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!

What is really fundamental in science?

What do we mean when we say something in science is fundamental? When is an entity or a theory more fundamental or less fundamental than something else? For example, are quarks and leptons more fundamental than atoms? Is statistical mechanics more fundamental than thermodynamics? Is physics more fundamental than chemistry or biology? In a fractional quantum Hall state, are electrons or the fractionally charged quasiparticles more fundamental?

Answers depend on who you ask. Physicists such as Phil Anderson, Steven Weinberg, Bob Laughlin, Richard Feynman, Frank Wilczek, and Albert Einstein have different views.

In 2017-8, the Foundational Questions Institute (FQXi) held an essay contest to address the question, “What is Fundamental?” Of the 200 entries, 15 prize-winning essays have been published in a single volume. The editors give a nice overview in the Introduction.

This post is mostly about the essay, Fundamental? of the first prize winner, Emily Adlam, a philosopher of physics. She contrasts two provocative statements.

Fundamental means we have won. The job is done and we can all go home.

Fundamental means we have lost. Fundamental is an admission of defeat.

This raises the question of whether being fundamental is objective or subjective.

Examples are given from scientific history to argue that what is considered to be fundamental has changed with time. The reductionism has led to the drive to explain everything in terms of smaller and smaller entities, that are deemed 'more fundamental". But we find that smaller does not always mean simpler.

Perhaps we should ask what needs explaining and what constitutes a scientific explanation. For example, Adlam asks whether explaining the fact that the initial state of the universe had a low entropy [the "past hypothesis"] is really possible or should be an important goal.

She draws on the issue of the distinction between objective and subjective probabilities. Probabilities in statistical mechanics are subjective: they are a statement about our own ignorance about the details of the motion of individual atoms and not any underlying randomness in nature. In contrast, probabilities in quantum theory reflect objective chance.

as realists about science we must surely maintain that there is a need for science to explain the existence of the sorts of regularities that allow us to make reliable predictions... but there is no similarly pressing need to explain why these regularities take some particular form rather than another. Yet our paradigmatic mechanical explanations do not seem to be capable of explaining the regularity without also explaining the form, and so increasingly in modern physics we find ourselves unable to explain either. 

It is in this context that we naturally turn to objective chance. The claim that quantum particles just have some sort of fundamental inbuilt tendency to turn out to be spin up on some proportion of measurements and spin down on some proportion of measurements does indeed look like an attempt to explain a regularity (the fact that measurements on quantum particles exhibit predictable statistics) without explaining the specific form (the particular sequence of results obtained in any given set of experiments). But given the problematic status of objective chance, this sort of nonexplanation is not really much better than simply refraining from explanation at all. 

Why is it that objective chances seem to be the only thing we have in our arsenal when it comes to explaining regularities without explaining their specific form? It seems likely that part of the problem is the reductionism that still dominates the thinking of most of those who consider themselves realists about science

In summary, (according to the Editors) Adlam argues that "science should be able to explain the existence of the sorts of regularities that allow us to make reliable predictions. But this does not necessarily mean that it must also explain why these regularities take some particular form." 

we are in dire need of another paradigm shift. And this time, instead of simply changing our attitudes about what sorts of things require explanation, we may have to change our attitudes about what counts as an explanation in the first place. 

Here, she is arguing that what is fundamental is subjective, being a matter of values and taste.

In our standard scientific thinking the fundamental is elided with ultimate truth: getting to grips with the fundamental is the promised land, the endgame of science. 

She then raises questions about the vision and hopes of scientific reductionists. 

In this spirit, the original hope of the reductionists was that things would get simpler as we got further down, and eventually we would be left with an ontology so simple that it would seem reasonable to regard this ontology as truly fundamental and to demand no further explanation. 

But the reductionist vision seems increasingly to have failed. 

When we theorise beyond the standard model [BSM] we usually find it necessary to expand the ontology still more: witness the extra dimensions required to make string theory mathematically consistent.

It is not just strings. Peter Woit has emphasised how BSM theories, such as supersymmetry, introduce many more particles and parameters.

