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

Emergence and continuous phase transitions in flatland

In two dimensions the phase transition that occurs for superfluids, superconductors, and planar classical magnets is qualitatively different from those which occur in higher  dimensions. Known as the Berezinskii-Kosterlitz-Thouless (BKT) transition, it involves several unique emergent phenomena. 

Novelty

The low-temperature state does not exhibit long-range-order or spontaneous symmetry breaking. Instead, the order parameter has power-law correlations, below a temperature T_BKT. Hence, it is qualitatively different from the high-temperature disordered state, which has correlations that decay exponentially. It is a distinct state of matter, with properties that are intermediate between the low- and high-temperature states normally associated with phase transitions. The power law correlations are similar to those at a conventional critical point, which decay in powers of the critical exponent eta. However, the BKT phase diagram can be viewed as having a line of critical points, consisting of all the temperatures below TBKT. Along this line, the critical exponent eta varies continuously with a value that depends on interaction strength. In contrast, at conventional critical points, eta has a fixed value determined by the universality class.

Sometimes it is stated that the low-temperature state has topological order, but I am not really sure what that means. Has this been made precise somewhere? 

The mechanism of the phase transition is qualitatively different from that for conventional phase transitions. It is driven by the unbinding of vortex and anti-vortex pairs by thermal fluctuations. In contrast, conventional phase transitions are driven by thermal fluctuations in the magnitude of the order parameter.

Discontinuity

There is a discontinuity in the stiffness of the order parameter at this transition temperature.

Unlike for conventional phase transitions the specific heat capacity is a continuous function of temperature. This is why the BKT transition is sometimes referred to as a continuous transition.

Toy model

A classical Heisenberg model for a planar spin, also known as the XY model, captures the essential physics.

Modularity at the mesoscale

The quasiparticles of the system that are relevant to understanding the transition are not magnons (for magnets) or phonons (for superfluids), but vortices, i.e., topological defects.  

These entities are usually on the mesoscale, i.e, there size is much larger than the lattice spacing. The relevant effective theory is not a non-linear sigma model. Thermal excitation of vortex-antivortex pairs determines the temperature dependence of physical properties and the transition at T_BKT.  There is an effective interaction between a vortex and an anti-vortex that is attractive and a logarithmic function of their spatial separation, analogous to a two-dimensional Coulomb gas. 

Universality

The BKT transition occurs in diverse two-dimensional models and materials including superfluids, superconductors, ferromagnets, arrays of Josephson junctions, and the Coulomb gas. The discontinuity in the order parameter stiffness at T_BKT has a universal value. 

The renormalisation group (RG) equations associated with the transition are the same as those of a multitude of other systems. The classical two-dimensional systems include the Coulomb gas, Villain model, Z_n model for large n, solid-on-solid model, eight vertex model, and the Ashkin-Teller model. They also apply to classical Ising chain with 1/r^2 interactions. Aside: Phil Anderson discovered these RG equations for the Ising chain before BKT derived their own equations.

Quantum models with the same RG equations include the anisotropic Kondo model, spin boson model, XXZ antiferromagnetic Heisenberg spin chain, and the sine-Gordon quantum field theory in 1+1 dimensions. In other words, all these models are in the same universality class.

Singularity

The correlation length of the order parameter is a non-analytic function of the temperature. 

This is related to the non-perturbative nature of the corresponding quantum models at their critical point. 

Personal aside: I first encountered this singularity (long ago) when working on a spin-Peierls model with quantum phonons.

Two-dimensional crystals

Similar physics is relevant to the solidification of two-dimensional liquids. However, the relevant toy model is not the classical XY model as one needs to include the effect of the discrete rotational symmetry of the lattice of the solid. The low-temperature state exhibits discrete rotational, but not spatial, symmetry breaking, with power-law spatial correlations. This state does not directly melt into a liquid, but into a distinct state of matter, the hexatic phase. It has short-range spatial order and quasi-long-range orientational (sixfold) order. The phase transitions are driven by topological defects, disclinations and dislocations.

Predictability

The BKLT transition, the quasi-ordered low-temperature state, and the hexatic phase were all predicted theoretically before they were observed experimentally. This is unusual for emergent phenomena but shows that unpredictability is not equivalent to novelty.

Steven Weinberg's radical change of mind

What is a fundamental theory? As we go to smaller and smaller distances and higher energies we keep finding new entities: atoms, electrons, nuclei, neutrons, protons, quarks, gluons, ...When will it stop?

If we look at a theory, such as a quantum field theory, at a particular energy and length scale, there may be hints that something is going on, such as the existence of new entities, at higher energies. One way to approach this problem is through the renormalisation group and to look at how coupling constants behave as the energy increases. If they start to blow up (diverge) is that a hint of something? But, this requires starting with a renormalisable theory...

An alternative approach is to start with an effective theory that one assumes [hypothesises] is valid at some limited energy scale. This goes against a previous dogma that one should only study renormalisable theories. Amongst elementary particle theorists, led by Steven Weinberg, there was a significant shift in perspective in the 1970s.

In a paper published in 2016, Effective field theory, past and future, Steven Weinberg reflected on how he changed his mind about renormalisability being a fundamental requirement for quantum field theories and how he came to the view that the Standard Model should be viewed as an effective field theory. Here are some quotes from the article. He first describes how in the 1960s he developed a field theory to describe the interactions of nucleons and pions.

"During this whole period, effective field theories appeared as only a device for more easily reproducing the results of current algebra. It was difficult to take them seriously as dynamical theories, because the derivative couplings that made them useful in the lowest order of perturbation theory also made them nonrenormalizable, thus apparently closing off the possibility of using these theories in higher order. 

