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.

Tony Leggett (1938-2026): condensed matter theorist

Tony Leggett died last week. The New York Times has a nice obituary. One measure of his influence on me is that more than 20 posts on this blog feature his work. He received the Nobel Prize in 2003 for developing the theory of superfluid 3He.

In 1972, a graduate student at Cornell, Doug Osheroff, discovered a phase transition around a temperature of 2 mK in liquid 3He. In the 1960s liquid 3He was established to be a Fermi liquid that was beautifully described by Landau's theory. Osheroff and his advisors, David Lee and Robert Richardson, incorrectly identified the phase transition as arising from antiferromagnetic order in the solid phase of 3He.

However, Leggett argued that it was actually due to superfluidity that there were two distinct superfluid phases, A and B, with different order parameters. 

Lee, Osheroff, and Richardson shared the Nobel Prize in 1996 for their discovery.

Leggett was primed to make rapid progress, as in 1965 and 1966 he had written three papers about superfluidity in liquid 3He, albeit assuming s-wave pairing. Indeed, by 1975 he wrote a comprehensive review article on the two superfluid phases.

For many reasons superfluid 3He was significant for the broader field of condensed matter. BCS showed that in elemental metals, superconductivity resulted from Cooper pairing of electrons due to an attractive electron-phonon interaction.  The order parameter (Cooper pair wave function) had s-wave spin singlet symmetry.

In contrast, superfluid 3He showed that Cooper pairing could also occur in a neutral Fermi liquid, and have non-trivial symmetry, i.e., p-wave symmetry and spin triplet. The order parameter has 18 components, compared to only 2 for elemental superconductors. There is spontaneous symmetry breaking of the local gauge symmetry, and spin or orbital rotational symmetries. 

The Cooper pairing in superfluid 3He is not due to a fermion-phonon interaction but due to spin fluctuations.

The fact that Cooper pairing was possible for different symmetries and mechanisms than for elemental superconductors was significant in that it meant it was reasonable to consider this possibility for superfluidity in neutron stars, and superconductivity in cuprates, strontium ruthenate, heavy fermions, and organic charge transfer salts.

There is rich physics associated with the symmetry breaking: 18 collective modes of the order parameter, textures such as boojums, and exotic vortex cores. For vortices, there is also some (controversial) connection to cosmic strings, including experiments that test the Kibble-Zurek mechanism and the electro-weak phase transition in the early universe.

Aside: My Ph.D. thesis was on the theory of the non-linear interaction of zero sound with the order parameter collective modes in the B-phase.

Leggett's development of the theory of superfluid 3He was amazing and certainly worthy of a Nobel. However, I think he made an even greater contribution to physics through his work on the theory of macroscopic quantum effects in Josephson junctions. This work was the basis for the experimental work that was honoured with the Nobel Prize last year.

With his student Amir Caldeira, Leggett performed concrete calculations of the effects of decoherence on quantum tunnelling in Josephson junctions.

[The NY Times obituary mistakenly says this work began after Leggett moved to Urbana. It was done while he was still at Sussex].

The formalism they developed involving the spectral density is the basis for most theoretical treatments of decoherence in superconducting qubits. A relevant toy model is the spin-boson model, and in 1987 Leggett published a seminal (but rather dense) review on the subject.

Leggett aided our understanding of cuprate superconductors. He contributed to the theoretical ideas that were the basis of the phase-sensitive measurements that established the d-wave nature of the order parameter. He also showed that experiments with inconsistent with  Anderson's interlayer tunneling theory.

I recommend reading Leggett's own scientific autobiography, Matchmaking Between Condensed Matter and Quantum Foundations, and Other Stories: My Six Decades in Physics and his book, The Problems of Physics

The legacy of 40 years of cuprate superconductivity

In February 1986, Bednorz and Müller made a stunning discovery: superconductivity at a temperature of 35 K in a doped copper oxide (cuprate). Arguably, this discovery changed condensed matter physics. In April 1986, they submitted their results to Z. Phys. B. Only nineteen months later, they were awarded the Nobel Prize in Physics, the shortest time ever between a discovery and the award. A nice and short review of the history is here.

One measure of my estimate of the influence of this discovery is that it received about 5 pages of coverage in my Condensed Matter Physics: A Very Short Introduction. (See Chapter 5, Adventures in Flatland).

How things have developed over the past forty years, for better and worse, may be representative of how science advances: discovery by serendipity, hype about applications, unexpected secondary benefits, foundational questions, new concepts, unification, and incremental advances.

Hype about technological applications

On March 20, 1987, The New York Times had a front-page article, DISCOVERIES BRING A 'WOODSTOCK' FOR PHYSICS, by James Gleick. This followed the 1987 APS March meeting. It began 

"Physicists from three continents converged on the New York Hilton for a hastily scheduled special conference on a string of discoveries that seem certain to produce a rapid cascade of commercial applications in electricity, magnetism and electronics.There are many things we know and understand that we did not when they were first discovered."

This has largely been unfulfilled. There are a few niche applications, but cuprates are not used in electricity distribution or even in the superconducting magnets in hospital MRI machines, which are probably the main commercial application of superconductors. One of the significant obstacles is that it is hard to make wires from these materials, as they are ceramics. This is an example of the common gap between research laboratory science and commercially viable technology.

After 40 years, do we have a successful theory?

It depends on who you ask. But I would say there is a lot we do understand.

We have a phenomenological theory for all the macroscopic phenomena associated with the superconducting state: Ginzburg-Landau theory!

Properties of the superconducting state are well-described by a BCS wavefunction with a d-wave order parameter and the associated Bogoliubov quasiparticles. [This is somewhat puzzling, as in the metallic state quasi-particles are not well defined].

Although not everyone agrees, I think it is fair to say that the essential physics is in a one-band Hubbard model, and the key physics is:

strong electronic correlations,

a doped antiferromagnetic Mott insulator,

d-wave pairing that is "mediated"/caused from some mixture/variant of antiferromagnetic spin fluctuations or RVB spin singlets,.....

We certainly don't understand the cuprates at the same level as elemental superconductors. But we do understand the essential physics.

What is harder to describe and understand are the states adjacent to the superconducting state in the phase diagram: the pseudogap state and the strange metal.

Strongly correlated electron materials became a large, vibrant and unified field

Before 1986, there were small, disconnected communities intermittently interested in transition metal oxides, rare earths, Kondo impurities, Mott metal-insulator transitions, organic superconductors, heavy fermions, and quantum antiferromagnets.

The discovery of the cuprates brought together these communities as they found common interests, challenges, questions, concepts, and techniques.

The discovery of superconductivity in strontium ruthenate, alkali fullerides, iron pnictides and chalcogenides, twisted bilayer graphene and more cuprates, organic charge-transfer salts, and heavy fermions has shown how rich these systems are. The challenge is to understand the similarities and differences between these chemically and structurally diverse systems. In many of them, superconductivity is proximate to a Mott insulating state.

The unity and excitement were probably stimulated and enhanced by the activities and ideas of high-profile theorists such as Anderson, Schrieffer, Scalapino, Pines, Rice, and Varma. On the other hand, their acrimonious disagreements probably did not help.

Secondary theoretical benefits

The things I list below were not new ideas when the cuprate discovery happened. However, interest in the cuprates led them to become major research themes and ideas.

Importance of phase diagrams, including as a function of interaction parameters in toy models

Highlighting the limitations of electronic structure methods based on Density Functional Theory with approximate Exchange-Correlation functionals (i.e., anything computational). In the presence of strong correlations, DFT methods have spectacular failures. For example, predicting a metallic state instead of the Mott insulator.