... the messiness deep down is a sign that the universe works not ‘bottom-up’ but rather ‘top-down,’ ... in many cases, things get simpler as we go further up.

Our best current theories are renormalisable, meaning that many different possible variants on the underlying microscopic physics all give rise to the same macroscopic physical theory, known as an infrared fixed point. This is usually glossed as providing an explanation of why it is that we can do sensible macroscopic physics even without having detailed knowledge of the underlying microscopic theories. 

For example, elasticity theory, thermodynamics and fluid dynamics all work without knowing anything about atoms, statistical mechanics, and quantum theory.

But one might argue that this is getting things the wrong way round: the laws of nature don’t start with little pieces and build the universe from the bottom up, rather they apply simple macroscopic constraints to the universe as a whole and work out what needs to happen on a more fine-grained level in order to satisfy these constraints.

This is rather reminiscent of Laughlin's views about what is fundamental.

Finally, I mention two other essays that I look forward to reading as I think they make particularly pertinent points.

Marc Séguin (Chap. 6) distinguishes "between epistemological fundamentality (the fundamentality of our scientific theories) and ontological fundamentality (the fundamentality of the world itself, irrespective of our description of it)."

"In Chap. 12, Gregory Derry argues that a fundamental explanatory structure should have four key attributes: irreducibility, generality, commensurability, and fertility."

[Quotes are from the Introduction by the Editors].

Some would argue that the Standard Model is fundamental, at least on some level. But it involves 19 parameters that have to be fixed from experiment. Related questions about the Fundamental Constants, have been explored in a 2007 paper by Frank Wilczek.

Again, I thank Peter Evans for bringing this volume to my attention.

Basic introductions to Condensed Matter Physics

Suppose a motivated and intelligent high school student or first year undergraduate comes to you and says, ``Condensed matter physics sounds really cool! What should I read or look at to learn more about it?"

Obviously, suggesting the student look at classic graduate texts such as Ashcroft and Mermin or Chaikin and Lubensky is not helpful. They need something that will inspire them to want to learn more as well as introduce them to some of the basic ideas and topics.

I would suggest the following.

David Pines, Unit 8 in Physics for the 21st Century, an on-line course
Emergent Behavior in Quantum Matter

Robert Laughlin, A Different Universe: Reinventing Physics from the Bottom Down

Stephen Blundell, Superconductivity: A Very Short Introduction

Rodney Cotterell, The Material World

But when then have read some of these it would be nice if the student could look at something more technical. To second year undergrads I give a series of lectures on Thermodynamics and Condensed Matter Physics. They don't need to know any quantum or stat. mech., just some thermo, and they can still get some of the flavour, excitement, and scope of the subject. But, I don't know a book that lays this material out clearly and simply. I draw on Schroeder, Thermal Physics, but it has no discussion of superfluids, order parameters, or symmetry breaking.

What do you think are good resources?

I thank Alex Agedah for asking this question.

Update. Here are some slides for a talk that Danielle McDermott gave on the subject. It lists many useful resources. (She mentions it in a comment below).

Quantum emergence is not strong emergence

Is there any difference in the nature of emergence in quantum and classical systems?
What is the difference between strong and weak emergence?

An emergent property of a system is one that is:
a. not present in the individual components of the system
b. difficult to predict a priori from a knowledge of the components and their interactions
c. independent of the finer details of the components

Equivalently emergent properties are
a. qualitatively different
b. usually discovered empirically and sometimes are given a reductionist explanation a posteriori
c. universal and stable to perturbations

This can be illustrated with the rigidity of a solid
a. the individual atoms that make up a solid are not rigid.
b. elasticity theory preceded crystallography
c. all solids are rigid, regardless of their chemical composition.

Emergence occurs in both quantum and classical systems.  The properties that emerge can be distinctly different.  Superconductivity  and superfluidity are intrinsically quantum.
However, the associated issues and challenges: scientific, methodological, and philosophical are essentially the same. Emergence in classical systems is just as fascinating and challenging as for quantum systems.

