## Magnets and Phases of Matter

### Absolute Zero

FUNDAMENTALS
There is a limit to how cold things can get, and physicists call this absolute zero. Temperature can be understood in terms of energy: if the particles of gas in a room have a lot of kinetic energy on average, then the temperature will be high. If their average kinetic energy is low, then the temperature is low. This suggests that at absolute zero, everything should stop moving. However this mechanism for temperature doesn’t consider quantum mechanics at all. When we consider the role that quantum mechanics plays in the behaviour of matter a low temperatures, we find that very strange and exciting phenomena occur—superfluids are just one example.

We all use the concept of temperature on a day-to-day basis. We describe the weather with temperatures, for example 25°C (77°F) is a hot day out, whereas if it’s 5°C (41°F) you need to wear a coat. We also use temperatures in cooking—water boils at 100°C (212°F) and cakes are baked at 180°C (356°F). Temperatures also extend far beyond the scales we encounter every day. The surface of the sun is 5500°C (9930°F), and the typical temperature at the South Pole in winter is -49°C (-56°F). But just how high or low can temperatures go?

To answer this question 19th century physicists had to think about what exactly is meant by temperature. Think about any system with lots of moving parts. This could be absolutely anything—the engine of a car, a human being, or even the whole universe—but physicists typically study simpler examples, such as a sealed box of gas or a magnet. The key ideas are: all the different parts of the system have a certain amount of energy, and they all interact with each other. In the example of a sealed box of gas (we look at magnets in the next section) the individual gas particles can have more or less energy by moving faster or slower respectively. They interact with each other by colliding, and in the process also change their individual energies (think of this like collisions between balls on a pool table—some will slow down, others will speed up, and they can all change direction).

Now if we were to increase the temperature of the box, on average every gas particle would be moving more quickly. Likewise, if we decreased the temperature, on average every particle would be moving less quickly. Raising the temperature of the box increases the total energy of the system, and vice versa. It turns out that the laws of physics allow us to add as much energy to a system as we like, but the most we can do to remove energy is to leave every particle stationary. This implies that there is a lowest possible temperature, which we cannot go below. This special temperature is called absolute zero. The technical definition of temperature means that absolute zero is the same for any possible system: it occurs at precisely -273.15°C (-459.67°F).

The Celsius and Fahrenheit temperature scales were both based around everyday temperatures. For example in Celsius, 0°C is the temperature at which water freezes, and 100°C is the temperature at which it boils. But physicists like to use a scale where 0 corresponds to absolute zero. This is called an absolute temperature scale, since there can be no temperatures less than zero on it. Physicists usually use the Kelvin scale, where 0K (said as zero Kelvin) is absolute zero. Going up or down 1 on the Kelvin scale is the same as going up or down 1 on the Celsius scale. This means we can convert from a temperature in Celsius to one in Kelvin by adding 273.15. For example room temperature, 25°C, is roughly 298K.

The laws of thermodynamics tell us we cannot actually get a physical system to absolute zero, but we can get very very close. On the Kelvin scale we’ve hit temperatures colder than about 10 billionths of a degree. In our example of a sealed box of gas a temperature of absolute zero would correspond to all the particles having stopped moving, but we haven’t considered quantum mechanics in our description. Strange and exciting things happen when we cool systems down to near absolute zero.

### Phase Transitions

FUNDAMENTALS

At school we learn that there are three phases of matter: solid, liquid, and gas. The most familiar type of matter which exists in all three phases is water. If we start with a tray of ice cubes in the freezer, we have water in its solid form. Take them out the freezer and we can readily observe the transition from the solid to liquid phase, i.e. melting. If we heat the water up further still, say with a kettle, it will boil and turn to steam. Boiling or evaporation is the transition from the liquid phase to the gas phase. If we had some way to trap all of the steam, we could then cool it back down and see it condense from gaseous steam to liquid water, and we’d get back the same amount of water we started with. Put the water back in the freezer and it will freeze into solid ice. Melting, evaporating, condensing, and freezing—and not just of water but of any material—are all examples of phase transitions.