Low dimensionality leads to qualitatively different behaviour, including the possibility of new types of order and quasiparticles. This is most dramatic in one dimension, where one has Luttinger liquids and spin-charge separation.

Spin liquids. Landau was wrong. Spontaneous symmetry breaking does not always occur in antiferromagnets.

Non-Fermi liquids. Landau was wrong. Not all metals are Fermi liquids.

Quantum criticality. Although this is a robust concept for certain toy models, whether it is relevant to the cuprates remains contentious.

Systematic improvements in approximation schemes and numerical techniques - exact diagonalisation, DMRG, DMFT, quantum Monte Carlo,...

Emergence. Chemical complexity and strong interactions can lead to new states of matter.

Secondary experimental benefits

Better probes. The desire to characterise the cuprates helped drive significant improvements in the resolution of ARPES (Angle-Resolved PhotoEmission Spectroscopy), STM (Scanning Tunnelling Microscopy), and inelastic neutron scattering. These advances have born fruit in the study of a wide range of other materials, beyond the cuprates.

Growth of single crystals. The early days of the cuprates produced a lot of junk experimental results because of the poor quality of the samples produced by "shake and bake". However, the involvement of solid-state chemists has improved things. The techniques have also led to the production of single crystals for a wide range of strongly correlated materials.

Why is there so little research on cuprates today?

Today, there is little research directly on cuprates, both theoretically and experimentally. It is hard to get funding to work on them, even though there is a lot we don't understand really well.

This is because of the problem of fashion in science. The low-lying fruit has been picked. There is a continuous new stream of materials being discovered with exotic properties, the latest being twisted bilayer van der Waals compounds.

What does learning to ride a bicycle teach us?

How do you learn to ride a bicycle? How do you teach someone to ride a bicycle? It is not easy to put this into words and that is an important point in itself. It may help to have some knowledge of the parts of the bicycle and their respective functions. It may help to know something about relevant physics such as inertia, the centre of gravity, and balance. It may help to have some practical advice about seat height, posture, the appropriate speed at which to pedal, and where to look when riding. 

Nevertheless, all that information may not help much. Some young children learn to ride without knowing any of this. They just watch other children doing it, get on bike, try it, and learn by trial and error. The more passionate they are about learning the more likely they may be to succeed.

The mind and body of a bicyclist focus on just a few things: looking where they are going, pedalling, steering, and a sense of balance. This information is integrated together, and the rider adjusts their direction, pedalling, and posture. Furthermore, that process of integration and adjustment involves much that is not the rider’s focus, and they may not even be directly aware of. A person’s sense of body awareness and coordination is shaped by biology, physique, experience, and training.

This example of bike riding illustrates several important things.  First, we can have the ability to do something without necessarily being able to articulate how we do it. Second, knowing requires personal commitment. It involves trust and risk. If a person is unwilling to trust or take risks, they may miss out on something good, such as the joy of riding a bicycle. Third, knowing requires integration of multifaceted information. Fourth, knowledge and understanding come from integrating our focus into an implicit background we may not even be aware of.

The example of riding a bike is valuable for understanding how we know (epistemology) because it is simpler and less fraught and emotionally charged than how we come to an understanding and make decisions about history, ethics, politics, religion, and the meaning of scientific knowledge. 

These observations draw on Michael Polanyi, including his book, The Tacit Dimension, published in 1966, but based on lectures he gave at Yale in 1962. He referred to the first point as tacit knowing, and the fourth point as the subsidiary-focal interaction. The relationship of the subsidiary and the focus is like the whole and the parts. Polanyi considered the idea of tacit knowledge his most important discovery.

Aside: Chapter 2 of The Tacit Dimension is entitled "Emergence" and discusses ideas similar to those that Phil Anderson promoted in 1972 in More is Different, without using the word "emergence." According to Google Scholar, The Tacit Dimension has been cited 45,000 times.

The role of superconductivity in development of the Standard Model

In 1986, Steven Weinberg published an article, Superconductivity for Particular Theorists, in which he stated

"No one did more than Nambu to bring the idea of spontaneously broken symmetries to the attention of elementary particle physicists. And, as he acknowledged in his ground-breaking 1960 article  "Axial Current Conservation in Weak Interactions'', Nambu was guided in this work by an analogy with the theory of superconductivity,..."

In the 1960 PRL, referenced by Weinberg, Nambu states that in the BCS theory, as refined by Bogoliubov, [and Anderson]

"gauge invariance, the energy gap, and the collective excitations are logically related to each other as was shown by the author. [Y. Nambu, Phys. Rev. 117, 648 (1960)] In the present case we have only to replace them by (chiral) (gamma_5) invariance, baryon mass, and the mesons." 

This connection is worked out explicitly in two papers in 1961. The first is

Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I

Y. Nambu and G. Jona-Lasinio

They acknowledge, 

"that the model treated here is not realistic enough to be compared with the actual nucleon problem. Our purpose was to show that a new possibility exists for field theory to be richer and more complex than has been hitherto envisaged,"

Hence, I consider this to be a toy model for an emergent phenomena.

The model consists of a massless fermion field with a quartic interaction that has chiral invariance, i.e., unchanged by global gauge transformations associated with the gamma_5 matrix. (The Lagrangian is given above.) At the mean-field level, this symmetry is broken. Excitations include massless bosons (associated with the symmetry breaking and similar to those found earlier by Goldstone) and bound fermion pairs. It was conjectured that these could be analogues of mesons and baryons, respectively. The model was proposed before quarks and QCD. Now, the fermion degrees of freedom would be identified with quarks, and the model illustrates the dynamical generation of quark masses. When generalised to include SU(2) or SU(3) symmetry the model is considered to be an effective field theory for QCD, such as chiral effective theory.

My review article on emergence

I just posted on the arXiv a long review article on emergence

Emergence: from physics to biology, sociology, and computer science

The abstract is below.

I welcome feedback. 

------

Many systems of interest to scientists involve a large number of interacting parts and the whole system can have properties that the individual parts do not. The system is qualitatively different to its parts. More is different. I take this novelty as the defining characteristic of an emergent property. Many other characteristics have been associated with emergence are reviewed, including universality, order, complexity, unpredictability, irreducibility, diversity, self-organisation, discontinuities, and singularities. However, it has not been established whether these characteristics are necessary or sufficient for novelty. A wide range of examples are given to show how emergent phenomena are ubiquitous across most sub-fields of physics and many areas of biology and social sciences. Emergence is central to many of the biggest scientific and societal challenges today. Emergence can be understood in terms of scales (energy, time, length, complexity) and the associated stratification of reality. At each stratum (level) there is a distinct ontology (properties, phenomena, processes, entities, and effective interactions) and epistemology (theories, concepts, models, and methods). This stratification of reality leads to semi-autonomous scientific disciplines and sub-disciplines. A common challenge is understanding the relationship between emergent properties observed at the macroscopic scale (the whole system) and what is known about the microscopic scale: the components and their interactions. A key and profound insight is to identify a relevant emergent mesoscopic scale (i.e., a scale intermediate between the macro- and micro- scales) at which new entities emerge and interact with one another weakly. In different words, modular structures may emerge at the mesoscale. Key theoretical methods are the development and study of effective theories and toy models. Effective theories describe phenomena at a particular scale and sometimes can be derived from more microscopic descriptions. Toy models involve minimal degrees of freedom, interactions, and parameters. Toy models are amenable to analytical and computational analysis and may reveal the minimal requirements for an emergent property to occur. The Ising model is an emblematic toy model that elucidates not just critical phenomena but also key characteristics of emergence. Many examples are given from condensed matter physics to illustrate the characteristics of emergence. A wide range of areas of physics are discussed, including chaotic dynamical systems, fluid dynamics, nuclear physics, and quantum gravity. The ubiquity of emergence in other fields is illustrated by neural networks, protein folding, and social segregation. An emergent perspective matters for scientific strategy, as it shapes questions, choice of research methodologies, priorities, and allocation of resources. Finally, the elusive goal of the design and control of emergent properties is considered.