Hence, last year I was surprised and disappointed to read the details of The Physics of Emergence program at the Templeton Foundation.
It appears to be based on two significant misunderstandings:
Emergence in quantum and classical systems is profoundly different.
In particular, quantum and classical emergence should be identified with strong and weak emergence, respectively.
I disagree with both the preceding two statements.

What is the difference between strong and weak emergence?
Some philosophers equate these with ontological and epistemological emergence.
For practical scientists the issue boils down to the following possible
answers to the question, "Is it possible to predict emergent properties?":
i. No. It is impossible.
ii. No. But, one can make postdictions, i.e., once the phenomena has been observed very smart people can construct reductionist models that explain the phenomena.  [BCS theory is an example].
iii. Yes. But, it is difficult. BECs and topological insulators give us hope.
iv. Yes. We just need a little more computer power and creativity.

The believer in strong emergence says i. All the other answers amount to weak emergence.
Different scientists will answer ii, iii, or iv.
I would probably go with ii.
The only scientist who I think might answer i. is Bob Laughlin on his more cantankerous days.
Yet i. appears to be serious option for many philosophers. This seems to be largely because of the thorny issue of consciousness.

A political metaphor for the correlated electron community

It is the conservatives vs. the radicals, the right vs. the left.

A colleague recently suggested to me that this is a good metaphor or analogue for describing and understanding the divisions in the physics community working on the theory of correlated electron materials.

In the USA political divisions have led to a "gridlock" that is stopping the country moving forward. Both conservatives and liberals have a rigid ideology that prevents them from seeing the merits
of their opponents concerns and from being willing to compromise. Conservatives believe one should never raise taxes. Liberals believe one should never cut social welfare programs.
Both "cherry pick" economic data to support their point of view.

Historically, political radicals believe that capitalism is a flawed and unstable system that must be replaced by some new, but unknown, system.

The world is more complex than political ideology concedes.
Both radicals and conservatives have extreme beliefs that I sometimes find simplistic.
"The only way to reduce crime is to put more people in prison."
vs.
"Crime is just a result of social injustice."
"Poverty can only be solved by economic growth. That means less taxes on big business and the wealthy."
vs.
"The poor are helpless. Government welfare programs will solve their problems."

I really like the book Poor Economics because the authors [MIT economists Abhijit Banerjee and Esther Duflo] do not see the developing world through the extreme eyes of the left [represented by Jeffrey Sachs] or the right [represented by William Easterly].
Instead, the authors actually do randomised trial experiments to obtain empirical data to see what does and does not work in poverty alleviation. They find that sometimes the right is correct and sometimes the left is correct. Sometimes neither. The world is complex.

So what does this have to do with the theory of correlated electron materials?
On one side we have the conservatives who believe that the key ingredients are atomistic detail, good density functionals, perturbation theory, mean-field theory, and the random-phase approximation.
New concepts and methods are not really needed. They have a good system [just like capitalism].
In particular, we don't need a revolution, just bigger computers!
Perhaps they are represented by Igor Mazin, David Singh, Warren Pickett, Olle Andersen, ... The former three all have a career connection to Naval Research Laboratory.
Unlike the radicals below, it is not clear to me that the conservatives have a clear ideological or inspirational leader.

The radicals believe in universality. Atomistic detail is largely irrelevant. It is all about collective behaviour [not individuality].
Completely new conceptual structures and techniques are needed. We must go beyond Landau's mean-field theory and Fermi liquid theory: topological order, quasi-particles with fractional quantum numbers, AdS-CFT, quantum criticality, ...
To the barricades!
Phil Anderson is an enduring inspiration for the radicals just as Marx still is for political radicals.
Indeed many of the radicals [Wen, Patrick Lee, Viswanath, Haldane, ...] have some historical connection to Princeton or Anderson.
I think Anderson is like Marx in that he defines the problems, and asks the hard questions; but I am not sure the answers are right.
But for some, even Marx is not radical enough.
Perhaps, Subir Sachdev is like Lenin with his AdS/CFT comrades. Anderson accuses them of "quasi-journalism". Perhaps, just like Leninists they consider propaganda is also good for their cause.
Bob Laughlin is an aging dis-illusioned radical who has become dis-engaged from the political process. He was a Berkeley undergrad, after all!
Perhaps, Piers Coleman is like a European Social Democrat.