What makes these phase transitions? No chemical changes take place, for example ice, water, and steam are all made up of H2O molecules, and the total number of these molecules always stays the same. The transitions can all be reversed, to return the material to it’s previous phase, as explained above for the water example. They all occur when the material reaches a certain pressure and temperature. For water this is 0°C (32°F) for melting / freezing and 100°C  (212°F) for evaporating / condensing. And for the above examples of melting, evaporating, condensing, and freezing, extra energy called the latent heat must be supplied at the phase transition to make the material change phase. This is most easily seen for boiling—a pot of water on the stove will quickly heat up to 100°C, but it will take much longer to boil all the water off into steam, and it will remain at the temperature of 100°C while this takes place. This is why in cooking, the stove can be turned right down once the water is boiling, since you only need to supply enough heat to the pot to match the rate at which it loses heat, otherwise all the water will boil off (which we term ‘boiling over’).

In fact it turns out this last property is not common to all types of phase transition. Physicists call the types for which it is true first order phase transitions. The main other type, called second order phase transitions, are the kind which physicists studying quantum matter are most interested in. Below we list some examples of phase transitions (and which type they are).

Phase transitions

• (First order) Melting, evaporating, condensing, freezing.
• (Second order) When iron is heated above the Curie temperature so that it can no longer be a permanent magnet (and vice versa).
• (Second order) When a metal cools down and becomes a superconductor (and vice versa).
• (Second order) When liquid helium is cooled down and becomes a superfluid.
• (Second order) Bose-Einstein condensation.

NOT phase transitions

• Baking bread, and most other processes that happen in cooking. The change in the material properties of the dough are due to thermally activated chemical reactions (although some evaporation will also take place).
• Firing clay, this is actually a combination of evaporation and a chemical reaction, happening in that order as the kiln heats up.
• Ionisation (when a gas becomes a plasma). This happens continuously as a gas is heated up.

Right: A temperature scale showing a number of important phase transitions and other temperatures for reference. Hover over the plus sign to find out what happens at each temperature. This temperature scale is logarithmic, which means each mark on the thermometer is for a temperature 10 times higher than the one below it. All temperature are given in Kelvin—take off 273 to convert to a temperature in Celsius (learn why we use this scale in our absolute zero section).

5772K – this is the temperature of the surface of the sun.
1043K – the Curie temperature of iron. Above this temperature iron cannot be a permanent magnet.
373K – the boiling point of water.
330K – the highest temperature ever recorded on Earth (Death Valley, California, 1913).
273K – water freezes.
234K – mercury, a metal which is liquid at room temperature, freezes.
184K – the lowest temperature ever recorded on Earth (Soviet Vostok Station, Antarctica, 1983).
90K – below this temperature the material yttrium barium copper oxide (YBCO) becomes a superconductor. YBCO is one of the so called high temperature superconductors.
77K – nitrogen, the element which makes up about 78% of air, condenses into a liquid at this temperature.
7K – lead becomes a superconductor. Lead is a rather poor conductor at normal temperatures, yet it’s the element which becomes a superconductor at the highest temperature. This is because electrons in lead interact strongly with the lattice – normally this leads to high resistance, but in the superconducting phase it is exactly these interactions which allow for the formation of Cooper pairs.
4K – helium condenses and becomes a liquid. This is the lowest temperature at which any known element condenses. Mercury is a superconductor below this temperature.
3K – this is the temperature of ‘deep space’, meaning very far away from any stars.
2K – helium-4, the heavier isotope of helium, becomes a superfluid.
1mK – helium-3, the lighter isotope of helium, becomes a superfluid.

### Magnets

INTERMEDIATE
Magnets are a quantum materials that were known about long before the development of quantum mechanics. Magnetism can be understood in terms of the spin of electrons, and magnets demonstrate a very important concept in physics: symmetry breaking. This is where a bunch of symmetric things come together to form something that isn’t symmetric. Symmetry in physics is a bit more abstract than the the type of symmetry most people are familiar with, but the idea is the same—if we change something in our system, is the physics still the same? If the answer is yes then there is a symmetry, defined by the thing we changed.

Lodestones—naturally occurring magnetic rocks—have been known to attract iron objects since ancient times. Since their invention in 11th century China, compasses have utilised the interaction of a magnetised iron needle with the weak magnetic field of the Earth for navigation purposes. Today we learn about magnets at primary school (remember placing a magnet on top of iron filings to see the field lines, or making a compass with a needle, cork, and bowl of water?) but explaining how they really work will require quantum mechanics.