Emergence in Chemistry

It is important to be clear what the system is. Most of chemistry is not really about isolated molecules. A significant amount of chemistry occurs in an environment, often within a solvent. Then the system is the chemicals of interest and the solvent. For example, when it is stated that HCl is an acid, this is not a reference to isolated HCl molecules but a solution of HCl in water, and then the HCl dissociates into H+ and Cl- ions. Chemical properties such as reactivity can change significantly depending on whether a compound is in the solid, liquid, or gas state, or on the properties of the solvent in which it is dissolved.

Scales

The time scales for processes, which range from molecular vibrations to chemical reactions, can vary from femtoseconds to days. Relevant energy scales, corresponding to different effective interactions, can vary from tens of eV (strong covalent bonds) to microwave energies of 0.1 meV (quantum tunnelling in an ammonia maser).

Other scales are the total number of atoms in a compound, which can range from two to millions, the total number of electrons, and the number of different chemical elements in the compound. As the number of atoms and electrons increases, so does the dimensionality of the Hilbert space of the corresponding quantum system.

Novelty

All chemical compounds are composed of a discrete number of atoms, usually of different type. For example, acetic acid, denoted CH3COOH, is composed of carbon, oxygen, and hydrogen atoms. The compound usually has chemical and physical properties that the individual atoms do not have.

Chemistry is all about transformation. Reactants combine to produce products, e.g. A + B -> C. C may have chemical or physical properties that A and B did not have.

Chemistry involves concepts that do not appear in physics. Roald Hoffmann argued that concepts such as acidity and basicity, aromaticity, functional groups, and substituent effects have great utility and are lost in a reductionist perspective that tries to define them precisely and mathematicise them.

Diversity

Chemistry is a wonderland of diversity, as it arranges chemical elements in a multitude of different ways that produce a plethora of phenomena. Much of organic chemistry just involves three different atoms: carbon, oxygen, and hydrogen.

Molecular structure

Simple molecules (such as water, ammonia, carbon dioxide, methane, benzene) have a unique structure defined by fixed bond lengths and angles. In other words, there is a well-defined geometric structure that gives the locations of the centres of atomic nuclei. This is a classical entity. This emerges from the interactions between the electrons and nuclei of the constituent atoms.

In philosophical discussions of emergence in chemistry, molecular structure has received significant attention. Some claim it provides evidence of strong emergence. The arguments centre around the fact that the molecular structure is a classical entity and concept that is imposed, whereas a logically self-consistent approach would treat both electrons and nuclei quantum mechanically.

The molecular structure of ammonia (NH3) illustrates the issue. It has an umbrella structure which can be inverted. Classically, there are two possible degenerate structures. For an isolated molecule, quantum tunnelling back and forth between the two structures can occur. The ground state is a quantum superposition of two molecular structures. This tunnelling does occur in a dilute gas of ammonia at low temperature, and an associated quantum transition (at a wavelength of 1.2 cm) is the basis of the maser, the forerunner of the laser. This example of ammonia was discussed by Anderson at the beginning of his seminal More is Different article to illustrate how symmetry breaking leads to well-defined molecular structures in large molecules. 

Figure is taken from here.

Born-Oppenheimer approximation 

Without this concept, much of theoretical chemistry and condensed matter would be incredibly difficult. It is based on the separation of time and energy scales associated with electronic and nuclear motion.  It is used to describe and understand the dynamics of nuclei and electronic transitions in solids and molecules. The potential energy surfaces for different electronic states define effective theory for the nuclei. Without this concept, much of theoretical chemistry and condensed matter would be incredibly difficult.

Singularity. The Born-Oppenheimer approximation is justified by an asymptotic expansion in powers of (m/M)^1/4, where m is the mass of an electron and M the mass of an atomic nucleus in the molecule. This has been discussed by Primas and Bishop.

The rotational and vibrational degrees of freedom of molecules also involve a separation of time and energy scales. Consequently, one can derive separate effective Hamiltonians for the vibrational and rotational degrees of freedom.

Qualitative difference with increase in molecular size

Consider the following series with varying chemical properties: formic acid (CH2O2), acetic acid (C2H4O2), propionic acid (C3H6O2), butyric acid (C4H8O2), and valerianic acid (C5H10O2), whose members involve the successive addition of a CH2 radical. The Marxist Friedrich Engels used these examples as evidence for Hegel’s law: “The law of transformation of quantity into quality and vice versa”.

In 1961, Platt discussed properties of large molecules that “might not have been anticipated” from properties of their chemical subgroups. Table 1 in Platt’s paper lists “Properties of molecules in the 5- to 50-range that have no counterpart in diatomics and many triatomics.” Table 2 lists “Properties of molecules in the 50- to 500-atom range and up that go beyond the properties of their chemical sub-groups.” The properties listed included internal conversion (i.e., non-radiative decay of excited electronic states), formation of micelles for hydrocarbon chains with more than ten carbons, the helix-coil transition in polymers, chromatographic or molecular sorting properties of polyelectrolytes such as those in ion-exchange resins, and the contractility of long chains.

Platt also discussed the problem of molecular self-replication. Until 1951, it was assumed that a machine could not reproduce itself,f and this was the fundamental difference between machines and living systems. However, von Neumann showed that a machine with a sufficient number of parts and a sufficiently long list of instructions can reproduce itself. Platt pointed out that this suggested there is a threshold for autocatalysis: “this threshold marks an essentially discontinuous change in properties, and that fully-complex molecules larger than this size differ from all smaller ones in a property of central importance for biology.” Thus, self-replication is an emergent property. A modification of this idea has been pursued by Stuart Kauffman with regard to the origin of life, that when a network of chemical reactions is sufficiently large, it becomes self-replicating.

Phase diagrams elucidate emergence

Phase diagrams have been ubiquitous in materials science for decades. They show what states of matter are thermodynamically stable depending on the value of external parameters such as temperature, pressure, magnetic field, or chemical composition. However, they are only beginning to be appreciated in other fields. Recently, Bouchaud argued that they needed to be used more to understand agent-based models in the social sciences.

For theoretical models, whether in condensed matter, dynamical systems, or economics, phase diagrams can show how the state of the system predicted by the model has qualitatively different properties depending on the parameters in the model, such as the strength of interactions. 

Phase diagrams illustrate discontinuities, how quantitative changes produce qualitative changes (tipping points), and diversity (simple models can describe rich behaviour). Phase diagrams show how robust and universal a state is, i.e., whether it only exists for fine-tuning of parameters. Theoretical phase diagrams can expand our scientific imagination, suggesting new regimes that might be explored by experiments. An example is how the phase diagram for QCD matter (shown below) has suggested new experiments, such as at the RHIC.

For dynamical systems, I recently illustrated this with the phase diagram for the Lorenz model. It shows for what parameter ranges strange attractors exist.