Each side is largely dis-engaged from the other and appears unwillingly to acknowledge the merits of their opponents point of view. This is bad for the field, just like uncompromising political divisions are bad for countries.

Gabi Kotliar has roots on the left, but is moving more towards the right as he grows older.  He and Andy Millis are probably a disappointment to both the left and the right, just like Obama!
I am also caught in the middle, with slightly more sympathy for the left than the right, just like in politics. As in politics, I am troubled at some of the extreme views I see on both the left and right. Inconvenient data is ignored.

Can the community move beyond ideology, see and respect others point of view, and work together?
It is interesting that topological insulators have actually led to some constructive dialogue and co-operation between left and right.

Aside: What about the theoretical chemistry community?
I feel it is dominated by the conservatives. Do they need some radicals to shake them up?

Note to university administrators: this is a personal blog and has a disclaimer at the bottom of the page.

Is meso the new nano?

There is an interesting article Emergent Physics the Mesoscale: Report from the special Kavli session at the 2012 APS March meeting by Sam Bader on The Back Page of the May edition of the American Physical Society News.

It sounds like there was a fascinating and contrasting series of talks by Bob Laughlin, Bill Phillips, Angela Belcher, Bill Bialek, and George Whitesides. Apparently Phillips "gave short shrift of the concept of emergence, discarding it mercilessly."

I would be interested to see copies of the talks. Has anyone seen them online?

Overall, this latest focus on the "meso" seems to be driven by hopes of a new burst of funding like what happened with nanotechnology in 2000. [See this brief piece in Science] In the end I think that initiative was a big disappointment scientifically. I feel the whole field was hijacked by people who just relabelled whatever they were doing as nanoscience or nanotechnology. To me it should have been all about control and manipulation at the nanoscale, e.g., single molecule electronics.

Desperately seeking a way to find order parameters

Much of condensed matter physics is concerned with finding the relevant order parameter for new phases of matter. Indeed this is a good way to win a Nobel Prize! This is much of what was done by Ginzburg, Neel, Leggett, de Gennes,....

A fundamental and controversial question is whether one can a priori predict new order parameters. Historically, the progression has always been:

1. Experimental discovery of a new phase of matter.
2. Proposal of an order parameter and a phenomenological (Ginzburg-Landau) theory to explain a range of experiments.
3. Proposal of an effective Hamiltonian which has a ground state with the desired spontaneous symmetry breaking and associated order parameter.
4. Justification of the effective Hamiltonian from so-called "ab initio" electronic structure calculations starting with Schrodinger's equation and the actual chemical composition of specific materials. 

The grand challenge is to invert this process, even just one step.

Laughlin and Pines seem to claim that this is essentially impossible.

There are some interesting fundamental philosophical questions as to whether the obstacles are ones of practical difficulty versus fundamental physics.

There is really interesting 2006 PRL, Systematic Derivation of Order Parameters through Reduced Density Matrices, by Shunsuke Furukawa, Grégoire Misguich, and Masaki Oshikawa.
Essentially they claim to have found a way to go from 3. to 2. above. In particular, given the results of an exact diagonalisation calculation of the low lying states of a lattice model they give a procedure to find the order parameter from looking at two nearly degenerate ground states.
They then apply the method to two concrete examples: a Heisenberg spin model on a ladder with ring exchange, and a quantum dimer model on the Kagome lattice. The method gives the correct order parameters in the first case and for the second shows there is no order parameter. I found this quite impressive and promising.

The PRL also promises future work generalising the method to more than two degenerate ground states and suggests application to frustrated two-dimensional quantum antiferromagnets. Unfortunately, I have not been able to find such work.