First let’s make a distinction between the different kinds of magnet. Electromagnets are made by running electrical current through a coiled wire (this is sometimes called a solenoid). As long as current flows, the coil will generate a magnetic field—but as soon as the current is switched off the coiled wire will no longer be a magnet. Electromagnets are the basis for technology such as electrical motors, generators, microphones, and loudspeakers, to name but a few. The theory that describes how they work, which we call classical electromagnetism, was developed by the Victorian physicists Michael Faraday and James Clerk Maxwell. According to classical electromagnetism, almost every material will become magnetic when it is placed in a magnetic field, but will revert back to normal when the field is turned off, a phenomena called paramagnetism. One of the main exceptions to this are ferromagnetic materials, which we will just call ferromagnets.

Ferromagnets (often called permanent magnets) are the kinds of magnets used to make compasses and fridge magnets. Simple examples of ferromagnets are iron, cobalt, and nickel. Once placed in a magnetic field, they will acquire a permanent magnetic field of their own—which we call the magnetisation of the material—even when the original field has been turned off. Actually, this isn’t strictly true. As you may have learnt in school, heating ferromagnets to a sufficiently high temperature, known as the Curie temperature (named after physicist Marie Curie) will destroy their permanent magnetism. This is an example of a phase transition because above the Curie temperature the material is in the paramagnetic phase, and below the Curie temperature it is in the ferromagnetic phase.

The nature of a ferromagnet’s phase transition is connected to its quantum mechanical properties—namely spin. The physicist Ernst Ising devised a simple model of a ferromagnet using quantum mechanics. In all solids, the constituent atoms form a lattice. The simplest version of this is a three dimensional grid, with an atom at each grid intersection (see diagram). The atom consists of a positively charged nucleus orbited by negatively charged electrons. The spin of the electrons gives each atom its own microscopic magnetisation, and will also allow the atoms to interact with any magnetic field they find themselves in. In the model devised by Ernst Ising, known as the Ising model, we ignore the details of the atoms and electrons and assign a value for the projection of the spin to each lattice position. What this means precisely is that we allow just two orientations for the microscopic magnetisation of each atom: ‘up’ and ‘down’.

In the interactive illustration below, spins at each lattice position are illustrated as either being up (an arrow pointing up) or down (arrow pointing down). All the spins can interact, but in the Ising model only spins which neighbour each other on the lattice are allowed to interact. The spins save energy if they point in the same direction as their neighbours. The illustration below shows a 2D lattice (of course real ferromagnets are 3D) simulation of the Ising model. Although it is very simple compared to a real ferromagnet, it shows that as the temperature is increased and decreased (move the temperature slider up and down) there is a phase transition between a phase in which the spins are ordered and a phase in which they are disordered.

An illustration of a simple lattice. The spheres represent atoms and the lines the bonds between them. This very simple example is in fact not commonly found in nature. The reason why can be explained in analogy to packing oranges in crates. Clearly packing the second layer such that each orange is directly above one in the bottom layer would be an inefficient use of space—this is also true for the arrangement of atoms into a lattice in solid materials.

An interactive simulation of the two-dimensional Ising model. The slider below the lattice controls the temperature. Move the slider and observe what happens at very low temperature vs. very high temperatures. When you move the slider all the way down to the lowest temperature, do the spins always align the same way? This works best on desktop browsers.

Because in the ordered (ferromagnetic) phase all the spins line up, there is an overall net magnetisation. This is exactly what happens in a permanent magnet. Above the Curie temperature in the disordered or paramagnetic phase, on average as many spins point down as up. This means if we add up all their magnetic fields, we get nothing overall—no net magnetisation. Above we said spins save energy if they line up—so why does this only happen below a certain temperature? It turns out that at higher temperatures, spins have more energy which makes them want to ‘flip’ direction. This competes with their tendency to align, and above the Curie temperature these so-called thermal effects win out.

#### Spontaneous Symmetry Breaking

There is much more to learn yet from the simple Ising model. Look at the interactive graphic above again. In the model used for the simulation, there is no preference for the spins to align up instead of down or vice versa in the ordered phase. This is called a symmetry—the physics of the system doesn’t depend on which direction is up and which is down.