Today, for theoretical models for strongly correlated electron systems it is common to map out phase diagrams as a function of the model parameters. However, this was not always the case. It was more common to just investigate a model for specific parameter values that were deemed to be relevant to specific materials. Perhaps, Anderson stimulated this new approach when, in 1961, he drew the phase diagram for the mean-field solution to his model for local moments in metals, a paper that was partly the basis of his 1977 Nobel Prize.

At a minimum, a phase diagram should show the state with the emergent property and the disordered state. Diagrams that contain multiple phases may provide hints for developing a theory for a specific phase. For example, for the high-Tc cuprate superconductors, the proximity of the Mott insulating, pseudogap, and non-Fermi liquid metal phases has aided and constrained theory development.

Phase diagrams constrain theories as they provide a minimum criterion of something a successful theory should explain, even if only qualitatively. Phase diagrams illustrate the potential and pitfalls of mean-field theories. Sometimes they get qualitative details correct, even for complex phase diagrams, and can show what emergent states are possible. Ginzburg-Landau and BCS theories are mean-field theories and work extremely well for many superconductors. On the other hand, in systems with large fluctuations, mean-field theory may fail spectacularly, and they are sometimes the most interesting and theoretically challenging systems.

Self-organised criticality and emergence in economics

A nice preprint illustrates how emergence is central to some of the biggest questions in economics and finance. Emergent phenomena occur as many economic agents interact resulting in a system with properties that the individual agents do not have.

The Self-Organized Criticality Paradigm in Economics & Finance

Jean-Philippe Bouchaud

The paper illustrates several key characteristics of emergence (novel properties, universality, unpredictability, ...) and the value of toy models in elucidating it. Furthermore, it illustrates the elusive nature of the "holy grail" of controlling emergent properties. 

The basic idea of self-organised criticality

"The seminal idea of Per Bak is to think of model parameters themselves as dynamical variables, in such a way that the system spontaneously evolves towards the critical point, or at least visits its neighbourhood frequently enough"

A key property of systems exhibiting criticality is power laws in the probability distribution of a property. This means that there are "fat tails" in the probability distribution and extreme events are much more likely than in a system with a Gaussian probability distribution.

Big questions

The two questions below are similar in that they concern the puzzle of how markets produce fluctuations that are much larger than expected when one tries to explain their behaviour in terms of the choices of individual agents.

A big question in economics

"A longstanding puzzle in business cycle analysis is that large fluctuations in aggregate economic activity sometimes arise from what appear to be relatively small impulses. For example, large swings in investment spending and output have been attributed to changes in monetary policy that had very modest effects on long-term real interest rates."

This is the "small shocks, large business cycle puzzle", a term coined by Ben Bernanke, Mark Gertler and Simon Gilchrist in a 1996 paper. It begins with the paragraph above. [Bernanke shared the 2022 Nobel Prize in Economics for his work on business cycles].

A big question in finance

The excess volatility puzzle in financial markets was identified by Robert Shiller: The volatility "is at least five times larger than it "should" be in the absence of feedback". In the views of some, this puzzle highlights the failings of the efficient market hypothesis and the rationality of investors, two foundations of neoclassical economics. [Shiller shared the 2013 Nobel Prize in Economics for this work]. 

"Asset prices frequently undergo large jumps for no particular reason, when financial economics asserts that only unexpected news can move prices. Volatility is an intermittent, scale-invariant process that resembles the velocity field in turbulent flows..." (page 2)

Emergent properties

Close to a critical point, the system is characterised by fat-tailed fluctuations and long memory correlations.

Avalanches. They allow very small perturbations to generate large disruptions.

Dragon Kings

Minsky moment

The holy grail: control of emergent properties

It would be nice to understand superconductivity well enough  to design a room-temperature superconductor. But, this pales in significance compared to the "holy grail" of being about to manage economic markets to prevent bubbles, crashes, and recessions.

Bouchaud argues that  the quest for efficiency and the necessity of resilience may be mutually incompatible. This is because markets may tend towards self-organised criticality which is characterised by fragility and unpredictability (Black swans).

The paper has the following conclusion

"the main policy consequence of fragility in socio-economic systems is that any welfare function that system operators, policy makers of regulators seek to optimize should contain a measure of the robustness of the solution to small perturbations, or to the uncertainty about parameters value.

Adding such a resilience penalty will for sure increase costs and degrade strict economic performance, but will keep the solution at a safe distance away from the cliff edge. As argued by Taleb [159], and also using a different language in Ref. [160], good policies should ideally lead to “anti-fragile” systems, i.e., systems that spontaneously improve when buffeted by large shocks."

Toy models

Toy models are key to understanding emergent phenomena. They ignore almost all details to the point that critics claim that the models are oversimplified. The modest goal of their proponents is simply to identify what ingredients may be essential for a phenomenon to occur. Bouchaud reviews several such models. All provide significant insight.

A trivial example (Section 2.1)

He considers an Ornstein-Uhlenbeck process for a system relaxing to equilibrium. As the damping rate tends to zero [κ⋆ → 0], the relaxation time and the variance of fluctuations diverge at the same rate. In other words, "in the limit of marginal stability κ⋆ →0, the system both amplifies exogenous shocks [i.e., those originating outside the system] and becomes auto-correlated over very long time scales."

The critical branching transition (Section 2.2)

The model describes diverse systems: "sand pile avalanches, brain activity, epidemic propagation, default/bankruptcy waves, word of mouth,..."

The model involves the parameter R0 which became famous during the COVID-19 pandemic. R0 is the average number of uninfected people who become infected due to contact with an infected individual. For sand piles R0 is the average number of grains that start rolling in response to a single rolling grain.

when R0 = 1 the distribution of avalanche sizes is a scale-free, power-law distribution 1/S^3/2, with infinite mean.

"most avalanches are of small size, although some can be very large. In other words, the system looks stable, but occasionally goes haywire with no apparent cause."

A generalised Lotka-Volterra model (Sections 3.3 and 4.2) 

This provides an analogue between economic production networks and ecology. Last year I reviewed recent work on this model, concerning how to understand the interplay of evolution and ecology.

A key result is how in the large N limit (i.e., a large number of interacting species/agents) qualitatively different behaviour occurs. Ecosystems and economies can collapse. 

 "any small change in the fitness of one species can have dramatic consequences on the whole system – in the present case, mass extinctions...

"most complex optimisation systems are, in a sense, fragile, as the solution to the optimisation problem is highly sensitive to the precise value of the parameters of the specific instance one wants to solve, like the Aij entries in the Lotka-Volterra model. Small changes of these parameters can completely upend the structure of the optimal state, and trigger large-scale rearrangements,..." 

Balancing stick problem (Section 3.4)

 The better one is able to stabilize the system, the more difficult it becomes to predict its future evolution! 

Propagation of production delays along the supply chain (Section 4.1)

An agent-based firm network model (Section 4.3)

This has the phase diagram shown below. The horizontal axis is the strength of forces counteracting supply/demand and profit imbalances. The vertical axis is the perishability of goods.

There are four distinct phases.

Leftmost region (a, violet): the economy collapses; 

Middle region (b, blue): the economy reaches equilibrium relatively quickly;

Right region (c, yellow): the economy is in perpetual disequilibrium, with purely endogenous fluctuations. 

The green vertical sliver (d) corresponds to a deflationary equilibrium

Phase diagrams illustrate how quantitative changes can produce qualitative differences.

Universality

The toy models considered describe emergent phenomena in diverse systems, including in fields other than economics and finance. 

Here are a few other recent papers by Bouchaud that are relevant to this discussion.