The most precise measurement of Planck's constant

Compared to some fields (e.g. biology, high energy physics, philosophy, history, movies, historical theology) I think most Wikipedia articles on condensed matter physics and theoretical chemistry are sporadic in quality. It would be wonderful if someone took the initiative and time to improve them. I keep trying to encourage students to do this but not succeeding. But, that is another story....
The actual purpose of this post is just to highlight some of the actual content of a really nice entry on the Magnetic flux quantum, [h/2e] which states:

The magnetic flux quantum may be measured with great precision by exploiting the Josephson effect. In fact, when coupled with the measurement of the von Klitzing constant R_K = h/e², this provides the most precise values of Planck's constant h obtained to date. This is remarkable since h is generally associated with the behavior of microscopically small systems, whereas the quantization of magnetic flux in a superconductor and the quantum Hall effect are both collective phenomena associated with thermodynamically large numbers of particles.

This point is also stressed and discussed at length in Bob Laughlin's book A Different Universe which this blog contains many quotes from.

Is space-time emergent?

On Friday at UQ we had a very stimulating colloquium A new view on quantum gravity and the origin of the Universe by Bei-Lok Hu (University of Maryland). A key aspect of this new view is that general relativity and space time should be viewed as emergent phenomena (more below).

There are six main points of experimental evidence in cosmology:
1. Hubble expansion of the universe.
2. Cosmic microwave background radiation (isotropy and uniformity).
3. Element abundance (+ nucleosynthesis)
4. Ratio of baryon/photon (entropy content of universe)
5. Structure: galaxy, clusters,...   hierarchy of scales
6. Cosmological constant ~ 0,  vacuum energy density

The fact that the night sky is dark implies a finite universe, and expansion or a hierarichial structure  (Olber's paradox).

Hu contrasted Two views of quantum gravity.

1. Bottom up view
Quantum gravity = quantisation of general relativity
This is the more traditional view and has been dominant.

2. Top down view
Gravity is emergent  and general relativity should be viewed as the "hydrodynamics" of some underlying "microscopic" theory.
This means that one must deal with a micro-macro transition as well as a quantum-classical one. This view has become more popular in the last 5 years.

Hu's advocacy of 2. is summarised in detail in a conference paper.

It should be pointed out that Bob Laughlin has also advocated such a perspective. His book, A Different Universe, has a chapter, The Fabric of Space-Time, which ends with the claim that if Einstein were alive today he would,

conclude that his beloved principle of relativity was not fundamental at all but emergent - a collective property of the matter constituting space-time that becomes increasingly exact at long length scales but fails at short ones. This is a  different idea from his original one but something fully compatible with it logically, and even more exciting and potentially important. It would mean that the fabric of space-time was not simply the stage on which life played out but an organizational phenomenon, and that there might be something beyond.

R.B. Laughlin, A Different Universe, p. 126

[See also his 2004 Perspective, The Cup of the Hand, in Science].

On monday we will have another colloquium, this one by Thanu Padmanabhan (IUCAA, Pune University), and advocating a similar view, and summarised in this conference paper.

Glueing together a theory

In elemental superconductors Cooper pairs form due to an attractive retarded electron-electron interaction that is mediated by exchange of phonons. A perennial and controversial question concerning superconductivity in strongly correlated electron materials (particularly the cuprates and organic charge transfer salts) is whether there is a similar "pairing glue". In a 2007 Science paper Phil Anderson claimed there is not. 

[Cartoons are by Bob Laughlin]

A nice PRL by Maier, Poilblanc, and Scalapino addresses the question in a precise manner. The abstract has a nice summary of the issue.

The question of whether one should speak of a “pairing glue” in the Hubbard and t-J models is basically a question about the dynamics of the pairing interaction. If the dynamics of the pairing interaction arises from virtual states, whose energies correspond to the Mott gap, and give rise to the exchange coupling J, the interaction is instantaneous on the relative time scales of interest. In this case, while one might speak of an “instantaneous glue”, this interaction differs from the traditional picture of a retarded pairing interaction. However, as we will show, the dominant contribution to the pairing interaction for both of these models arises from energies reflecting the spectrum seen in the dynamic spin susceptibility. In this case, the basic interaction is retarded, and one speaks of a spin-fluctuation glue which mediates the d-wave pairing.