My thinking about this began to change in 1976. I was invited to give a series of lectures at Erice that summer, and took the opportunity to learn the theory of critical phenomena by giving lectures about it. In preparing these lectures, I was struck by Kenneth Wilson’s device of “integrating out” short-distance degrees of freedom by introducing a variable ultraviolet cutoff, ...

Non-renormalizable theories, I realized, are just as renormalizable as renormalizable theories.

For me in 1979, the answer involved a radical reconsideration of the nature of quantum field theory.

The advent of effective field theories generated changes in point of view and suggested new techniques of calculation that propagated out to numerous areas of physics, some quite far removed from particle physics. Notable here is the use of the power-counting arguments of effective field theory to justify the approximations made in the BCS theory of superconductivity. Instead of counting powers of small momenta, one must count powers of the departures of momenta from the Fermi surface. Also, general features of theories of inflation have been clarified by re-casting these theories as effective field theories of the inflaton and gravitational fields. 

Perhaps the most important lesson from chiral dynamics was that we should keep an open mind about renormalizability. The renormalizable Standard Model of elementary particles may itself be just the first term in an effective field theory that contains every possible interaction allowed by Lorentz invariance and the SU (3) × SU (2) × U (1) gauge symmetry, only with the non-renormalizable terms suppressed by negative powers of some very large mass M...

... we should not despair of applying quantum field theory to gravitation just because there is no renormalizable theory of the metric tensor that is invariant under general coordinate transformations. It increasingly seems apparent that the Einstein–Hilbert Lagrangian √gR is just the least suppressed term in the Lagrangian of an effective field theory containing every possible generally covariant function of the metric and its derivatives...

it is usually assumed that in the quantum theory of gravitation, when Λ reaches some very high energy, of the order of 10^15 to 10^18 GeV, the appropriate degrees of freedom are no longer the metric and the Standard Model fields, but something very different, perhaps strings...

But maybe not..."

In 2021 Weinberg gave a talk, with a similar point of view, which inaugurated an international seminar series [online during covid-19]. 

In response to that talk, Peter Woit has a blog post where he objects to Weinberg's point of view that the Standard Model is "just" an effective theory, only valid at low energies.

Reviews of Modern Physics recently published a review that discussed how Weinberg's perspective is worked out in detail.

The standard model effective field theory at work

Gino Isidori, Felix Wilsch, and Daniel Wyler

The discussion above fits naturally with an emergentist perspective: reality is stratified. Effective theories at one strata may have singularities around boundaries between strata, and new entities emerge, both physically and theoretically, as one moves to the next higher or lower strata.

Why do deep learning algorithms work so well?

I am interested in analogues between cognitive science and artificial intelligence. Emergent phenomena occur in both, there have been some fruitful cross-fertilisation of ideas, and the extent of the analogues is relevant to debates on fundamental questions concerning human consciousness.

Given my general ignorance and confusion on some of the basics of neural networks, AI, and deep learning, I am looking for useful and understandable resources.

Related questions are explored in a nice informative article from 2017 in Quanta magazine, New Theory Cracks Open the Black Box of Deep Learning by Natalie Wolchover.

Like a brain, a deep neural network has layers of neurons — artificial ones that are figments of computer memory. When a neuron fires, it sends signals to connected neurons in the layer above. During deep learning, connections in the network are strengthened or weakened as needed to make the system better at sending signals from input data — the pixels of a photo of a dog, for instance — up through the layers to neurons associated with the right high-level concepts, such as “dog.” 

After a deep neural network has “learned” from thousands of sample dog photos, it can identify dogs in new photos as accurately as people can. The magic leap from special cases to general concepts during learning gives deep neural networks their power, just as it underlies human reasoning, creativity and the other faculties collectively termed “intelligence.” 

Experts wonder what it is about deep learning that enables generalization — and to what extent brains apprehend reality in the same way.

The article describes work by Naftali Tishby and collaborators that provides some insight into why deep learning methods work so well. This was first described in purely theoretical terms in a 2000 preprint

The information bottleneck method, Naftali Tishby, Fernando C. Pereira, William Bialek 

The idea is that a network rids noisy input data of extraneous details as if by squeezing the information through a bottleneck, retaining only the features most relevant to general concepts.

Tishby was stimulated in new directions in

2014 after reading a surprising paper by the physicists David Schwab and Pankaj Mehta

 An exact mapping between the Variational Renormalization Group and Deep Learning 

[They] discovered that a deep-learning algorithm invented by Geoffrey Hinton called the “deep belief net” works, in a particular case, exactly like renormalization [group methods in statistical physics... When they]. applied the deep belief net to a model of a magnet at its “critical point,” where the system is fractal, or self-similar at every scale, they found that the network automatically used the renormalization-like procedure to discover the model’s state. 

Although this connection was a valuable new insight, the specific case of a scale-free system, is not relevant to many deep learning situations.

Tishby and Ravid Shwartz-Ziv discovered that 

Over the course of training, common patterns in the training data become reflected in the strengths of the connections, and the network becomes expert at correctly labeling the data, such as by recognizing a dog, a word, or a 1.

...layer by layer, the networks converged to the information bottleneck theoretical bound: a theoretical limit derived in Tishby, Pereira and Bialek’s original paper that represents the absolute best the system can do at extracting relevant information. At the bound, the network has compressed the input as much as possible without sacrificing the ability to accurately predict its label...

...deep learning proceeds in two phases: a short “fitting” phase, during which the network learns to label its training data, and a much longer “compression” phase, during which it becomes good at generalization, as measured by its performance at labeling new test data.

What these new discoveries teach us about the relationship between learning in humans and in machines is contentious and explored briefly in the article. Although neural nets were inspired by the structure of the human brain the connection with the neural nets used today is tenuous.