Navigating through Economic Complexity: Phase Diagrams & Parameter Sloppiness

From statistical physics to social sciences: the pitfalls of multi-disciplinarity

This includes the opening address from a workshop on "More is Different" at the College de France in 2022.

John Hopfield on what physics is

A decade ago John Hopfield reflected on his scientific life in Annual Reviews in Condensed Matter Physics, Whatever Happened to Solid State Physics?

"What is physics? To me—growing up with a father and mother who were both physicists—physics was not subject matter. The atom, the troposphere, the nucleus, a piece of glass, the washing machine, my bicycle, the phonograph, a magnet—these were all incidentally the subject matter. The central idea was that the world is understandable, that you should be able to take anything apart, understand the relationships between its constituents, do experiments, and on that basis be able to develop a quantitative understanding of its behavior. 

Physics was a point of view that the world around us is, with effort, ingenuity, and adequate resources, understandable in a predictive and reasonably quantitative fashion. Being a physicist is a dedication to the quest for this kind of understanding."

He describes how this view was worked out in his work in solid state theory and moved into biological physics and the paper for which he was awarded the Nobel Prize. 

"Eventually, my knowledge of spin-glass lore (thanks to a lifetime of interaction with P.W. Anderson), Caltech chemistry computing facilities, and a little neurobiology led to the first paper in which I used the word neuron. It was to provide an entryway to working on neuroscience for many physicists..."

After he started working on biological physics in the late 1970s he got an offer from Chemistry and Biology at Caltech and Princeton Physics suggested he take it. 

"In 1997, I returned to Princeton—in the Molecular Biology Department, which was interested in expanding into neurobiology. Although no one in that department thought of me as anything but a physicist, there was a grudging realization that biology could use an infusion of physics attitudes and viewpoints. I had by then strayed too far from conventional physics to be courted for a position in any physics department. So I was quite astonished in 2003 to be asked by the American Physical Society to be a candidate for vice president. And, I was very happy to be elected and ultimately to serve as the APS president. I had consistently felt that the research I was doing was entirely in the spirit and paradigms of physics, even when disowned by university physics departments."

The multi-faceted character of emergence (part 2)

In the previous post, I considered five different characteristics that are often associated with emergence and classified them as being associated with ontology (what is real and observable) rather than epistemology (what we believe to be true). 

Below I consider five more characteristics: self-organisation, unpredictability, irreducibility, contextuality and downward causation, and intra-stratum closure.

6. Self-organisation

Self-organisation is not a property of the system but a mechanism that a theorist says causes an emergent property to come into being. Self-organisation is also referred to as spontaneous order. 

In the social sciences self-organisation is sometimes referred to as an endogenous cause, in contrast to an exogenous cause. There is no external force or agent causing the order, in contrast to order that is imposed externally. For example, suppose that in a city there is no government policy about the price of a loaf of sliced wholemeal bread or on how many loaves that bakers should produce. It is observed that prices are almost always in the range of $4 to $5 per loaf, and that rarely are there bread shortages. This outcome is a result of the self-organisation of the free-market, and economists would say the price range and its stability has an endogenous cause. In contrast, if the government legislated the price range and the production levels that would be an exogenous cause. Friedrich Hayek emphasised the role of spontaneous order in economics. In biology, Stuart Kaufmann equates emergence with spontaneous order and self-organisation.

In physics, the periodicity of the arrangement of atoms in a crystal is a result of self-organisation and has an endogenous cause. In contrast, the periodicity of atoms in an optical lattice is determined by the laser physicist who creates the lattice and so has an exogenous cause.

Self-organisation shows how local interactions can produce global properties. In different words, short-range interactions can lead to long-range order. After decades of debate and study, the Ising model showed that this was possible. Other examples of self-organisation, include flocking of birds and teamwork in ant colonies. There is no director or leader but the system acts “as if” there is. 

7. Unpredictability

Ernst Mayr (This is Biology, p.19) defines emergence as “in a structured system, new properties emerge at higher levels of integration that could not have been predicted from a knowledge of the lower-level components.” Philip Ball also defines emergence in terms of unpredictability (Quanta, 2024).

More broadly, in discussions of emergence, “prediction” is used in three different senses: logical prediction, historical prediction, and dynamical prediction.

Logical prediction (deduction) concerns whether one can predict (calculate) the emergent (novel) property of the whole system solely from a knowledge of all the properties of the parts of the system and their interactions. Logical predictability is one of the most contested characteristics of emergence. Sometimes “predict” is replaced with “difficult to predict”, “extremely difficult to predict”, “impossible to predict”, “almost impossible to predict”, or “possible in principle, but impossible in practice, to predict.” 

As an aside, I note that philosophers distinguish between epistemological emergence and ontological emergence. They are associated with prediction that is "possible in principle, but difficult in practice" and "impossible in principle" respectively.

After an emergent property has been discovered experimentally sometimes it can be understood in terms of the properties of the system parts. In a sense “pre-diction” then becomes “post-diction.” An example is the BCS theory of superconductivity, which provided a posteriori, rather than a priori, understanding. In different words, development of the theory was guided by a knowledge of the phenomena that had already been observed and characterised experimentally. Thus, a keyword in the statement above about logical prediction is “solely”. 

Historical prediction. Most new states of matter discovered by experimentalists were not predicted even though theorists knew the laws that the microscopic components of the system obeyed. Examples include superconductivity (elemental metals, cuprates, iron pnictides, organic charge transfer salts, …), superfluidity in liquid 4He, antiferromagnetism, quasicrystals, and the integer and fractional quantum Hall states.

There are a few exceptions where theorists did predict new states of matter. These include are Bose-Einstein Condensates (BECs) in dilute atomic gases and topological insulators, the Anderson insulator in disordered metals, the Haldane phase in even-integer quantum antiferromagnetic spin chains, and the hexatic phase in two dimensions. It should be noted that prediction of BECs and topological insulators were significantly helped that theorists could predict them starting with Hamiltonians of non-interacting particles. Furthermore, all of these predictions involved working with effective Hamiltonians. None started with microscopic Hamiltonians for specific materials.

Dynamical unpredictability concerns what it means in chaotic dynamical systems, where it relates to sensitivity to initial conditions. I do not see this as an example of emergence as it can occur in systems with only a few degrees of freedom. However, some authors do associate dynamical unpredictability with complexity and emergence.

8. Irreducibility and singularities

An emergent property cannot be reduced to properties of the parts, because if emergence is defined in terms of novelty, the parts do not have the property. 

Emergence is also associated with the problem of theory reduction. Formally, this is the process where a more general theory reduces in a particular mathematical limit to a less general theory. For example, quantum mechanics reduces to classical mechanics in the limit where Planck’s constant goes to zero. Einstein’s theory of special relativity reduces to Newtonian mechanics in the limit where the speeds of massive objects become much less than the speed of light. Theory reduction is a subtle philosophical problem that is arguably poorly understood both by scientists [who oversimplify or trivialise it] and philosophers [who arguably overstate the problems it presents for science producing reliable knowledge]. Subtleties arise because the two different theories usually involve language and concepts that are "incommensurate" with one another. 

Irreducibility is also related to the discontinuities and singularities associated with emergent phenomena. As emphasised independently by Hans Primas and Michael Berry, singularities occur because the mathematics of theory reduction involves singular asymptotic expansions. Primas illustrates this by considering a light wave incident on an object and producing a shadow. The shadow is an emergent property, well described by geometrical optics, but not by the more fundamental theory of Maxwell’s electromagnetism. The two theories are related in the asymptotic limit that the wavelength of light in Maxwell’s theory tends to zero. This example illustrates that theory reduction is compatible with the emergence of novelty. Primas also considers how the Born-Oppenheimer approximation, which is central to solid state theory and quantum chemistry, is associated with a singular asymptotic expansion (in the ratio of the mass of an electron to the mass of an atomic nuclei in the system). 