We can see this by writing down the Hamiltonian for the Ising model (if you don’t know what this is think of it as the energy of a particular configuration of the system at any moment in time, or read more about it in our quantum mechanics article):

$$H=-J\sum_{\langle ij\rangle}\sigma_i\sigma_j$$

Here J is the strength of the coupling between neighbouring spins, and σi denotes the spin at the lattice site labelled i. The unusual notation in the sum just means that we only sum over i and j if they refer to neighbouring sites. Now if we were to relabel up spins as down, and down spins as up, we would have to switch out σ for –σ in the Hamiltonian above. But clearly this would introduce two minus signs, which cancel out, and therefore H will be left unchanged. We say that H is symmetric when σi →-σi.

And yet, as the temperature is increased or decreased, the spins sometimes all order up and sometimes all order down, but will always pick one or the other. This is called spontaneous symmetry breaking – at the phase transition, the symmetry which the underlying physics has is broken by the spins choosing to all order in a specific direction. It turns out that the type of phase transition which occurs is called a second order phase transition. Instead of being associated with latent heat, like in first order phase transitions such as melting and evaporation, these phase transitions are just associated with an order parameter (in the Ising model, this just means the average direction of the spins) which spontaneously breaks a symmetry of the system’s underlying physics below the temperature of the phase transition.

A variant of ferromagnetism, called antiferromagnetism, favours spins anti-aligning with their neighbours. The system still has a phase transition, below which it is ordered, but there is no net magnetisation. Antiferromagnets can be examples of spin liquids.

### Order and Superconductors

INTERMEDIATE
Magnetic phases of matter are characterised in terms of spin. In most materials, when the temperature is lowered to approaching absolute zero all the spins go from disordered to ordered meaning that they all point in the same direction. There are some materials, known as spin liquids that don’t do this. The spins remain disordered at temperatures approaching absolute zero. This is known as the spin liquid phase. In the spin liquid phase, we can probe the materials and find emergent particles which we can measure the properties. It turns out that these particles can values of spin, for example, that are forbidden to all non-emergent particles that we know of.

In our section on magnets we introduced the idea of an order parameter. We can assign a number that indicates the average orientation of spins, which serves as an order parameter. If this number is zero the spins are aligned randomly so that on average there are just as many pointing up as there are pointing down. This is the case with a ferromagnet. It characterises a particular kind of phase transition, a second order one, found in many different types of quantum matter. At a temperature above the critical temperature at which the phase transition happens, the order parameter is zero. Below the critical temperature, the order parameter will take a non-zero value. The reason it does this is in order to minimise a quantity called the free energy. The free energy of a system can be thought of as its useful energy—for example in a car engine, the free energy of the fuel is the energy that will go into turning the wheels, whilst the rest is lost as heat. It’s not immediately obvious why the free energy should be minimised—the fact that it should be is actually a statement of the second law of thermodynamics (it’s also quite a general principle in physics that things should minimise their energy).

In the context of the above, the critical temperature of a phase transition is seen to be the temperature below which the system saves free energy by having the order parameter take a non-zero value. When this happens, there might be different values the order parameter can take which will give the same free energy, and the system must choose a single one. This is spontaneous symmetry breaking. Above the critical temperature of a ferromagnet, flipping all the spins does not change the net direction of spin; there are still as many pointing up as there are down. The symmetry is broken in the ordered phase (below the critical temperature) because flipping the spins in this phase does change the net direction of spins. For example, if all the spins were pointing up, flipping them all means that they then all point down. The symmetry breaking is spontaneous because there is no reason the spins should choose one direction to align in over the other. Describing phase transitions in terms of order parameters is called Landau theory, after the Russian physicist Lev Landau.