The mystery of how brains sift signals from our senses and elevate them to the level of our conscious awareness drove much of the early interest in deep neural networks among AI pioneers, who hoped to reverse-engineer the brain’s learning rules. AI practitioners have since largely abandoned that path in the mad dash for technological progress, instead slapping on bells and whistles that boost performance with little regard for biological plausibility.

Philosophers of science on which theories are fundamental

What is real? What is true? These big questions are central to philosophy and issues in the philosophy of science.

Emergent properties of complex systems raise similar philosophical questions such as  "What is fundamental?" and "Are quasiparticles real?".

Robert Batterman is a philosopher of science who is the author of the book,

The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence

In 2017 Batterman wrote an article in an edition of the Journal of Statitiscal Physics that was in memory of Leo Kadanoff. 

Philosophical Implications of Kadanoff’s Work on the Renormalization Group

Below I reproduce some of the text as it provides a helpful (and disturbing) summary of how the philosophy of science has evolved.

There are very few natural philosophers anymore. The fields of philosophy and science parted company at the end of [the nineteenth] century. Philosophers more and more began to turn toward the disciplines of logic and the analysis of language, and their examination of the enterprise of science began to follow a different, less-engaged-with-scientific-detail, direction. They began to try to determine the logic and structure of scientific theorizing in a way that was much more arm-chair and much less concerned with details about individual theories. The aim was to construct or reconstruct the proper logical structures of scientific explanation, confirmation, and theory choice. The philosophical reconstructions were, by and large, designed to fit all empirical science. For example, an explanation in physics should share the same general (logical) form as explanations in biology, chemistry, or sociology. 

I find this problematic because how physicists and biologists do science and the knowledge that they produce is quite different. In fact, similar differences exist between elementary particle physics and condensed matter physics. That also applies to Batterman's next claim.

I think it is fair to say that from a philosophy of science point of view, physical theories are supposed to reflect our best attempts to understand nature. Philosophers are also enamored with the idea that theories have a certain logical structure—they can be written down in some kind of axiomatic form from which, given certain inputs, various features of physical systems (future states, e.g.) can be derived using logic and reasonably straightforward mathematics.

Furthermore, philosophers often distinguish fundamental from nonfundamental (or “phenomenological”) theories. This latter distinction presupposes the idea that fundamental theories are the ones that tell us really what nature is actually like at “bottom.” These presumably include, quantum theory, quantum field theory, maybe a theory of quantum gravity, etc.

In contrast, Bob Laughlin, argues that certain emergent properties are exact [such as quantisation of magnetic flux in a superconductor, hydrodynamics, sound waves] and so they are more fundamental than microscopic theories. [A Different Universe, pp. 36-40].

Batterman continues

Nonfundamental theories such as thermodynamics, continuum mechanics, and fluid dynamics, on the other hand, while pragmatically useful, are in a certain sense (exactly what sense is a matter of serious contention) superfluous. We could, in principle, solve problems involving the elastic bending of beams by starting from the fundamental atomic and subatomic theories of the constituents of the beam.

Nonfundamental theories don’t get nature right. Steel beams are not really the continua whose bending behaviors are described by the Navier–Cauchy equations. Gases are not continuous blobs of stuff. The important theories, according to many philosophers and, I believe, according to many physicists, are those that get the ontology right. In part, the (often unarticulated) reason for preferring fundamental theories over phenomenological theories is a realist presupposition that physical theories must accurately describe the world the way the world really is. 

 Perhaps the view that atoms are real but solids are not is reflected by Bertrand Russell in the opening paragraph of his book, The ABC of Atoms, published in 1923 and intended for popular audiences.

Phenomenological theories are often good for calculating, but they don’t accurately describe the world and so must, in a sense, play second fiddle to their fundamental partners.

This is also contentious. Thermodynamics, elasticity theory, and fluid dynamics are perfectly accurate and never wrong within their domain of validity. Many courses and texts on thermodynamics begin with the following quote from Einstein.

 A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.

I should stress that Batterman is not agreeing with or promoting the views I have questioned above. Rather, he is trying to characterise what many philosophers believe.

Fifty years ago: three big discoveries in condensed matter

For the marketing plan for my Very Short Introduction, I was recently asked whether there were any significant anniversaries happening in condensed matter physics (and associated conferences). This is not something I normally think about.

I realised that fifty years ago there were three big discoveries. All eventually led to Nobel Prizes. Each discovery had a profound effect on the formation of condensed matter as a distinct discipline built around a few unifying concepts. At the time the discoveries and ideas appeared quite independent, but there are deep connections between them.

Renormalisation group and critical phenomena

In 1971 Ken Wilson published two papers  [PRB 4, 3174, and PRB 4, 3184] laying the foundations, followed by two PRLs in 1972, including one with the provocative title, Critical Exponents in 3.99 Dimensions. 

Wilson received the Nobel Prize in 1982. This work had many implications and applications. 

Explained universality in critical phenomena.

Highlighted how spatial dimensionality changes physics.

Illustrates why effective Hamiltonians work (so well).

Showed the power of quantum field theory techniques.

Defined concepts of scaling and fixed points.

Superfluidity in liquid 3He

In 1972,  Osheroff, Richardson, and Lee reported new phase transitions in liquid/solid 3He. Tony Leggett identified these transitions as due a superfluid phases and also identified the order parameters. The experimentalists shared the Nobel Prize in 1996 and Leggett in 2003. The discovery was significant for many reasons, beyond just being a new state of matter.

It provided a rich example of a state of matter with multiple broken symmetries. The order parameter has eighteen components, which can be viewed as a combined superfluid, ferromagnet, and liquid crystal.

The rich order parameter led to an exploration of diverse topological defects, from superfluid vortices with magnetic cores to boojums. This highlighted the concepts of broken symmetry, rigidity, and topological defects.

