Macroscopic quantum effects in superconductors and superfluids

Quantisation of magnetic flux in a superconductor

Magnets and electrical currents produce magnetic fields, regions of space where other magnets and electrical wires experience a mechanical force. For a circle of wire in the presence of a magnetic field the magnetic flux is defined as the strength of the magnetic field passing through the circle multiplied by the area of the circle. A law of electromagnetism states that if the field varies with time, then a voltage is produced in the wire with a magnitude that is proportional to the rate at which the magnetic flux through the circle changes. This is the physics behind all electrical motors and electrical generators. In the everyday world magnetic flux can have any value and can be varied continuously by changing the strength of magnetic field. In the quantum world that is not the case. Magnetic flux is quantised.

In 1961, two experimental groups independently reported the first observation of a macroscopic quantum effect, the quantisation of the magnetic flux passing through a superconducting cylinder (Figure 30). One team was Bascom Deaver and William Fairbank and the other Robert Doll and Martin Nabauer. A tall thin cylinder made of tin was placed in a magnetic field and cooled down to a low enough temperature that it entered the superconducting state. The magnetic flux passing through the cylinder was then measured as the magnetic field was varied. The resulting graph has four noteworthy features. First, there are clear steps, showing that the magnetic flux has discrete values. In contrast, in the normal metallic state the graph was a straight line. Secondly, the magnitude of the steps was the same, to within about one per cent, suggesting quantisation of a single unit of magnetic flux. Thirdly, the value of this quantum of magnetic flux was equal to the value of h/2e. Thus, it was completely determined by the two fundamental constants, h and e, Planck’s constant and the charge on an electron, respectively. And fourthly, graphs with the same three features noted above were later observed in other superconducting materials and cylinders. This showed that flux quantisation is independent of details such as the chemical composition and dimensions of the cylinder. This flux quantisation is a macroscopic quantum effect. It is macroscopic because the system is macroscopic, and the magnetic flux is a macroscopic property. It is quantum as and the magnitude of the quantisation is determined by Planck’s constant.

                                                                       (b)

Figure 30. Quantisation of magnetic flux in a superconducting cylinder. (a) A tall thin cylinder of tin was placed in a magnetic field. (b) The graph shows the value of the magnetic flux passing through the cylinder as the magnetic field was varied. Note the step like structure, showing quantisation of the flux.

The quantum of magnetic flux is denoted Φ0 (= h/2e) and has the value 2.067833848...×10−15   tesla (metre)2. This number also determines the scale of quantum interference effects between two superconductors, as we will see shortly. The flux quantum is also relevant to vortices that form when some superconductors are placed in a magnetic field (Figure 20). A persistent electrical current flows around the vortex and the magnetic field penetrates the core of the vortex. It can also be shown, both theoretically and experimentally, that the magnetic flux associated with each vortex is exactly equal to one quantum of flux. Something similar happens in superfluids.

Macroscopic quantum effects in superfluids

When a cylinder containing a fluid is rotated about an axis passing down the centre of the cylinder the fluid will also rotate. The faster the cylinder is rotated the faster the fluid rotates. A physical quantity known as the circulation is proportional to the speed of rotation and the diameter of the cylinder. With a variable speed motor, the rotation speed can be continuously varied and in normal fluids the circulation has continuous values. But not in a superfluid, as shown in a beautiful experiment done by W.F. Vinen in 1961 using liquid 4He. He observed that when the liquid was cooled below the superfluid transition temperature that the circulation could only take on discrete values. Furthermore, these discrete values are multiples of h/M where h is Planck’s constant and M is the mass of one atom of helium. This value was predicted by Lars Onsager in 1949 who identified h/M with the circulation of a single vortex in the superfluid. This is another macroscopic quantum effect.

The quantisation of magnetic flux in superconductors and of circulation in superfluids showed that both superconductors and superfluids can be classified as quantum states of matter. The close similarity of these quantum phenomena, even though superconductivity occurs in solids and superfluidity in liquids. This indicates a deep underlying unity, demonstrated through the study of condensed matter physics. 

This is an extract from Chapter 7, Quantum Matter, in Condensed Matter Physics: A Very Short Introduction.

Rudolph Marcus (1923-2026): theoretical chemical physicist

Rudolph Marcus died last week. He was 102. There is a nice obituary in The New York Times. He was best known for his theory of electron transfer, for which he was the sole recipient of the Nobel Prize in Chemistry in 1992.