#### Landau Theory

Let’s take a look at the maths of Landau theory. The free energy F is a function of the order parameter φ (technically it as an analytic function, but don’t worry about this). Near the critical temperature it can be written as a Taylor expansion in the order parameter—meaning we can express it as a sum of φ to an integer power. And since the order parameter φ is small (it has to be, because it’s zero above the critical temperature), φ to a large power is really small, so we can just pretend it isn’t there and this won’t have a big impact on our maths. We need to multiply each of these terms by a yet to be determined number (which could depend on temperature T, so we write these as α, β etc.

$$F=\alpha(T)\varphi^2+\beta(T)\varphi^4+…$$

Ellipses mean we are ignoring the terms with bigger powers. Why have we not included odd powers? This is due to symmetry, as including an odd power would mean the order parameter would always choose the same value below the critical temperature, which as we discussed above isn’t what happens (try sketching F with and without a φ3 term. You will see the two minima of F don’t have the same free energy. α needs to be negative for this to work).

Along with another Russian physicist, Vitaly Lazarevich Ginzburg, Landau extended this theory to superconductors. Unlike in ferromagnets, there is no obvious candidate for the order parameter (there are no spins choosing a direction). They developed this so-called Ginzburg-Landau theory anyway, and it wasn’t until later that the nature of the order parameter was fully understood. It is in fact interpreted as a macroscopic wave function. Originally in quantum mechanics, wave functions were envisaged as describing a single particle and only being relevant at a subatomic scale. Remarkably a macroscopic wave function not only describes the entire superconductor (a superconductor sample could have as many as 1023 electrons), but the variation of this macroscopic wave function throughout the superconductor leads to phenomena which we can literally see (admittedly only through a microscope as the scale is about 10μm, comparable to the width of a human hair. Contrast this with the scale of an atom which is 0.05nm).

The most notable such effect is the existence of vortices in some superconductors, which were predicted using the Ginzburg-Landau theory before they were discovered. Theoretically these vortices are lines through the superconductor where the order parameter goes to zero, which means they contain a quantised amount of magnetic flux (don’t worry about what this is, the significance is that it’s a macroscopic quantity which is quantised). All of the so called high temperature superconductors (with critical temperatures above 77K, the boiling point of liquid nitrogen) contain vortices, so understanding what they are might help us discover even higher temperature superconductors.

(A brief note on the history of the theory of superconductors—the Ginzburg-Landau theory and the BCS theory discussed in our other section on superconductors were both developed independently in the 1950s, but it was later shown that Ginzburg-Landau theory emerged in the limit of BCS theory.)

#### The Meissner Effect

As well as having no electrical resistance, there is another piece of bizarre physics that happens in superconductors, called the Meissner Effect. Any normal material placed in a magnetic field will have the magnetic field penetrate inside it (although it may be weaker inside the material than it is outside it). A superconductor however will not allow any magnetic field inside it. Even if we start with a material above the superconducting phase transition, with a magnetic field penetrating into it, and cool it down such that it becomes superconducting, the magnetic field will be completely expelled. The flux lines of the magnetic field will pass around the outside of the superconductor instead (see diagram). This effect is responsible for “magnetic levitation” such as in the image below.

A diagram of the Meissner effect. The red arrows represent a magnetic field. No matter which order the material is made superconducting and placed in a magnetic field, we have the same outcome: the magnetic field is completely expelled (bottom of the diagram). This has physical significance as it shows the Meissner effect is thermodynamic (the final state is independent of how we got there).

A superconductor levitates above a magnet due to the Meissner effect. The superconductor has been cooled in liquid nitrogen (at a temperature of 77K)—it is nitrogen which can be seen evaporating off of the sample (it looks like steam). Photo: Steven Blundell and Andrew Boothroyd, Oxford University.

#### The Meissner Effect and the Higgs Boson

Why does the Meissner effect happen? The answer reveals a connection with particle physics—the study of interactions between high energy particles as carried out at institutions like CERN. This is where the now famous Higgs boson was discovered in 2012 [1]. The Higgs boson exists as a result of the Higgs mechanism and Higgs field. When we say field here, we’re referring to a concept from particle physics; for our purposes we can think of a field as an order parameter for the fundamental physics of the whole universe. As the universe cooled down after the Big Bang (which was extremely hot), it underwent a phase transition at a temperature around 1015K [2]. Spontaneous symmetry breaking of the Higgs field occurs below this temperature, just like it does in a superconductor. But something else happened too—the W and Z bosons (fundamental force carriers), which were originally massless, spontaneously gained mass. The exact same thing happens in superconductors below their phase transition, except now it is photons which gain mass. Because photons are normally massless, the electromagnetic force which they carry has an infinite range (light, an electromagnetic wave, will travel forever in a vacuum). Massive photons have a limited range (this should make some intuitive sense), which is very very small in a superconductor, so the magnetic field which is carried by photons cannot penetrate into a superconducting material.