This was the first example of an unconventional fermionic superfluid. Specifically, it could be described by BCS theory, but not with s-wave pairing nor with the pairing mechanism of the electron-phonon interaction in elemental superconductors. This showed the adaptability of BCS theory. It laid the groundwork for understanding unconventional superconductivity in heavy fermions, organics, and cuprates.

Berezinskii-Kosterlitz-Thouless phase transitions

In Berezinskii published papers in 1970 and 1971, and Kosterlitz and Thouless published papers in 1972 and 1973. This work was significant for reasons including the following.

It showed states of matter and phase transitions were qualitatively different in two and three dimensions.

New concepts such as topological order, quasi-long-range order, essential singularities, and defect-mediated phase transitions were introduced.

Like that of Wilson, this work highlighted universality. There were connections between superfluids, superconductors, and XY magnets.

Scaling equations provided insight.

Kosterlitz and Thouless were awarded the Nobel Prize in 2016

We should celebrate!

Wow! Quite the Golden Jubilee!

Does anyone know of any conferences, events, or books that are planned to mark these anniversaries?

Kondo effect in the New York Times!

The Kondo effect is a paradigm for quantum many-body physics. It has so much: non-perturbative effects, scaling, emergent energy scales, Bethe ansatz solution, asymptotic freedom, Fermi liquid, ...

The Kondo model is a benchmark for testing many approximations and numerical methods.

Furthermore, it connects to so many other things: Anderson single impurity model, Dynamical Mean-Field Theory, Kosterlitz-Thouless transition, heavy fermions, ...

Nevertheless, outside the strongly correlated electron community, it is not widely known, and particularly not in popular discussions of science.

I never thought it would feature at the beginning of the New York Times article, unless Jun Kondo (now 90 years old) was awarded a belated Nobel Prize.

I was pleasantly surprised to see a long profile of Myriam Sarachik that began with her experimental work on the Kondo effect back in 1963.

The article also chronicles some of the sexism she faced in her career and the very limited employment options there were for women in physics. The article also describes how she was not very "productive" for a decade due to recovering from the personal tragedy of the murder of her daughter. Yet, as her mental health recovered she made significant contributions: quantum tunneling in single molecule magnets and the metal-insulator transition in semiconductor heterostructures.

There is a longer autobiographical piece in Annual Reviews.

The intellectual legacy of Phil Anderson

I am looking forward to reading Andrew Zangwill's book, A Mind Over Matter: Philip Anderson and the Physics of the Very Many, that should be available in January 2021.

Andy recently gave a beautiful talk at an ICAM meeting on the life and science of Phil Anderson. I highly recommend it. Yesterday, at the UQ condensed matter theory group meeting we watched it and discussed it.

A few things that stood out to me, partly because some were new to me.

``PWA was a brilliant intuitionist who did more than any other person to transform the patchwork of ideas and techniques of what was formerly called solid state physics into the deep, subtle, and intellectually coherent discipline know as condensed matter physics.''

Phil's wife, Joyce, had an MA in English literature and edited all his prose pieces. This may explain how well written his writing for general audiences, such as Physics Today columns and book reviews in The Times Higher Education Supplement were so well written. In contrast, Phils talks and some papers were rather obscure.

PWA was a contrarian. He did not follow the pack. This is embodied in the fact that he chose to work on his PhD at Harvard with van Vleck, rather than Schwinger, who was chosen by eleven of his peers! van Vleck said "follow the data". During this time he was a friend of Tom Lehrer, a mathematics graduate student who became famous for writing and performing satirical songs with a strong social justice theme.

Phil did a BS in Electronic Physics (essentially Radio Engineering) and did not learn any modern physics. He did a PhD in chemical physics. It was only at Bell Labs that he started working on condensed matter problems. There he had three significant mentors: Conyers Herring, Gregory Wannier, and Charles Kittel.

Phil's 1952 paper on antiferromagnetism contained the idea of spontaneous symmetry breaking. But, this was not appreciated for a decade.

Phil's 1957 localisation paper and his 1961 magnetic impurities paper [the two works cited for his Nobel Prize] were both stimulated by talking to experimentalists at Bell Labs [George Feher and Berndt Matthias, respectively].

Concepts in Solids, based on his graduate lectures at Cambridge in 1961-2, was revolutionary for the time because the focus was on the properties of model Hamiltonians, rather than detailed phenomenology.

Phil's criticisms of high energy physics, its reductionism and drawing resources away from "tabletop" science, began as early as 1971, when he wrote a New Scientist article on the subject. 

But there is a lot more. Watch the video!

The Critical Point

I have now finished my first draft of chapter 6, of Condensed Matter Physics: A Very Short Introduction. The main purpose of the chapter is to introduce the idea of the critical point and the renormalisation group.

 I welcome comments and suggestions. However, bear in mind that my target audience is not the typical reader of this blog, but rather your non-physicist friends and family.

I think it still needs a lot of work, particularly to be less technical.

The goal is for the chapter to be interesting, accessible, and bring out the excitement and importance of condensed matter physics.

Aside. Don't think that because I am posting two chapters one week apart that I am writing one per week. I wish I could. I sometimes move onto a new chapter before getting it into a draft form good enough to post.

Introducing universality and particularity

Every person is unique. No two people on earth are identical. We differ in physical appearance, personality, fingerprints, heartbeat, gait, and DNA. Such differences are used to identify criminals and in the 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. Arguably, on some level, we all have common aspirations: to survive, to be loved, to be happy, and to find meaning and purpose. Yet these aspirations find many particular expressions. All humans have certain universal qualities and properties, from the biomolecular to the social. Yet at a finer level of detail, there is a particularity of each of these properties. This paradox of ``same and not the same’’ can be viewed as a tension between universality and particularity.