Although the theory was proposed for electron transfer in a polar solvent it applies to a wide range of other systems, where two quantum states are coupled to one another and to an environment. One example is for Forster transfer of excitons between molecules, which is central to photosynthesis.

I will give a physics perspective, based on a talk I gave in Slovenia back in 2013. Slides are here.

Basically, Marcus proposed an effective Hamiltonian for two diabatic states and calculated a transition rate in a limit that is relevant to most chemical contexts.

The Hamiltonian can be viewed as the spin-boson model, in which a two-level system is coupled to a bath of harmonic oscillators. 

[The dense book by Weiss on Quantum Dissipative Systems makes the connection explicit in detail. A more accessible treatment may be chapter 16 in the book Chemical Dynamics in Condensed Phases by Nitzan.]

The Hamiltonian is 

This defines a spectral density, which is important for quantum decoherence, but not so much here, except it defines a timescale that determines the classical limit, which Marcus assumed.

This can be used to define a quantity central to Marcus' theory, the reorganisation energy.

The transition rate between the two quantum states is given by

I consider this to be one of the most important equations in chemical physics, particularly for the understanding and design of functional materials.

Aside. Much of this is equivalent to Holstein's 1959 treatment of incoherent polaron transport (see Mahan, Many-body physics).

A key experiment by John R. Miller, Lidia T. Calcaterra and Gerhard L. Closs in 1984 showed how good the theory was and how its predictions were counter-intuitive. The authors considered a family of molecules that allowed them to tune epsilon, the energy difference between the two electronic states. [On the graph below epsilon = - Delta G].

The vertical scale is the reaction rate on a logarithmic scale. It varies by four orders of magnitude.

What is surprising? As stated at the Nobel prize award ceremony:

"The quadratic equation predicts that electron transfer reactions will occur more slowly the larger the driving force of the reaction is. This phenomenon received its own name, “the inverted region.” To a chemist, the phenomenon is just as unexpected as when a skier finds himself gliding more slowly down a slope the steeper it is."

The theory implies an important design principle for functional materials: if optimising functionality means maximising the reaction rate, then tune the energy difference epsilon to equal the reorganisation energy E_R.

The theory illustrates two important aspects of emergence: effective theories and universality. Many different systems can be described by the same theory. The environment may involve many degrees of freedom and its coupling to the system is characterised by many parameters (the M_alpha above). However, only one parameter matters, the reorganisation energy.

For Australians, there is some ambivalence about the way Marcus' name is often solely associated with electron transfer theory. We often refer to it as Marcus-Hush or Hush-Marcus theory because Noel Hush did similar work around the same time. Some of the history is recounted here by Ian Rae and Jeff Reimers. There are also subtle debates about whether the electron transfer is adiabatic or non-adiabatic.

My only personal interaction with Marcus was in 2011 when I visited the chemistry department at Caltech. Marcus kindly took me to lunch at the faculty club, along with his research group. Then he was 84 years old. He kept publishing papers until he died.

Philosophical perspectives on the emergence of molecular structure

 In philosophical discussions of emergence and reductionism in chemistry, molecular structure has received significant attention and debate. Sometimes molecular structure is used to justify strong emergence, i.e., that molecular cannot be predicted, even in principle, solely from quantum theory.

Primas was one of the first to claim that molecular structure could not be reduced to quantum physics. Consider the following statements.

“From a physical point of view the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one…

[the structure of benzene] does not exist in a full quantum-theoretical description since electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations. The concept of molecular structure does not follow from first principles - all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence. In a quantum theoretical description the molecular shape emerges by abstracting from the actually existing Einstein-Podolsky–Rosen correlations between the electrons and the nuclei. Historically, the structure concept has been introduced into quantum chemistry by the so called Born-Oppenheimer approximation. But this terminology is misleading since the main issue is not an approximation, but the breaking of a holistic symmetry.”

I have italicised claims that are contestable and that I will discuss further below.

Cartwright has given philosophical arguments as to why chemistry cannot be reduced to physics. In this context, she claims (pp. 106-7)

“The typical method of quantum-mechanical treatment of molecules begins with the Born–Oppenheimer approximation…

This approximation treats the atomic nucleus as a classical particle. But this fundamentally violates quantum mechanics which, following the Heisenberg uncertainty principle, maintains that we cannot have a simultaneous assignment of fixed positions and fixed momenta. The approximations that provide the reduction violate the very theory that the chemistry is being reduced to… the success of quantum chemistry relies fundamentally on assumptions that belong to classical chemistry.” 

