Even though our world is three dimensional, we can get a two dimensional system by making the electrons within a material need an extremely high level of energy to move in the third dimension, so that they are restricted to moving in two dimensions. The anyons will also be confined to move in two dimensions and this is important because if they were not, braiding their worldlines would not give the correct topological objects. Anyons are generally classified as abelian or non-abelian and it is non-abelian anyons that are needed for topological quantum computers. So what exactly do we mean then by braided world lines?

The world lines of the anyons are braided by moving them around on the two dimensional surface, which is shown in the diagram below.

For non-abelian anyons, the order in which worldlines are braided together does affect the outcome of an experiment.

The reason that topological quantum computers are much more stable to perturbations is that the computations only depend on the topological properties of the braided worldlines. A small perturbation from the environment is a bit like squashing a sphere slightly—it’s still the same topologically. Much more than small interactions are needed to change the topology of the braided wordlines, which means that topological quantum computers don’t suffer from the sensitivity that regular quantum computers do.

In practice the physical systems in which we can find non-abelian anyons have not been realised in experiment, however there are a few candidates predicted by theory. There has been some experimental work done by physicists Robert Willett and his colleagues which could bring us a step closer to realising these systems if their results are convincingly verified [1].

1. Steve Simon writing for Physics World: https://physicsworld.com/a/quantum-computing-with-a-twist/

Topology and the Classification of Matter

The modern periodic table is perhaps the most successful endeavour to classify and organise in science. Because of a breakthrough by Henry Moseley in 1913, who devised an experiment to measure the atomic number—the number of protons in an atom—using x-rays, we can use mathematics, albeit very simple mathematics, to organise the elements.

Knowing precisely the number of protons in each element means that we can assign a whole number (this has to be a whole number because we can’t have a fraction of a proton in an atom) to each element, which can be thought of as a mathematical mapping. Organising the elements is now a matter of counting. As of 2018, we know for sure that we have all the elements between atomic number 1 and 118 and any new elements need to have atomic numbers greater than 118 because we’ve seen atoms with 1, 2, 3 … up to 118 protons. Before Moseley’s experiment, the ordering of the elements was based on the atomic mass but we can’t conclude from the atomic mass alone whether we have all the elements or whether there are some missing. For example, if you had two elements next to each other and all you knew were their atomic masses you wouldn’t be able to tell whether an undiscovered element should come in between them or not—you would just know that the mass of one element is larger than the mass of the other. If you knew their atomic numbers, and say one had 7 protons and the other had 9 protons, you would then be able to conclude that an element with 8 protons should exist. This all sounds more like chemistry than physics but it is a very important story to tell because many physicists are now working towards ways of classifying and organising all types of matter in a similar way to which the elements have been classified and organised in the periodic table. There have been over 300 million compounds identified, which are made up of all sorts of combinations of the 118 elements, so where do we even begin? In recent years physicists have been guided by ideas from topology, a branch of mathematics.

Here, we will talk about the topological classification of insulators. These are materials with a gap of forbidden energies between the conduction band and the valence band (explained in the metals, insulators and semiconductors article). In these materials, all the quantum states in the valence band are filled with electrons and they don’t have enough energy to cross the gap and fill the empty quantum states in the conduction band, which would allow them to conduct. Each insulator is mathematically treated as a quantum system described by a Hamiltonian, which is a mathematical function that contains all the interactions of the system. We define a topological class by saying two insulators are in the same topological class if the Hamiltonian describing one can be continuously deformed into the Hamiltonian describing the other and vice versa (see topology in condensed matter article). Within each class, there is a certain set of properties that all the insulators have, which is different from the properties shared by the insulators in a different class.

For topological objects, continuous deformation means stretching, shrinking, bending and twisting as long as there is no tearing or sticking. If we tear or stick together parts of the object, then we have gone through a singularity and have changed our object into one that is topologically different. For a given Hamiltonian, continuous deformation means changing the parameters that define that Hamiltonian without turning it into a Hamiltonian that describes a system in which the gap between the valence and the conduction band is closed i.e. a conductor. If the gap is closed at any point during the deformation then we have gone through a metal state, which is the singularity in the context of Hamiltonians, and have changed the topological class of the material. Some of these deformations correspond to things we can actually do to the material such as applying pressure and applying and varying the strength of an electromagnetic field. Some are more abstract, like changing the masses of particles. Mathematically we can write down anything we want but the important thing to take away is that if we can make the Hamiltonian of one insulator look like the Hamiltonian of another (by changing the parameters in all sorts of crazy ways) without closing the energy gap between the valence and conduction band in the process, then those two insulators are in the same topological class.

In two dimensions, it turns out that classes of insulators can be indexed by a single integer, which we can call N (this is just a placeholder for the actual numbers). This integer is a topological invariant, a bit like the number of handles on a topological object (a sphere has zero handles, a coffee cup/doughnut has one…). The N=0 insulators don’t conduct electricity at all—of course, they’re insulators. However, the materials we classify as insulating materials are much more interesting than was thought before the 1980s. The materials in the N≠0 classes (the symbol means ‘not equal to’) share a remarkable property (although there are properties specific to the insulators within each class): they insulate everywhere, except on their edges where they conduct. This phenomenon is known as edge conductance. We refer to N=0 insulators (the ones which behave in the expected way) as trivial topological insulators and we refer to the N≠0 insulators as non-trivial topological insulators. One of the most remarkable features of non-trivial insulators in two dimensions is that they can exhibit edge conductance while breaking time-reversal symmetry.