One thing that I found interesting and impressive about the paper is that it shows how one can define a dynamical pairing function in a non-perturbative manner. The relevant three equations are below. The Gorkov pairing (anomalous) Greens function F is defined by

which then leads to the pairing function 

 where G is the normal Greens function. This function has a real and imaginary part which can be related by Kramers-Kronig (or Cauchy) relations. The static part gives the pairing which can be written

The paper concludes:

The question regarding whether there is a “pairing glue” is then a question of whether the dominant contribution to φ1(kA,ω = 0) comes from the integral of φ2(kA, ω)/ω. From the results presented here we conclude that both the t-J and Hubbard models have spin-fluctuation “pairing glue”. However, they also exhibit a smaller, non-retarded contribution. For the cuprate materials, the relative weight of the retarded and non-retarded contributions to the pairing interaction remains an open question. Thus the continuing experimental search for a pairing glue in the cuprates is important and will play an essential role in determining the origin of the high Tc pairing interaction.

Physics hubris?

There is an interesting piece on the New York Times web site which quotes numerous scientists slamming Bob Laughlin's recent article about climate change. I wish he would go back to doing regular science....

On a more positive note Laughlin has a worthwhile project that is digitising Conrad Herring's famous file card box.

On his website he also has all the cartoons from his wonderful book, The Emergent Universe.

Broken symmetry, broken heart

A week ago, we had a very interesting Physics department colloquium by Marcelo Gleiser about his recent book, A Tear at the Edge of Creation: A Radical New Vision for Life in an Imperfect Cosmos. 
He discussed his growing disillusion with string theory and the search for a "theory of everything" which is based on the predominance of symmetry. He gave important examples of symmetry breaking in nature including CP violation in the electro-weak interactions [which because of the CPT theorem implies time reversal invariance] and the unique chirality of amino acids in proteins.

Although it was a nice talk I thought it was all a bit sad to see someone who had become so enamoured with the propaganda of the reductionism in the high-energy physics community that it was painful when doubts emerged. Most of the points Gleiser was making seem to me to have been made long ago (in a more constructive sense) by Anderson in his 1972 "More is Different" article in Science, and more passionately and more recently by Laughlin in A Different Universe. 

Our tendency to scientific fantasy not reality

More great quotes from Bob Laughlin, A Different Universe: Reinventing Physics from the Bottom Down

“The great power of science is its ability, through brutal objectivity, to reveal to us truth we did not anticipate.”
(p. xvi)

``mythologies are immensely powerful things, and sometimes we humans go to enormous lengths to see the world as we think it should be, even when the evidence says we are mistaken.’’
(p. 114)

“ideologies preclude discovery. All of us see the world as we wish it were rather than as it actually is.”
(p. 116).

There are similarities to the cautions of Walter Kauzmann, in his Reminiscences of a Life in Protein Chemistry.

Hiding the truth

In their wonderful (and provocative) PNAS article, The Theory of Everything, Laughlin and Pines introduced the term protectorate to describe the insensitivity of higher level laws (organising principles) to the details of lower level laws. For example, the laws of thermodynamics are the same regardless of whether the microscopic dynamics of the constituent particles is quantum or classical. Universality in the theory of continuous phase transitions is another important example. Near the liquid-vapour critical point the critical exponents are independent of the chemical composition of the system or of the interatomic forces involved.

Emergence versus reductionism

In the final chapter of his book, A Different Universe, Bob Laughlin states:

while a simple and absolute law, such as hydrodynamics, can evolve from the deeper laws underneath, it is at the same time independent of them, in that it would be the same even if the deeper laws were changed.
Thinking through these effects seriously moves one to ask which law is the more ultimate, the details from which everything flows or the transcendent, emergent law they generate. That question is semantic and thus has no absolute answer, but it is clearly a primitive version of the moral conundrum raised by the alleged subordination of the laws of living to the laws of chemistry and physics. It shows allergorically how a person could easily master one and learn nothing whatsoever about the other. The epistemological barrier is not mystical but physical.
The conflict between these two conceptions of the ultimate: the laws of the parts or the laws of the collective is very ancient and not resolvable in a few minutes’ reflection or a casual conversation. One might say it represents the tension between two poles of thought, which drives the process of understanding the world the way the tension between the tonic and dominant drives a classical sonata. At any one time in history a given pole may be stronger than the other, but its predominance is only temporary, for the essence of the plot is the conflict itself.