Berry considers several other examples of theory reduction, including going from general to special relativity, from statistical mechanics to thermodynamics, and from viscous (Navier-Stokes) fluid dynamics to inviscid (Euler) fluid dynamics. He has discussed in detail how the caustics that occur in ray optics are an emergent phenomena and are associated with singular asymptotic expansions in the wave theory.

The philosopher of science Jeremy Butterfield showed rigorously that theory reduction occurred for four specific systems that exhibited emergence, defined by him as a novel and robust property. Thus, novelty is not sufficient for irreducibility.

9. Contextuality and downward causation

Any real system has a context. For example, it has boundary and an environment, both in time and space. In many cases the properties of the system are completely determined by the parts of the system and their interactions. Previous history and boundaries do not matter. However, in some cases the context may have a significant influence on the state of the system. An example is Rayleigh-Bernard convection cells and turbulent flow whose existence and nature are determined by the interaction of the fluid with the container boundaries. A biological example concerns what factors determine the structure, properties, and function that a particular protein (linear chain of amino acids) has. It is now known that the only factor is not just the DNA sequence that encodes for the amino acid sequence, in contradiction to some versions of the Central Dogma of molecular biology.  Other factors may be the type of cell that contains the protein and the network of other proteins in which the particular protein is embedded. Context sometimes matters.

Supervenience is the idea that once the micro level is fixed, macro levels are fixed too. The examples above might be interpreted as evidence against supervenience. Supervenience is used to argue against “the possibility for mental causation above and beyond physical causation.” 

Downward causation is sometimes equated with emergence, particularly in debates about the nature of consciousness. In the context of biology, Denis Noble defines downward causation as when higher level processes can cause changes in lower level properties and processes. He gives examples where physiological effects can switch on and off individual genes or signalling processes in cells, including maternal effects and epigenetics.

10. Intra-stratum closure: informational, causal, and computational

The ideas described below were recently developed by Rosas et al. from a computer science perspective. They defined emergence in terms of universality and discussed its relationship to informational closure, causal closure, and computational closure. Each of these are given a precise technical definition in their paper. Here I give the sense of their definitions. In considering a general system they do not pre-define the micro- and macro- levels of a system but consider how they might be defined so that universality holds, i.e., so that properties at the macro-level are independent of the details of the micro-level (i.e., are universal).

Informational closure means that to predict the dynamics of the system at the macroscale an observer does not need any additional information about the details of the system at the microscale. Equilibrium thermodynamics and fluid dynamics are examples. 

Causal closure means that the system can be controlled at the macroscale without any knowledge of lower-level information. For example, changing the software code that is running on a computer allows one to reliably control the microstate of the hardware of the computer regardless of what is happening with the trajectories of electrons in the computer.

Computational closure is a more technical concept, being defined in terms of “a conceptual device called the ε-(epsilon) machine. This device can exist in some finite set of states and can predict its own future state on the basis of its current one... for an emergent system that is computationally closed, the machines at each level can be constructed by coarse-graining the components on just the level below: They are, “strongly lumpable.” "

Rosas et al., show that informational closure and causal closure are equivalent and that they are more restrictive than computational closure. It is not clear to me how these closures relate to novelty as a definition of emergence.

In summary, emergence means different things to different people. I have listed ten different characteristics that have been associated with emergent properties. They are not all equivalent and so when discussing emergence it is important to be clear about which characteristic one is using to define emergence.

The multi-faceted character of emergence (part 1)

There is more to emergence than novel properties, i.e., where a whole system has a property that the individual components of the system do not have. Here I focus on emergent properties, but in most cases “property” might be replaced with state, phenomenon, or entity. I now discuss ten characteristics often associated with emergence, beyond novelty. Some people include one or more of these characteristics in their definitions of emergence. However, I do not include them in my definition because as I explain some of the characteristics are contentious. Some may not be necessary or sufficient for novel system properties.

The first five characteristics discussed below might be classified as objective (i.e., observable properties of the system) and the second five as subjective (i.e., associated with how an investigator thinks about the system). In different words, the first five are mostly concerned with ontology (what is real) and the second five with epistemology (what we know). The first five characteristics concern discontinuities, universality, diversity, mesoscales, and modification of parts. The second five concern self-organisation, unpredictability, irreducibility, downward causation, and closure. 

1. Discontinuities 

Quantitative changes in the system can become qualitative changes in the system. For example, in condensed matter physics spontaneous symmetry breaking only occurs in the thermodynamic limit (i.e., when the number of particles of the system becomes infinite). More is different. Thus, as a quantitative change in the system size occurs the order parameter becomes non-zero. In a system that undergoes a phase transition at a non-zero temperature, a small change in temperature can lead to the appearance of order and to a new state of matter. For a first-order phase transition, there is discontinuity in properties such as the entropy and density. These discontinuities define a phase boundary in the pressure-temperature diagram. For continuous phase transitions the order parameter is a continuous function of temperature, becoming non-zero at the critical temperature. However the derivative with respect to temperature may be discontinuous and/or thermodynamic properties such as the specific heat and susceptibility associated with the order parameter may approach infinite as the critical temperature is approached.

Two different states of a system are said to be adiabatically connected if one can smoothly deform one state into the other and all the properties of the system also change smoothly. The case of the liquid-gas transition illustrates subtle issues about defining emergence. A discontinuity does not imply a qualitative difference (novelty). On the one hand, there is a discontinuity in the density and entropy of the system as the liquid-gas phase boundary is crossed in the pressure-temperature diagram. On the other hand, there is no qualitative difference between a gas and a liquid. There is only a quantitative difference: the density of the gas is less than the liquid. Albeit sometimes the difference is orders of magnitude. The liquid and gas state can be adiabatically connected. There is a path in the pressure-temperature phase diagram that can be followed to connect the liquid and gas states without any discontinuities in properties.

The ferromagnetic state also raises questions, as illustrated by a debate between Rudolf Peierls and Phil Anderson about whether ferromagnetism exhibits spontaneous symmetry breaking. Anderson argued that it did not as, in contrast to the antiferromagnetic state, a non-zero magnetisation (order parameter) occurs for finite systems and the magnetic order does not change the excitation spectrum, i.e., produce a Goldstone boson. On the other hand, singularities in properties at the Curie temperature (critical temperature for ferromagnetism) only exist in the thermodynamic limit. Also, a small change in the temperature, from just above the Curie temperature to below, can produce a qualitative change, a non-zero magnetisation.

2. Universality

Properties often referred to as emergent are universal in the sense that it is independent of many of the details of the parts of the system. There may be many different systems that can have a particular emergent property. For example, superconductivity is present in metals with a diverse range of crystal structures and chemical compositions. 

Robustness is related to universality. If small changes are made to the composition of the system (for example replacing some of the atoms in the system with atoms of different chemical element) the novel property of the system is still present. In elementary superconductors, introducing non-magnetic impurity atoms has no effect on the superconductivity.