This claim that the BOA violates the Heisenberg uncertainty principle was rebutted in an earlier post and discussed in more detail by Huggett et al. Nevertheless, Lombardi et al. are not satisfied by the rebuttal.

Hendry claimed molecular structure is evidence of strong emergence and/or downward causation. In a similar spirit, Ellis and Drossel argued that crystal structures in solid state physics are evidence of strong emergence. The arguments of Hendry have been criticised by Seifert. The arguments centre around the fact that the molecular structure is a classical entity and a concept that is imposed, whereas a logically self-consistent approach would treat both electrons and nuclei quantum mechanically. It is claimed that the existence of molecular structures is assumed and not derived in quantum chemistry calculations as they assume the Born-Oppenheimer approximation (BOA). 

Scerri responded to these arguments claiming that chemistry (particularly the concept of molecular structure) is irreducible to quantum physics. He claimed these arguments are not valid because they misunderstand the role of the BOA. It does not violate the uncertainty principle and in practice chemists can and do perform non-BOA approximations. 

Fortin et al. rejected the view that decoherence can explain isomerism, as decoherence does not resolve issues associated with the quantum measurement problem. 

Franklin and Seifert claim “the problem of molecular structure just is the quantum measurement problem.” This is debatable. Most molecular structures can be understood in terms of the nuclear probability density having a unique maximum and decoherence then is not relevant. Decoherence and the collapse of the nuclear wavefunction are only relevant in systems such as ammonia and stereoisomers in which there are molecular structures with equal energy and separated by high energy barriers.

I now respond to some of the contestable claims of Primas.

“electrons and nuclei are entangled by Einstein-Podolsky–Rosen correlations”

It is possible to quantify and calculate the amount of entanglement between electrons and nuclei in a specific quantum state. In an EPR state the entanglement is maximal. In the BOA wavefunction entanglement is present, but is absent in the crude BOA. The entanglement has been estimated in benzene and is very small. The only molecules where the entanglement may be significant are those, such as isomers, where there are two degenerate molecular structures and the overlap of the associated nuclear wavefunctions is small (i.e., the tunnel splitting is small). But then, in most chemical situations decoherence will wash out this entanglement.

“all molecules with the same empirical formula have the same Schrödinger equation, so that, at this level, the shape of a molecule as the main feature of molecular chemistry is simply not in evidence.”

This is the problem of isomers. It is resolved because isomers are present in the solution to the Schrödinger equation, as I argued earlier.

“the crucial point of a Born–Oppenheimer description is not a simplification of the mathematical problem, but the replacement of the basic theory by a related but qualitatively new one… the main issue is not an approximation, but the breaking of a holistic symmetry.”

This seems subjective to me. I see the BOA as just a well-justified approximation. The electrons and nuclei are treated differently because they are. They have vastly different masses and this affects how they can be treated in any solution to the full Hamiltonian. Nevertheless, the BOA is a quantum theory and the nuclei are described by a wavefunction.

As discussed earlier, for small molecules in practise it is no longer necessary to use the BOA and the electrons and nuclei can be treated on an equal footing. Molecular structure is present in solutions to the full Schrödinger equation.

I wonder if the objection to use of the BOA is any different to the use of approximations in other theories? For example, consider theoretical treatments of the motion of planets in the solar system. The effects of all the planets are not treated on an equal footing. The effect of other planets on a planet of interest are treated perturbatively.

In conclusion, the arguments that molecular structure is evidence of strong emergence are weak. Some confusion may partly arise from misinterpreting the significance of the following valid observations.

i. Molecular structures were first conjectured before quantum theory was proposed.

ii. The BOA was proposed to explain molecular structure from quantum theory.

iii. Today, almost all calculations of molecular structure assume BOA.

iv. Chemists talk about molecular structures largely in classical not quantum terms.

However, the scientific reality is that for small molecules their structure, can be understood, described, and calculated in purely quantum terms. Yet, that understanding provides a strong justification for the validity of the BOA and for the convenience of using classical language to describe molecular structure.

I stress that the weakness of the arguments for the strong emergence of molecular structure, does not mean that an emergent perspective is not relevant to chemistry. Molecular structure is emergent, when defined in terms of novelty. This leads to effective theories defined in terms of potential energy surfaces. Furthermore, as the next section argues chemistry exhibits novel properties, concepts, and theories that are not present in physics.

This post is extracted from Section 15, of my review article "Emergence: from physics to biology, sociology, and computer science."