In superconductors this is called the Anderson-Higgs mechanism. A quasiparticle (a particle which can be found in some materials, but not by itself) like the Higgs Boson exists in superconductors too. This is just one of many surprising connections between high energy physics and the physics of quantum matter.

1. CERN webiste, Higgs Boson, https://home.cern/topics/higgs-boson
2. Hyperphysics, Electroweak Unification, http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/unify.html

### Spin Liquids

Magnetic phases of matter are characterised in terms of spin. In most materials, when the temperature is lowered to approaching absolute zero all the spins go from disordered to ordered (all pointing in the same direction). There are some materials that don’t do this: the spins remain disordered at temperatures approaching absolute zero—known as the spin liquid phase. In this phase, we can probe the materials to find emergent particles which we can measure the properties of. It turns out that these particles can have values of spin, for example, that are forbidden to all non-emergent particles that we know of.

The central aim in condensed matter physics is to understand and classify possible phases of matter [1]. The phases of matter you are likely to be familiar with are solids liquids and gasses which are characterised by their arrangement of atoms in space. However not all the materials we come across in nature fall neatly into these three categories and there are other properties we can use to define a phase of matter. Here we will discuss spin liquids which are a phase of matter, defined by spin, that magnetic materials can exist in. It will be useful to have read the articles on spin and magnetism because we will treat the atoms that make up the magnetic material as spin-half particles, which can point up or down (discussed in the spin article).

In a magnetic material at high temperature the spins are thermally disordered and the material is in the paramagnetic phase, which is the magnetic equivalent of a gas. At low temperature, many magnetic materials have ordered arrangements of spin orientations, i.e. most of the spins pointing in the same direction, analogous to the regular positions of atoms in a crystal. If you move the temperature slider on the interactive Ising model you can see the change from an ordered configuration of spins (low temperature) to a disordered configuration of spins (high temperature).

At sufficiently low temperature, we know that everything becomes a solid except for the two isotopes of Helium. The helium atoms cannot form a solid lattice (basically a grid of atoms) because they cannot sit still—there are quantum fluctuations in their positions. This is related to the Heisenberg uncertainty principle which states that we can’t precisely know both the momentum and the position of a particle at the same time. As we get closer to absolute zero, the precision to which we know the momentum of each atom increases, and the precision to which we know the position of each atom decreases. The quantum fluctuations in helium are large and keep the atoms in a liquid phase meaning that a solid lattice cannot form at ordinary pressures, even at the absolute zero of temperature. Just as the two isotopes of Helium don’t become solid at zero temperature, some magnetic materials, known as spin liquids, don’t reach ordered arrangements of spin orientations at zero temperature because large quantum fluctuations allow the materials to maintain a thermally disordered state of spins, even at zero temperature.

We expect some of these spin liquid phases to have what are known as fractionalised excitations. This means that the constituents of the material correlate their motion with one another so strongly that we can no longer describe the behaviour of the material in terms of individual electron spins (we will just call these spins). In effect, the original particles dissolve into a quantum soup and are lost. In their place, when we probe the material we find new emergent particles with quantum numbers corresponding to fractions of the original constituents [1]. The fact that the emergent particles have fractional quantum numbers is quite striking. For elementary and composite particles like electrons and protons, which do not emerge as consequences of complex interactions of many particles, these fractional quantum numbers would be forbidden. For example the spin of every known, non-emergent particle takes one of the values 1/2, 1, 3/2, 2, 5/2… but there are emergent particles whose spins are not in this set [2].

At present, there are only a few fully certified examples of phases of matter that show fractionalisation and the hope is that spin liquids will give many more examples of fractionalisation. To find spin liquid materials, we should avoid systems that favour a limited number of regular, solid-like, spin configurations. What we want instead is geometrical frustration, where each spin wants to be anti parallel to its neighbours, because this is the lowest energy state, but the geometric arrangement of the spins makes this impossible. Quantum fluctuations are more likely to dominate in geometrically frustrated materials, which makes them excellent places to look for spin liquid phases [1].