Symmetry is an extremely important principle in theoretical physics. When physicists talk about symmetry, we ask this question: if we take the maths that describes out system (in our case this is the Hamiltonian) and transform one of the parameters, do we end up with the same physics? This is actually the same as the more intuitive idea of symmetry—if we transform a picture, typically by a reflection, does it look the same? The question to ask when it comes to time reversal symmetry is this: if we ran time backwards and observed a system as time ran backwards, could we tell the difference? If the answer is no, then our system is said to possess time reversal symmetry. You can actually test time reversal symmetry for yourself in some physical systems. Film a ball bouncing up and down and see if you can tell the difference between the video running backwards and forwards. If you can tell a difference, why?

We actually need to combine symmetry and topology for a more complete classification of materials. We use the idea of a symmetry protected topological class, which we define by saying that two insulators are in the same symmetry protected topological class if the Hamiltonian describing one insulator can be continuously deformed into the Hamiltonian describing the other insulator without going through a singularity (as defined above) and without breaking the given symmetry.

The principles of symmetry and topology together are extremely powerful. The classification scheme that has been developed has allowed us to predict phases of matter, which physicists are now doing experiments to find, just like how Moseley’s experiment systematically allowed us to look for elements. We can go on to add more complex symmetries than the ones listed in the table above to the classification scheme and even go on to classify metals in the same way.

The Quantum Hall Effect

The story of the quantum Hall effect begins with Edwin Herbert Hall in 1879. Using a thin gold leaf taped to a glass plate, Hall discovered what we now call the classical Hall effect: he applied a magnetic field in a direction perpendicular to the plane of the gold leaf and observed a voltage perpendicular to the direction of the flowing current. The existence of this transverse voltage is what we call the classical Hall effect.

Associated with this transverse voltage is the Hall conductance, defined as the ratio of the current flowing along the sheet to the voltage measured in the perpendicular direction (see diagram). The conductance is a measure of how easily current passes through a given object (e.g. a metal wire has a high conductance, whereas a wire made from rubber will have a low conductance).  While studying the Hall conductance of two dimensional materials at very low temperatures, Klaus von Klitzing discovered the quantum Hall effect and was awarded the Nobel prize in 1985 for his discovery. Classically, as the magnetic field is continuously increased, the Hall conductance continuously increases too. In his experiment von Klitzing found that as the magnetic field was increased continuously, the Hall conductance also increased but did so in discrete steps. In essence, the quantum Hall effect is the quantisation of the Hall conductance.

The graph above shows that as the strength of the magnetic field is increased, the Hall conductance stays the same until the strength of the field is changed enough so that the conductance can jump to the next allowed value. The formula for the Hall conductance is

\(\frac{e^2}{h}\nu\)

The letter 𝑒 is the symbol for the charge of an electron, ℎ is Planck’s constant and most importantly 𝜈 can take the values 1,2,3… and so on. Each allowed value of the Hall conductance corresponds to a value of 𝜈. It turns out that what von Klitzing discovered is the integer quantum Hall effect. There are other quantum Hall effects and perhaps the most important to mention—which resulted in the 1998 Nobel Prize in Physics awarded to Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui for its discovery—is the fractional quantum Hall effect which occurs in certain two dimensional systems. Again, a Hall conductance can be measured and its value increases in steps as the magnetic field is increased continuously and it even has the same formula as above. The difference is that 𝜈 takes the values 1/3, 2/5, 3/7, 2/3… Basically 𝜈 can be any fraction that you can write down which can’t be simplified further. Unlike in the integer quantum Hall effect, fractionalised emergent particles can arise in systems exhibiting the fractional quantum Hall effect—these are particles that can have quantum numbers which are fractions of the quantum numbers of electrons and other non-emergent particles.

One of the most striking things about von Klitzing’s result is the fact the Hall conductance of a material does not depend on any material properties or the shape or size of the sample. Also impurities in the sample do not affect the result of the experiment. In fact, the measurement of the Hall conductance has been repeated many times and it has been shown that the result is consistent and precise to one part in 10¹⁰. This is like measuring the distance from London to Los Angeles to within a fraction of a millimetre. Although the experimental results were clear and undisputed, the theories of quantum matter at the time of its discovery failed to explain the quantum Hall effect. The theoretical understanding of the quantum Hall effect that has been developed since then has some deep connections to topology.

One of the key ideas from topology used in explaining the quantum Hall effect is that of topological invariants, which are properties that stay the same as an object is continuously deformed. When talking about materials, what we actually deform is the mathematics describing the material in a particular state by changing some parameters in the mathematical expressions. These parameters can be changed in such a way that some of the properties of the material remain that same throughout the change—such a change is called a continuous deformation. The properties that stay the same as the material, or rather the mathematics, is continuously deformed are called topological invariants of the material. Of course, not all materials have topologically invariant properties and a given material may not have topologically invariant properties in all phases that it can exist in—materials existing in different phases should be familiar to you. The phases of a material that do have topologically invariant properties are called topological phases of matter.

It turns out that materials exhibiting the quantum Hall effect are in topological phases. Changing the strength of the applied magnetic field is a way of deforming the material and the Hall conductance is the topological invariant. Small changes in the magnetic field or the addition of impurities or changing the shape or size of the material all correspond to continuous deformations because they do not change the value of the Hall conductance. If the change in the magnetic field is large enough, then the value of the Hall conductance will change—it will jump to the next step. Going from one step to another is a bit like tearing a hole in a sphere to make a doughnut, which are two topologically different objects.

Each value of the Hall conductance corresponds to a different topological phase, so going from one step to another corresponds to changing from one topological phase to another.

The discovery of the quantum Hall effect in the 1980s is one of the most important in the field of condensed matter physics. With hindsight, we see that it was the discovery that kickstarted the study of topological materials, which has become one of the main research areas studied by quantum condensed matter physicists today.