Universality is both a blessing and a curse for theory. Universality can make it easier to develop successful theories because it means that many details need not be included in a theory in order for it to successfully describe an emergent phenomenon. This is why effective theories and toy models can work even better than might be expected. Universality can make theories more powerful because they can describe a wider range of systems. For example, properties of elemental superconductors can be described by BCS theory and by Ginzburg-Landau theory, even though the materials are chemically and structurally diverse. The curse of universality for theory is that universality illustrates the problem of “under-determination of theory”, “over-fitting of data” and “sloppy theories” [Sethna et al.]. A theory can agree with the experiment even when the parameters used in the theory may be quite different from the actual ones. For example, the observed phase diagram of water can be reproduced, sometimes with impressive quantitative detail, by combining classical statistical mechanics with empirical force fields that assume water molecules can be treated purely being composed of point charges.

Suppose we start with a specific microscopic theory and calculate the macroscopic properties of the system, and they agree with experiment. It would then be tempting to think that we have the correct microscopic theory. However, universality suggests this may not be the case.

For example, consider the case of a gas of weakly interacting atoms or molecules. We can treat the gas particles as classical or quantum. Statistical mechanics gives exactly the same equation of state and specific heat capacity for both microscopic descriptions. The only difference may be the Gibbs paradox [the calculated entropy is not an extensive quantity] which is sensitive to whether or not the particles are treated as identical or not. Unlike the zeroth, first, and second law of thermodynamics, the third law does require that the microscopic theory be quantum. Laughlin discusses these issues in terms of “protectorates” that hide “ultimate causes” .  

In some physical systems, universality can be defined in a rigorous technical sense, making use of the concepts and techniques of the renormalisation group and scaling. These techniques provide a method to perform coarse graining, to derive effective theories and effective interactions, and to define universality classes of systems. There are also questions of how universality is related to the robustness of strata, and the independence of effective theories from the coarse-graining procedure.

3. Diversity

Even when a system is composed of a small number of different components and interactions, the large number of possible stable states with qualitatively different properties that the system can have is amazing. Every snowflake is different. Water is found in 18 distinct solid states. All proteins are composed of linear chains of 20 different amino acids. Yet in the human body there are more than 100,000 different proteins and all perform specific biochemical functions. We encounter an incredible diversity of human personalities, cultures, and languages. A stunning case of diversity is life on earth. Billions of different plant and animal species are all an expression of different linear combinations of the four base pairs of DNA: A, G, T, and C.

This diversity is related to the idea that "simple models can describe complex behaviour". One example is Conway’s Game of Life. Another example is how simple Ising models with a few competing interactions can describe a devil's staircase of ground states or the multitude of different atomic orderings found in binary alloys.

Goldenfeld and Kadanoff defined complexity [emergence] as “structure with variations”. Holland (VSI) discusses “perpetual novelty” giving the example of the game of chess, where are typical game may involve the order of 1050 move sequences. “Motifs” are recurring patterns (sequences of moves) in games. 

Condensed matter physics illustrates diversity with the many different states of matter that have been discovered. The underlying microscopics is “just” electrons and atomic nuclei interacting according to Coulomb’s law.

The significance of this diversity might be downplayed by saying that it is just a result of combinatorics. But such a claim overlooks the issue of the stability of the diverse states that are observed. In a system composed of many components each of which can take on a few states the number of possible states of the whole system grows exponentially with the number of components. For example, for a chain of ten amino acids there are 1013 different possible linear sequences. But this does not mean that all these sequences will produce a functional protein, i.e., a molecule that will fold rapidly (on the timescale of milliseconds) into a stable tertiary structure and perform a useful biochemical function such as catalysis of a specific chemical reaction or signal transduction.

4. Simple entities at the mesoscale 

A key idea in condensed matter physics is that of quasi-particles. A system of strongly interacting particles may have excitations, seen in experiments such as inelastic neutron scattering and Angle Resolved PhotoElectron Spectroscopy (ARPES), that can be described as weakly interacting quasi-particles. These entities are composite particles, and have properties that are quantitatively different, and sometimes qualitatively different, from the microscopic particles. Sometimes this means that the scale (size) associated with the quasi-particles is intermediate between the micro- and the macro-scales, i.e., it is a mesoscale. The existence of quasi-particles leads naturally to the technique of constructing an effective Hamiltonian [effective theory] for the system where effective interactions describe the interactions between the quasi-particles.

The economist Herbert Simon argued that a characteristic of a complex system is that the system can be understood in terms of nearly decomposable units. Rosas et al., argue that emergence is associated with there being a scale at which the system is “strongly lumpable”. Denis Noble has highlighted how biological systems are modular, i.e., composed of simple interchangeable components.

5. Modification of parts and their relationships

Emergent properties are often associated with the state of the system exhibiting patterns, order, or structure, terms that may be used interchangeably. This reflects that there is a particular relationship (correlation) between the parts which is different to the relationships in a state without the emergent property. This relationship may also be reflected in a generalised rigidity. For example, in a solid applying a force on one surface results in all the atoms in the solid experiencing a force and moving together. The rigidity of the solid defines a particular relationship between the parts of the system.

Properties of the individual parts may also be different. For example, in a crystal single-atom properties such as electronic energy levels change quantitatively compared to their values for isolated atoms. Properties of finite subsystems are also modified, reflecting a change in interactions between the parts. For example, in a molecular crystal the frequencies associated with intramolecular atomic vibrations are different to their values for isolated molecules. However, emergence is a sufficient but not a necessary condition for these modifications. In gas and liquid states, novelty is not present but there are still such changes in the properties of the individual parts.

As stated at the beginning of this section the five characteristics above might be associated with ontology (what is real) and objective properties of the system that an investigator observes and depend less on what an observer thinks about the system. The next five characteristics might be considered to be more subjective, being concerned with epistemology (how we determine what is true). In making this dichotomy I do not want to gloss over the fuzziness of the distinction or of two thousand years of philosophical debates about the relationship between ontology and epistemology, or between reality and theory.

In the next post, I will discuss the remaining five characteristics: self-organisation, unpredictability, irreducibility, contextuality and downward causation, and intra-stratum closure.

Thanks for reading this far!

Do arrows of explanation point down or up?

The figure above shows the stratification of objects that interest physicists. As one goes down the chain length and time scales get smaller and energy scales get larger.

A reductionist seeks to explain the objects at each strata in terms of the objects that occur at the next lower strata.

In 1987 Steven Weinberg gave a talk at the University of Cambridge at the Tercentenary Celebration of Newton's Principia.

The talk was published in Nature. Newtonianism, Reductionism, and the Art of Congressional Testimony

Part of the talk is about Weinberg's testimony to a US Congressional Committee making the case for the construction of the SSC (Superconducting Super Collider). Phil Anderson spoke against the SSC.

Weinberg argued that the SSC should be built because particle physics is "in some sense more fundamental than other areas of physics." He claims that this is because "the arrows of explanation point down", as in the diagram shown above.

A contrasting perspective is that of Andrew Steane. His book, Science and Humanity, contains the figure below.

In his picture of the explanatory relationship between physics, chemistry, and biology, Steane draws arrows pointing in both directions. The up arrow is denoted “supports [allows and physically embodies the expression of]” and the down arrow is denoted “enarches [exhibits the structures and behaviours that make sense in their own terms and are possible within the framework of].”

Weinberg's article is worth reading in full. It has many insights about science and physics worth considering, including the relationship between emergence and reductionism.

Aside: It is also reproduced in his book of essays, Facing Up: Science and Its Cultural Adversaries, published in 2001.

Condensed matter physics in 15 minutes!

Oxford University Press has a nice podcast on Very Short Introductions. 

In each episode, an author of a specific volume has 10-15 minutes to introduce themself and answer several questions.

What is X [the subject of the VSI]?

What got you first interested in X?