A new version of my review article on emergence

On the arXiv, I have posted a new version of my review article, Emergence: from physics to biology, sociology, and computer science.

I have added expanded sections on molecular structure, quantitative measures of causal emergence, and biological evolution.

There are also many minor additions and corrections. I hope the hyperlinked Table of Contents is helpful.

I welcome feedback and suggestions. I am sure there is much more to do.

What is a quasiparticle?

 An example of emergent entities in condensed matter physics are quasiparticles. The concept can be described with the following analogue. When a horse gallops through the desert it stirs up a dust cloud that travels with it. The motion of the horse cannot be separated from the accompanying dust cloud. They act as one entity. Similarly, in a system consisting of many interacting particles, when one particle moves it carries with it a “cloud” of other particles. This composite entity is referred to as a quasiparticle. It turns out to be easiest to understand the whole system of particles in terms of the quasiparticles rather than in terms of the individual particles.

Quasiparticles are composite objects. Like the constituent particles in the system, quasiparticles each have properties such as charge, mass, and spin. However, these properties of a single quasiparticle may be different from those of the individual particles of which it is constituted. An example is holes in semiconductors; the many electrons in a crystal act collectively to produce a hole (the absence of a single electron), a quasiparticle with the opposite charge to that of a single electron. A more striking example is for the fractional quantum Hall states; the charge of the quasiparticles can be a fraction of the charge on a single electron.

Different musical instruments produce distinct sounds because they are made of different materials, and they vibrate in different ways in response to different stimuli. In general, the vibrations of a medium reflect something about the medium itself. Chapter 3 discussed how in a crystal the number of distinct ways that sound can travel through a crystal reflects the symmetry and ordering of the atoms in the crystal.

When the skin on a drum is hit by a drumstick the skin vibrates at particular frequencies. Similarly, a state of matter responds to external stimuli such as light, sound or heat, by oscillating at particular frequencies. These vibrations travel through the matter as waves. The properties of these waves reflect the particular order present in the state of matter. Here is a specific example. When a neutron with a particular energy and momentum is absorbed by a ferromagnetic crystal the interaction of the magnetism of the neutron with that of the atoms in the crystal produces a collective oscillation of the magnetic state of the crystal in time and space. Known as a spin wave, this oscillation has a particular frequency and wavelength. In quantum theory, waves and particles are equivalent to one another. The energy and momentum of a particle are related to the waves’ frequency and wavelength, respectively. Particles equivalent to light waves are known as photons; particulate equivalents of sound waves are known as phonons. And similarly, the particle equivalent of a spin wave is known as a magnon. These collective excitations are quasiparticles. Whereas the particles in a system may interact strongly with one another, the quasiparticles may interact weakly with one another. This makes analysis and understanding of the relevant theories more tractable.

The quasiparticle concept is a powerful theoretical tool in condensed matter physics. It is the basis for the construction of models that enable emergent phenomena to be understood in terms of the effective interactions between components such as quasiparticles, rather than in terms of the actual constituent particles and their interactions. This approach requires profound physical insight in order to discern what the truly essential components of a system are. Lev Landau was one of the first theoretical physicists to take this approach, introducing the idea of quasiparticles in his theories of superfluidity in 4He and of liquid 3He. This approach was also central to the BCS theory of superconductivity. Phil Anderson was also a master of the approach, using intuition to propose models that were simple enough for analysis and yet complex enough to capture the essential physics associated with a particular state of matter. In 1977 he was awarded the Nobel Prize for work using this approach to understand two specific systems: magnetic atoms in metals and the motion of electrons in materials that are not crystals and are dirty in the sense of containing many impurities.

An extract from Chapter 9, "Emergence: More is Different", in Condensed Matter Physics, A Very Short Introduction

A more detailed and technical discussion is in Section 8.2 of my review article on emergence.

Biological evolution and emergence

 The theory of evolution explains the origin of biological diversity and levels of similarity between species. A characteristic of emergence is that many iterations of a simple law (natural selection of the fittest to reproduce) can produce novel, diverse and rich structures. In biological evolution many generations in a population can produce new traits and species. 

Many of the most debated issues about evolution relate to the different characteristics of emergence and are briefly discussed below.

Scales

Central to emergence are the ideas of “many” and of scales. The former can take two forms: a system composed of many interacting components, or a system that undergoes many iterations according to a rule that is repeated many times. For evolution, both forms of “many” are relevant and have several dimensions. Evolution occurs in a population, i.e., a community of many members of a species living in a specific environment. Each member of the population has a specific genotype (many genes), which largely determines biological characteristics, from proteins to organs, defined as the phenotype. The environment also consists of many interacting species. Natural selection can act at multiple levels: on genes, cells, organisms, species, and groups of species.