All academic disciplines search for universals; they are used to categorise, to conceptualise, and to theorise. Biologists classify species of plants and animals and types of cells and viruses. All of these different biological systems make use of the same biomolecules (DNA, RNA, and proteins), and biochemical reactions. The same genetic code uses the information encoded in a piece of DNA to make proteins with a specific function. Anthropologists study the immense diversity of human cultures and societies. This diversity can be understood in terms of certain universal concepts such as kinship, sexual relations, family, ritual, community, economics, and religion. Linguists study the structure and grammar of the thousands of different human languages.  Given the diverse world that we live in many of us find the universality that different academic disciplines have discovered over the past century surprising and exciting.

Condensed matter physicists study the incredible diversity of different states of matter and the transitions between them. A surprising discovery is that there is much more universality than might be expected. In this chapter, I will discuss the nature of this universality, the different associated length scales associated with phase transitions, and how this universality emerges. The insight of Landau (chapter 4) was correct: 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 (universality classes) can be made. Even superconducting and superfluid and a subset of some magnetic transitions are in the same class. The determinants of the universality classes are the symmetry of the order in a state of matter and the spatial dimensionality of the system.

In society today there is significant public debate about morality; what is universal and what is particular to specific individuals, societies, or situations? Philosophers have debated universals for centuries. Many academic disciplines have discovered certain universal patterns, yet struggle to understand how universality does or does not emerge in the presence of particularity.  This struggle is due to the complexity of the systems of interest, including the many different scales present. Condensed matter physics provides a concrete and beautiful example where we do understand how to relate universality and particularity.

Would your non-scientist friends and relatives find this interesting? comprehensible?

I welcome suggestions.

The central role of scales in condensed matter

An important concept in condensed matter is the role played by scales, i.e. how big or small physical quantities are. Length, time, energy, and temperature are all physical quantities.

For example, there are many different length scales associated with a piece of material, say a block of copper, ranging from centimetres to a fraction of a nanometer. This covers lengths varying by a factor of a trillion, i.e., twelve orders of magnitude. The piece of copper may have dimensions of a centimetre. But it may be composed of small metallic grains of micron (micrometer) dimensions, and that can only be seen with a microscope. On an even smaller scale is the size of the individual copper atoms that make up the material, with dimensions less than a nanometer. Using different experimental techniques a scientist can ``zoom in and out'' and examine the properties of a material at different length scales.

Similarily one can investigate properties of a material at different time scales. This is similar to how one may use a high-speed movie camera to observe something and then replay it in slow motion. In a metal there are different time scales associated with different phenomena: the vibration of an atom, the time between collisions of electrons with each other, the period of the collective oscillation of all of the electrons.

There are also different energy scales associated with a material. Examples include the energy required to move a single atom a particular distance, the energy required to remove a single electron from the crystal, the kinetic energy of an electron inside the material, and the energy required to compress the whole material by a certain amount.

In quantum theory, energy and time are related by a proportionality factor known as Planck's constant. Thus, the energy scale and time scale associated with a specific phenomenon are related to each other.

The magnitude or scale of the temperature is also important. Temperature is related to energy via heat. Using clever refrigeration techniques materials can be cooled down to temperatures of less than one-thousands of a degree above absolute zero. This means that the properties of a material can be studied over a temperature range varying by about a factor of one million (six orders of magnitude).

This wide range of length, time, energy, and temperature scales is central for condensed matter physics in several respects. Overall, it means that phenomena, experimental techniques, theories, and concepts are relevant to a particular scale.

Experimental techniques have to be designed to investigate and ``probe'' the relevant phenomena at the relevant scale. Theories are also constructed with a concern with the relevant scales. Perhaps this is obvious.

There are also three profound and unanticipated aspects of the role of scales in condensed matter. 

a. Whereas, the existence of the atomic and macroscopic scales is obvious, due to collective behaviour (emergence) there are intermediate scales of length and time associated with particular phenomena. Before, I have discussed examples of emergent energy scales and length scales.

b. In distinct systems, the same phenomena can occur at scales that differ by many orders of magnitude. A striking example is the occurrence of superfluidity in liquid 3He at temperatures below one-thousandth of a one degree Kelvin and in neutron stars at temperatures below one hundred thousand degrees.

c. Through a highly sophisticated theoretical method, known as the renormalisation group and scaling, it is possible to make concrete connections between the properties of a system at different scales.

It is worth considering whether this wide range of scales and the central role they play occurs in other academic disciplines. In biology, this is certainly true, with a hierarchy of scales from biomolecules to protein networks to cells to organs. In economics, one goes from individual consumers to microeconomics to macroeconomics. The size of personal incomes, businesses, and government debt can also range of many orders of magnitude. In sociology, there is also a range of scales. Indeed, emergence does shape many of the big questions of many disciplines.
Arguably, what is really unique about CMP is b. and c. above.

I thank my son for asking me to clarify this central role of scales in condensed matter.

What was the greatest discovery of the past decade?

To my readers my best wishes for the New Year and the new decade!

It is worth reflecting on what has been achieved in condensed matter and chemical physics over the past decade. Which discovery or achievement would you rate as the most exciting, surprising, or significant?

To benchmark things, this is what I would say about previous decades, with regard to hard condensed matter, with a personal bias towards strongly correlated electron systems.

1970s: scaling and the renormalisation group

1980s: quantum Hall effects, cuprate superconductivity, heavy fermions, scanning tunneling microscopy (STM)

1990s: Dynamical Mean-Field Theory (DMFT), Kondo effect in quantum dots, superconducting qubits, (Angle-Resolved PhotoElectron Spectroscopy) ARPES advances, DMRG

2000s: iron-based superconductors, graphene, DMFT+DFT, topological insulators

2010s: ?

To be honest, I am worried that with each decade the discoveries are somewhat less exciting or significant. On the other hand, incremental advances, particularly steady ones over several decades should not be looked down on. An example is computational electronic structure methods and increases in the energy and momentum resolution associated with inelastic neutron scattering spectroscopy and ARPES.