More fundamentalism

Which is more fundamental? Elementary particles or elastic solids?

In his book A Different Universe, Bob Laughlin considers the implications of the observation that quantities such as the Josephson constant and the von Klitzing resistance are known will incredible accuracy, the latter to one part in ten billion (pp.15-16):

``Paradoxically, the existence of these highly reproducible experiments leads us to think in two mutually incompatible ways about what is fundamental. One is that exactness reveals something about the primitive building blocks out of which our complicated, uncertain world is made…. The other is that exactness is a collective effect that comes into existence because of a principle of organization…. There is no way to reconcile these two ideas; they are exact opposites. Yet we use the fundamental to describe both.”

Laughlin points out:

“The fractional quantum Hall effect reveals that ostensibly indivisible quanta—in this case the electron charge e—can be broken into pieces through self-organization of phases. The fundamental things, in other words, are not necessarily fundamental.”

He further claims that in quantum field theory the vacuum state is not fundamental but is “an emergent phenomena characteristic of a phase of matter” (p.110-115).

Laughlin claims that if Einstein were alive today then he would,

“conclude that his beloved principle of relativitiy was not fundamental at all but emergent – a collective property of the matter constituting space-time that becomes increasingly exact at long length scales but fails at short ones.’’ (p. 126)

What is an explanation? What is the ultimate cause?

George Ellis points out that the question, ``Why does an aircraft fly?’’ has several different answers, ranging from ``because it is on the airlines schedule’’ to ``because the air molecules produce a differential force between the top and bottom of the wings.’’ Which answer is ``correct’’ depends on the context of the original question. Furthermore, different individuals and different social groups will have different standards and values that will determine what constitutes a ``satisfactory’’ explanation.

In his book, A Different Universe, Bob Laughlin states,

``microscopic laws are true and could plausibly cause phases; therefore we are sure they do cause them, even though we cannot prove this deductively. The argument does have the strange effect of giving the word ``cause’’ a meaning it does not customarily have. One could say that the laws of chemistry ``caused’’ the destruction of Tokyo, but what really did it was Godzilla.’’

``Symmetries are caused by things, not the cause of things.’’ (p.124)

He further claims that ``protection’’ obscures ultimate causes,

``The elastic rigidity of the solid state, hides the existence of atoms, because the elastic properties are universal consequences of ordering and would be the same if the solids were made of something else. (p. 144)

The theory of everything

This is the provocative title of a very nice paper by Laughlin and Pines in PNAS back in 2000. They point out that in principle Schrodinger's equation from quantum mechanics and Coulomb's law of electrostatics is ‘The Theory of Everything’ since these equations determine all of chemistry and all the properties of all matter that we encounter everyday. Yet, due to limited computational resources even the most powerful supercomputer can only solve these equations and make predictions for systems containing at most ten particles. However, even if we had a supercomputer that could treat Avogadro's number (i.e., 10^23) of particles that would not help. Such a computer would require more atoms than there are in the universe.

First, doing the calculations would be just like doing an experiment. It would be a ‘black box’ that would give little insight into the origin of the phenomena. Morever, such calculations on finite systems cannot predict phenomena such as broken symmetry and the exact quantisation of quantities such as the quantum Hall resistance, the magnetic flux associated with a vortex in a type II superconductor, or the circulation associated with a vortex in superfluid Helium. If we ‘know the answer’, i.e., expect broken symmetry, then we can ‘jig’ the equations so we can get the answer out. But this is a posteriori not a priori reasoning.

Phenomena such as the quantisation of magnetic flux of vortices in type II superconductors present a problem for methodological reductionism. Even though Ginsburg-Landau theory is only approximate and does not require a detailed knowledge of the underlying quantum dynamics of the constituent atoms and electrons in a superconducting metal it predicts exactly the value of the magnetic flux. This is because of the principle of broken symmetry.