Microscopic and macroscopic scales can also manifest in different ways. In terms of length, the micro- and macro- scales can be defined in terms of genotypes and phenotypes, respectively. In terms of time, microevolution and macroevolution roughly correspond to directly observable timescales and geological timescales, respectively. They are associated with the emergence of new traits within a species and new species, respectively.

Novelty

Development of new traits and species occurs over many generations, due to the repetition of the rule of natural selection.

Evolution theory uses concepts such as natural selection, survival of the fittest, niches, and hierarchical trees, that are not present in chemistry and physics. 

Connecting micro- and macro- properties

As for other systems, this is one of the great challenges of emergence. Genotypes and phenotypes are extremely well characterised. Genotype-phenotype maps seek to connect these micro- and macro- levels. A detailed understanding of how microevolution leads to macroevolution is a challenge.

Discontinuities

In microevolution, new traits occur within a species due to (continuous) adaptation to the environment. In contrast, in macroevolution, new organs and species can occur suddenly (at least on geological timescales). An example is the Cambrian explosion of new life forms. Extinctions can also represent discontinuities.

Evolution of a population occurs in response to changes in an environment. New traits, new species, and extinctions can be viewed as qualitative changes due to quantitative changes. For example, small changes in the oxygen concentration in the atmosphere is one (among many) hypotheses for the cause of the Cambrian explosion.

Using techniques from statistical physics, the transition of a species from survival to extinction can be viewed as a non-equilibrium phase transition to an absorbing state. The order parameter is the population and a toy model is directed population.1

Diversity with limitations

All species are based on the same biochemistry of DNA and proteins. Yet from these same building blocks there is an incredible diversity: more than 8 million distinct species, including more than 10,000 species of birds and more than 15,000 species of ants. Darwin said nature produces “endless forms most beautiful.”

But there are limitations. For example, the number of species with more than one head, brain, heart, or liver is limited. There are many more genotypes than phenotypes. 

The dominant view is that evolution is driven by random genetic mutations. Debates have arisen about how much evolution is limited (constrained) by morphology and environment.

Ball stated: (p. 332)

“convergent evolution is often regarded as a sign that certain shapes or structures are ideal adaptations to particular environments for physical reasons: wings consisting of flat, thin membranes are best for flying, torpedo-shaped bodies a streamlined for efficient swimming, and so on… There is a tendency in evolutionary biology to regard natural selection as a process with an infinite palette: anything is possible so long as it doesn't break the laws of physics. But the laws of physics might impose more constraint than that, precisely because biology uses rather than merely suffers them.”

Universality

Not all mutations produce a change in phenotype. There are neutral mutations. There are many more genotypes than phenotypes. In other words, genotype-phenotype maps are many-to-one.

Species that are unrelated or distantly related (in the tree of life) sometimes have traits or behaviours that are similar. Convergent evolution is the hypothesis that natural selection produced the same outcome in a different context. 

Modularity at the mesoscale

The economist Simon pointed out that evolution can occur on much faster time scales than might be expected because of modularity. According to Clune et al.

“A long-standing, open question in biology is how populations are capable of rapidly adapting to novel environments, a trait called evolvability [1]. A major contributor to evolvability is the fact that many biological entities are modular, especially the many biological processes and structures that can be modelled as networks, such as metabolic pathways, gene regulation, protein interactions and animal brains [1–7].”

Ball highlighted how domains in proteins provide functional modules that evolution uses: (pp. 174-5)

“the evolution of metazoan proteins is not so much a slow affair of letting random genetic mutations change one amino acid for another and seeing what effect it produces. Rather, it constitutes a reshuffling of already functional modules to produce multidomain molecules with new potential - a strategy much more likely to yield successful results…  the “unit” of molecular evolution here is not really the base pair of DNA or the amino acid or protein, or the gene itself, by the peers at a scale intermediate between the two: the module of a domain. It seems that this shuffling, rather than the slow mutation of primary base sequences, is what has driven the evolution of animals.”

Johnston et al. considered an algorithmic picture of evolution that 

“suggests that symmetric structures preferentially arise not just due to natural selection but also because they require less specific information to encode and are therefore much more likely to appear as phenotypic variation through random mutations… many genotype–phenotype maps are exponentially biased toward phenotypes with low descriptional complexity. A preference for symmetry is a special case of this bias… Lower descriptional complexity also correlates with higher mutational robustness, which may aid the evolution of complex modular assemblies of multiple components.”