What would you nominate for the past decade?

Emergence and complexity in social systems

Emergent phenomena occur in social systems. For example, self-organisation, power laws, networks, aggregation/segregation, political polarisation, political revolutions...
Can lessons from condensed matter physics help at all in understanding and modeling of social systems? Can analogies from social systems help non-scientists understand some of the basic ideas in condensed matter?

In two months I am giving a  seminar in a new UQ multi-disciplinary seminar series, Futures of International Order. In preparation, I am slowly engaging with relevant literature, particularly the work of Scott Page, including his course on Model Thinking at Coursera. The NetLogo software is helpful for exploring a range of simple models.
However, before plunging in here are a few tentative thoughts of ideas that might connect with condensed matter, in the vein of reviews such as

Physics and financial economics (1776–2014): puzzles, Ising and agent-based models 
Didier Sornette

Statistical physics of social dynamics 
Claudio Castellano, Santo Fortunato, and Vittorio Loreto

Emergence occurs in systems with many interacting components. In social systems, the components are human agents. They can aggregate into emergent entities such as neighbourhoods, institutions, and communities. Associated with these new entities are new scales of size (number of agents), length, time, and connectivity. New effective interactions between entities can also emerge. Even knowing all the details of the system components and the interactions it can be very difficult to predict the properties of the whole system. Surprises are common. Humility is needed.

Qualitative changes can occur due to small quantitative changes in a system parameter.
In condensed matter examples are phase transitions between different states of matter.  Furthermore, these changes can be directly seen as discontinuities or singularities in observables. Order parameters can quantify the changes. In social systems, similar phenomena are sometimes called tipping points.

Universality versus particularity
Close to a critical point for a phase transition most of the details of the system components and their interactions do not matter. Properties such as critical exponents are independent of most details. This is wonderful for theory because one can describe large classes of diverse systems with the same model/theory and one does not have to know all the details of the system.
Similar issues of universality are also relevant when one considers phenomena at different length scales. For example, one does not need to know anything about the atoms (even their existence!) in a crystal to develop a theory of elasticity or the propagation of sound waves.
When it comes to social systems there are a wide range of phenomena that can be potentially described by the same model. For example, Miller and Page point out that the essence of the standing ovation problem is how a binary choice (sit or stand) is influenced by the behaviour of one's neighbours. This is similar to choices as to whether to join a riot, take illegal drugs, or whether to vote of political party A or B.

Why is condensed matter physics important and interesting?

I am trying to get some momentum in writing A Very Short Introduction to Condensed Matter Physics. The intended audience is the intellectually curious person with little background in science. My goal is to convince them that CMP is important and interesting. I can think of several reasons.

1. CMP is intimately connected with everyday technology ranging from liquid crystal displays to computer chips.
2. CMP comprises the majority of physics (employees, papers, conferences, ...) and has significant interaction with areas of science and engineering.
3. CMP is a rich source of creative ideas, concepts, and techniques that represent a significant intellectual achievement and are relevant to many other intellectual endeavors.
4. CMP is full of surprises. We keep discovering new unanticipated phases of matter.
5. CMP presents significant scientific challenges: theoretical, computational, and experimental (from characterisation to sample synthesis).

I am going to focus on 3.
However, it is interesting that the traditional route is 1. Furthermore, different people (including reviewers of the book proposal) are quite divided about 1. versus 3.
[More on that later following this article].

What are the big picture ideas of condensed matter, that are significant intellectual achievements in their own right and particularly relevant to other endeavors?
Here are a few suggestions. It provides very concrete systems to address, at both the mathematical and experimental level, the following issues, which turn out to be often inter-related.

A. Qualitative distinctions are defined by discontinuities. (Different phases of matter).

B. Simple models of complex systems. (Landau theory of phase transitions; Effective Hamiltonians).

C. Universality versus particularity. (Universality classes for critical phenomena).

D. Emergence and the hierarchal nature of reality. (Effective interactions. Renormalisation.)

What do you think are the great intellectual achievements of condensed matter that people need to know about?

Universality, probability, and the growth of rough surfaces

On Friday there was a nice UQ Maths and Physics Colloquium, Beyond the Gaussian Universality Class, given by Ivan Corwin,
The talk was a very nice example of synergy between fundamental physics and maths research.
There are interesting connections with simple one-dimensional models for surface growth, the Kardan-Parisi-Zhang equation, the KPZ universality class, traffic models, random matrix theory, directed polymers in random media, ....

What can students learn from an Ising model simulation?

Computer simulations can provide significant insight into different physical phenomena. Two decades ago the best one could do in a class or seminar was show screen shots of simulations and try and explain what was going on. Now one can show a simulation live and even vary parameters in real time to provide insight. I have done this quite a bit with Solid State Simulations.

One simulation I like but have never used effectively is that of the Ising model.
See for example, Daniel Schroeder's simulation or James Sethna or Matt Bierbaum.
What does it help me understand?
The main ideas are the concept of symmetry breaking, the correlation length, and the divergence of the correlation length at the critical point.

1. Watching the different configurations changing with time illustrates the notion of an ensemble.
2. At high temperatures one sees the paramagnetic phase where the spins are independent of each other and so there are no domains.
3. As the temperature approaches the critical temperature (T=2.27J) from above the correlation length increases and large fluctuating domains form.
4. Below the critical temperature large domains form and fluctuate less and less as the temperature lowers.
5. The ferromagnetic ground state (blue or yellow, up or down spin) in zero external field depends on the history. This illustrates symmetry breaking

Any other things?

Solid State or Condensed Matter Physics?