What are the key aspects of X that you would like everyone to know?

The ones I have listened to and particularly liked are Infinity, Philosophy of Science, Evangelicalism, Development, Consciousness, Behavioural Economics, and Modern China.

Tomorrow, I am recording an episode for Condensed Matter Physics: A Very Short Introduction.

Here is a practise version of the audio and the draft text is below. 

I welcome feedback.

VSI Podcast 

I am Ross McKenzie. I am an Emeritus professor of physics at the University of Queensland in Brisbane, Australia. I have spent the past forty years learning, teaching, and researching condensed matter physics. I really love the Very Short Introduction series and so I am delighted to share my experience by writing Condensed Matter Physics: A Very Short Introduction.

What is condensed matter physics? It is all about states of matter. At school, you were probably taught that there are only three states of matter: solid, liquid, and gas. This is wrong. There are many more states such as liquid crystal, glass, superconductor, ferromagnet, and superfluid. New states of matter are continually, and often unexpectedly, being discovered. Condensed matter physics investigates how the distinct physical properties of states of matter emerge from the atoms of which a material is composed.

What first got me interested in condensed matter physics?

After I finished an undergraduate degree in theoretical physics in Australia in 1982, I would not have been able to answer the question, “what is condensed matter physics?”, even though it is the largest sub-field of physics. I then went to Princeton University in the USA to pursue a Ph.D. in and I took an exciting course on the subject and began to interact with students and faculty working in the field. 

At Princeton was Phil Anderson, who had won a Nobel Prize in physics for work in condensed matter. At the time I did not appreciate his much broader intellectual legacy. In his recent biography of Anderson, Andrew Zangwill states “more than any other twentieth-century physicist, he [Anderson] transformed the patchwork of ideas and techniques formerly called solid-state physics into the deep, subtle, and intellectually coherent discipline known today as condensed matter physics.” Several decades later, my work became richer as Anderson gave me an appreciation of the broader scientific and philosophical significance of condensed matter physics, particularly its connection to other sciences, such as biology, economics, and computer science. When do quantitative differences become qualitative differences? Can simple models describe rich and complex behaviour? What is the relationship between the particular and the universal? How is the abstract related to the concrete?

So what are the key aspects of condensed matter physics that I would like everyone to know?

First, there are many different states of matter. It is not just solid, liquid, and gas. Consider the “liquid crystals” that are the basis of LCDs (Liquid Crystal Displays) in the screens of televisions, computers, and smartphones. How can something be both a liquid and a crystal? A liquid crystal is a distinct state of matter. Solids can be found in many different states. In everyday life, ice means simply solid water. But there are in fact eighteen different solid states of water, depending on the temperature of the water and the pressure that is applied to the ice. In each of these eighteen states, there is a unique spatial arrangement of the water molecules and there are qualitative differences in the physical properties of the different solid states.

Condensed matter physics is concerned with characterising and understanding all the different states of matter that can exist. These different states are called condensed states of matter. The word “condensed’’ is used here in the same sense as when we say that steam condenses into liquid water. Generally, as the temperature is lowered or the pressure is increased, a material can condense into a new state of matter. Qualitative differences distinguish the many different states of matter. These differences are associated with differences in symmetry and ordering.

Second, condensed matter physics involves a particular approach to understanding properties of materials. Every day we encounter a diversity of materials: liquids, glass, ceramics, metals, crystals, magnets, plastics, semiconductors, and foams. These materials look and feel different from one another. Their physical properties vary significantly: are they soft and squishy or hard and rigid? Shiny, black, or colourful? Do they absorb heat easily? Do they conduct electricity? The distinct physical properties of different materials are central to their use in technologies around us: smartphones, alloys, semiconductor chips, computer memories, cooking pots, magnets in MRI machines, LEDs in solid-state lighting, and fibre optic cables. Why do different materials have different physical properties? 

Materials are studied by physicists, chemists, and engineers, and the questions, focus, goals, and techniques of researchers from these different disciplines can be quite different. The focus of condensed matter physics is on states of matter. Condensed matter physics as a research field is not just defined by the objects that it studies (states of matter in materials), but rather by a particular approach to the study of these objects. The aim is to address fundamental questions and to find unifying concepts and organizing principles to understand a wide range of phenomena in materials that are chemically and structurally diverse. 

The central question of condensed matter physics is, how do the properties of a state of matter emerge from the properties of the atoms in the material and their interactions? 

Let’s consider a concrete example, that of graphite and diamond. While you will find very cheap graphite in lead pencils, you will find diamonds in jewelery. Both graphite and diamond are composed solely of carbon atoms. They are both solid. So why do they look and feel so different?  Graphite is common, black, soft, and conducts electricity moderately well. In contrast, diamond is rare, transparent, hard, and conducts electricity very poorly. We can zoom in down to the scale of individual atoms using X-rays and find the spatial arrangement of the carbon atoms relative to one another. These arrangements are qualitatively different in diamond and graphite.. Diamond and graphite are distinct solid states of carbon. They have qualitatively different physical properties, at both the microscopic and the macroscopic scale. 

Third, I want you to know about superconductivity, one of the most fascinating states of matter. I have worked on it many times over the past forty years. Superconductivity occurs in many metals when they are cooled down to extremely low temperatures, close to absolute zero (-273 ºC). In the superconducting state, a metal can conduct electricity perfectly; without generating any heat. This state also expels magnetic fields meaning one can levitate objects, whether sumo wrestlers or trains. 

The discovery of superconductivity in 1911 presented a considerable intellectual challenge: what is the origin of this new state of matter? How do the electrons in the metal interact with one another to produce superconductivity? Many of the greatest theoretical physicists of the twentieth century took up this challenge but failed. The theoretical puzzle was only solved 46 years after the experimental discovery. The theory turns out also to be relevant to liquid helium, nuclear physics, neutron stars, and the Higgs boson. New superconducting materials and different superconducting states continue to be discovered. A “holy grail” is to find a material that can superconduct at room temperature. 

I find superconductivity even more interesting when considering quantum effects. By 1930 it was widely accepted that quantum theory, in all its strangeness, describes the atomic world of electrons, protons, and photons. However, this strangeness does not show itself in the everyday world of what we can see and touch. You cannot be in two places at the same time. Your cat is either dead or alive. However, condensed matter physicists have shown that the boundary between the atomic and macroscopic worlds is not so clear cut. A piece of superconducting metal can take on weird quantum properties, just like a single atom, even though the metal is made of billions of billions of atoms. It is in two states at the same time, almost like Schrodinger’s famous cat.

Fourth, condensed matter physics is all about emergence; the whole is greater than the sum of the parts. A system composed of many interacting parts can have properties that are qualitatively different from the properties of the individual parts. Water is wet, but a single water molecule is not. Your brain is conscious, but a single neuron is not. Such emergent phenomena occur in many fields, from biology to computer science to sociology, leading to rich intellectual connections. Condensed matter physics is arguably the field with the greatest success at understanding emergent phenomena in complex systems, particularly at the quantitative level. This is not because condensed matter physicists are smarter than sociologists, economists, or neuroscientists. It is because the materials we study are much “simpler” than societies, economies, and brains. 

Finally, condensed matter physics is one of the largest and most vibrant sub-fields of physics. For example, in the past thirty years, the Nobel Prize in Physics has been awarded thirteen times for work on condensed matter. In the past twenty years, eight condensed matter physicists have received the Nobel Prize in Chemistry. 

I hope I have sparked your interest in condensed matter physics. I invite you to learn more about why I consider this field of science significant, beautiful, and profound.