Self-organisation

Complex biological structures, from proteins to organisms, have formed spontaneously due to evolution over millions of years. Their intricacy and functionality have led to claims of purpose and design. However, this is argued to be an “apparent” design, just like an economy whose self-organisation appears “as if” it is guided by an “invisible hand.”

Kauffman claimed that self-organisation is as important as natural selection in driving evolution.

Unpredictability

A contested question about evolution is the role of contingency (historical accidents) and whether the evolution of complex life forms, particularly humans, was an accident of history or inevitable.

Irreducibility

Until recently, evolutionary biology has been dominated by a reductionist gene-centric view, popularised by Dawkins. However, recent discussions about systems biology, evo-devo, and epigenetics have questioned this view. Some characterise these alternative views as a form of structuralism.

Complexity

An algorithmic picture of evolution suggests that simplicity spontaneously emerges as many genotype-phenotype maps may be biased towards phenotypes with low descriptional complexity. 

Toy models

An earlier post discussed the key role that toy models, such as “bean bag” genetics, have played in evolutionary theory.

Cross-fertilisation of fields

Ideas from evolution have stimulated the development of genetic algorithms in computer science.

Drossel has reviewed connections between evolution and statistical physics, including a wide range of toy models. Examples include spin glass models that give rise to rugged landscapes for fitness and can describe hierarchical structures, comparable to Darwin’s tree of life. Goldenfeld and Woese argued that evolution can be viewed as a collective phenomenon far from equilibrium. The toy model central to their discussion is directed percolation.

I welcome comments. My knowledge of biology is limited, and scientifically some the ideas above can be contentious. (Never mind philosophy, politics, or theology!)

Critical points in condensed matter illuminate universality

Every person is unique. No two people are identical. We differ in physical appearance, personality, fingerprints, heartbeat, gait, and DNA. Such differences are used to identify criminals and in video surveillance of citizens by nation states. Yet in other ways all humans are the same. We all have brains, hearts, and lungs. All our bodies use the same biochemistry to stay alive: whether to breathe oxygen, digest food, or fight infections. On some level we have common aspirations: to survive, to be loved, to be happy, and to find meaning and purpose. Yet these aspirations find many expressions. Humans have certain universal qualities and properties, yet at a finer level of detail there is a particularity of each of these properties. They are at one level the same but are not the same at another level. 

All academic disciplines search for universals; they develop categories, concepts, and theories that overarch particularities. Biologists classify species of plants and animals and types of cells and viruses. All biological systems use the same molecules (DNA, RNA, and proteins) and chemical reactions. The same genetic code uses the information encoded in a piece of DNA to make proteins with specific functions. Anthropologists study the immense diversity of human cultures and societies. This diversity can be described in terms of universal concepts such as kinship, family, ritual, community, economics, law, and morality. Linguists study the common structures and grammars of the thousands of different human languages.  Although the world we live in is diverse, disciplines have each discovered some universals.

Condensed matter physicists study diverse states of matter and the transitions between them. A surprising discovery is that there is much more universality than might be expected, particularly given the chemical and structural diversity of materials. In this chapter, I will discuss the nature of this universality, how it emerges, and the length scales associated with transitions between different states of matter. Landau’s great insight was that many of the chemical and structural details of materials are irrelevant to understanding phase transitions. Furthermore, a precise classification of different types of phase transitions, into what are called universality classes, can be made. For example, superconducting, superfluid, and a subset of magnetic transitions are in the same class. The determinants of the universality classes are the symmetry of the state and the spatial dimensionality of the system. None of the other details matter.

Many phase diagrams (such as the Figure above) include a critical point, located at the end of a boundary between two different states of matter. A common example is the critical point that occurs at a specific temperature and pressure for a transition between a liquid and a gas. Understanding the physical properties of a material close to its critical point was a great challenge for theoretical physics, lasting a hundred years, and was only solved in the 1970s. The powerful theoretical ideas and techniques that were developed provide a quantitative way to relate the properties of a system at one length scale to properties at a different length scale. These techniques also have application to a wide range of other problems and fields including elementary particle physics, chaos theory, fractals, polymers, and machine learning. New insights were gained into universality and emergent phenomena.

An extract from "The Critical Point," chapter 6, Condensed Matter Physics: A Very Short Introduction