The two terms are often used interchangeably, but that is not appropriate. Condensed matter physics does not just involve solids but also phenomena in liquids, liquid crystals, superfluids, and polymer melts.  Solid state physics is a subset of condensed matter physics. The latter term was arguably coined by Phil Anderson, when he and Mott renamed their research group at Cambridge in the 1970s. One can view research fields or course titles as a list of topics or as a way of thinking about certain parts of reality. Solids exhibit rich phenomena including magnetism and superconductivity. However, it is best to actually view the solids as (an almost irrelevant) substrate for the phenomena.

Like many things, this perspective arguably started with Landau. His theory of phase transitions in the 1930s did not consider atomic structure or chemical composition. Even structural phase transitions were viewed in terms of symmetry change, not in terms of explicit microscopic details. In 1950 this led to the Ginzburg-Landau theory of superconductivity. This all suggested a unified approach to phase transitions.
Furthermore, Landau's Fermi liquid theory papers were originally concerned with understanding liquid 3He, not electrons in metallic crystals.

This idea was further highlighted in the 1970s with the study of critical phenomena and the associated idea of universality. Specifically, the critical behaviour of an XY magnet, a superconductor, and a superfluid, are the same (i.e. they have the same critical exponents). The critical behaviour of the liquid-gas transition, an Ising magnet, and the order-disorder transition in a binary alloy are the same. The view that the solid state might actually not be the key feature for understanding and describing superconductivity was highlighted in the 1950s by Fritz London in his two-volume book, Superfluids, which suggested the two phenomena were intimately connected. Beginning in 1968, De Gennes took a condensed matter perspective in applying order parameters and scaling ideas to “soft matter”: liquid crystals, polymers, wetting, …

The important element to this conceptual view of condensed matter is that it provides a unifying perspective on phenomena in a diverse range of materials. It also brings to the fore how a wide suite of powerful theoretical and experimental tools (esp. neutron and x-ray scattering) can be used to study diverse materials. One of the key theoretical strategies is that of effective Hamiltonians, which is not unique to condensed matter, because it just reflects the hierarchy of energy, length, and times scales that result from emergence. This then leads to an intellectually rich interchange of ideas and techniques from other fields of physics, particularly quantum field theory.

More recently, this unity is illustrated by ultracold atomic gases which can be used to study some phenomena that had previously only been studied in solids.

Low energy scales near the orbital-selective Mott transition

One of the most fundamental and profound concepts in quantum many-body theory is the emergence of low energy scales that are much smaller than the energy scales in the "bare" Hamiltonian.
For example, in a metallic phase near the Mott transition in a single band system, there is the energy scale associated with a Fermi liquid. Studies using Dynamical Mean-Field Theory (DMFT) have shown how this scale is associated with ``kinks'' in the quasi-particle dispersion relation and is related to the energy scale for spin fluctuations.

The problem of the Mott transition in multi-band systems (degenerate orbitals) is fascinating and of renewed interest since the discovery of iron-based superconductors. A basic question concerns how the Mott transition is qualitatively different from in single band systems. More specifically, how does a Hund's rule coupling change things?

One new concept is that of an orbital-selective Mott transition. This is where one or more of the bands remains metallic but others become Mott insulators. This concept was originally introduced to explain the intriguing properties of Ca_xSr_2-xRuO4 with x ~ 0.5: it is metallic but has localised spin-1/2 magnetic moments.
[For a critical discussion see the  nice review Strong correlations from Hund's coupling by Antoine Georges, Luca de' Medici, and Jernej Mravlje.]

One might expect that near this transition there are separate low energy scales associated with each of the bands and that these scales are quite different for the bands that become insulator.
However, this is not the case.

There is a nice paper
Emergence of a Common Energy Scale Close to the Orbital-Selective Mott Transition 
Markus Greger, Marcus Kollar, and Dieter Vollhardt

They use DMFT to study a two-band Hubbard model with different bandwidths. They calculate the one-electron spectral functions, the electronic self energy, and the dynamical spin susceptibilities.

The left panel below shows the spectral functions for the two bands. Note how for one the quasi-particle peak width is much smaller than the other.
The right panel (top) shows the energy dependence of the real part of the self-energy for the two bands. Surprisingly, the kink occurs at the same energy.
Furthermore, the bottom of the right panel shows that this peak corresponds to the peak in the dynamical spin susceptibility for both bands.

The figure below shows that "If the Hund’s rule coupling is sufficiently strong, one common energy scale emerges which characterizes both the location of kinks in the self-energy and extrema of the diagonal spin susceptibilities."

The authors then give a physical explanation of this energy scale from a two-impurity Kondo model.

Statistical properties of networks

Today I am giving a cake meeting talk about something a bit different. Over the past year or so I have tried to learn something about "complexity theory", including networks. Here is some of what I have learnt and found interesting. The most useful (i.e. accessible) article I found was a 2008 Physics Today article, The physics of networks by Mark Newman.

The degree of a node, denoted k, is equal to the number of edges connected to that node. A useful quantity to describe real-world networks is the probability distribution P(k); i.e. if you pick a random node it gives the probability that the node has degree k.

Analysis of data from a wide range of networks, from the internet to protein interaction networks, finds that this distribution has a power-law form,

This holds over almost four orders of magnitude.
This is known as a scale-free network, and the exponent is typically between 2 and 3.
This power law is significant for several reasons.
First, it is in contrast to a random network for which P(k) would be a Poisson distribution, which decreases exponentially with large k, with a scale defined by the mean value of k.
Second, if the exponent is less than three, then the variance of k diverges in the limit of an infinite size network, reflecting large fluctuations in the degree.
Third, the "fat tail" means there is a significant probability of "hubs", i.e. nodes that are connected to a large number of other nodes. This reflects the significant spatial heterogeneity of real-world networks. This property has a significant effect on others properties of the network, as I discuss below.

An important outstanding question is do real-world networks self-organise in some sense to lead to the scale-free properties?

A question we do know the answer to is, what happens to the connectivity of the network when some of the nodes are removed?
It depends crucially on the network's degree distribution P(k).
Consider two different node removal strategies.
a. Remove nodes at random.
b. Deliberately target high degree nodes for removal.
It turns out that a random network is equally resilient to both "attacks".
In contrast, a scale-free network is resilient to a. but particularly susceptible to b.

Suppose you want to stop the spread of a disease on a social network. What is the best "vaccination" strategy?  For a scale-free network that will be to target the highest degree nodes in the hope of producing "herd immunity".

It's a small world!
This involves the popular notion of "six degrees of separation". If you take two random people on earth then on average one has to just go six steps in "friend of a friend of a friend ..." to connect them. Many find this surprising but, this arises because your social network increases exponentially with the number of steps you take and something like (30)^6 gives the population of the planet.
Newman states that a more surprising result is that people are good at finding the short paths, and Kleinberg showed that the effect only works if the social network has a special form.

How does one identify "communities" in networks?
A quantitative method is discussed in this paper and applied to several social, linguistic, and biological networks.

A topic which is both intellectually fascinating and of great practical significance concerns

This is what epidemiology is all about. However, until recently, almost all mathematical models assumed spatial homogeneity, i.e. that the probability of any individual being infected was equally likely. In reality, it depends on how many other individuals they interact with, i.e. the structure of the social network.

The crucial parameter to understand whether an epidemic will occur turns out to not be the mean degree but the mean squared degree. Newman argues

Consider a hub in a social network: a person having, say, a hundred contacts. If that person gets sick with a certain disease, then he has a hundred times as many opportunities to pass the disease on to others as a person with only one contact. However, the person with a hundred contacts is also more likely to get sick in the first place because he has a hundred people to catch the disease from. Thus such a person is both a hundred times more likely to get the disease and a hundred times more likely to pass it on, and hence 10 000 times more effective at spreading the disease than is the person with only one contact.

I found the following Rev. Mod. Phys. from 2015 helpful.
Epidemic processes in complex networks 
Romualdo Pastor-Satorras, Claudio Castellano, Piet Van Mieghem, Alessandro Vespignani

They consider different models for the spread of disease. A key parameter is the infection rate lambda which is the ratio of the transition rates for infection and recovery. This is the SIR [susceptible-infectious-recover] model proposed in 1927 by Kermack and McKendrick [cited almost 5000 times!]. This was one of the first mathematical models for epidemiology.

Behaviour is much richer (and more realistic) if one considers models on a complex network. Then one can observe "phase transitions" and critical behaviour. In the figure below rho is the fraction of infected individuals.

In a 2001 PRL [cited more than 4000 times!] it was shown using a "degree-based mean-field theory" that the critical value for lambda is given by

In a scale-free network the second moment diverges and so there is no epidemic threshold, i.e. for an infinitely small infection rate and epidemic can occur.

The review is helpful but I would have liked more discussion of real data about practical (e.g. public policy) implications. This field has significant potential because due to internet and mobile phone usage a lot more data is being produced about social networks.

Emergence frames many of the grand challenges and big questions in universities

What are the big questions that people are (or should be) wrestling within universities?
What are the grand intellectual challenges, particularly those that interact with society?

Here are a few. A common feature of those I have chosen is that they involve emergence: complex systems consisting of many interacting components produce new entities and there are multiple scales (whether length, time, energy, the number of entities) involved.

Economics
How does one go from microeconomics to macroeconomics?
What is the interaction between individual agents and the surrounding economic order?
A recent series of papers(see here and references therein) have looked at how the concept of emergence played a role in the thinking of Friedrich Hayek.

Biology
How does one go from genotype to phenotype?
How do the interactions between many proteins produce a biochemical process in a cell?

The figure above shows a protein interaction network and taken from this review.

Sociology
How do communities and cultures emerge?
What is the relationship between human agency and social structures?

Public health and epidemics
How do diseases spread and what is the best strategy to stop them?

Computer science
Artificial intelligence.
Recently it was shown how Deep learning can be understood in terms of the renormalisation group.

Community development, international aid, and poverty alleviation
I discussed some of the issues in this post.

Intellectual history
How and when do new ideas become "popular" and accepted?

Climate change

Philosophy
How do you define consciousness?

Some of the issues are covered in the popular book, Emergence: the connected lives of Ants, Brains, Cities, and Software.
Some of these phenomena are related to the physics of networks, including scale-free networks. The most helpful introduction I have read is a Physics Today article by Mark Newman.

Given this common issue of emergence, I think there are some lessons (and possibly techniques) these fields might learn from condensed matter physics. It is arguably the field which has been the most successful at understanding and describing emergent phenomena. I stress that this is not hubris. This success is not because condensed matter theorists are smarter or more capable than people working in other fields. It is because the systems are "simple" enough and the presence (sometimes) of a clear separation of scales that they are more amenable to analysis and controlled experiments.

Some of these lessons are "obvious" to condensed matter physicists. However, I don't think they are necessarily accepted by researchers in other fields.

Humility.
These are very hard problems, progress is usually slow, and not all questions can be answered.

The limitations of reductionism.
Trying to model everything by computer simulations which include all the degrees of freedom will lead to limited progress and insight.

Find and embrace the separation of scales.
The renormalisation group provides a method to systematically do this. A recent commentary by Ilya Nemenman highlights some recent progress and the associated challenges.

The centrality of concepts.

The importance of critically engaging with experiment and data.
They must be the starting and end point. Concepts, models, and theories have to be constrained and tested by reality.

The value of simple models.
They can give significant insight into the essentials of a problem.

What other big questions and grand challenges involve emergence?

Do you think condensed matter [without hubris] can contribute something?