Quantum states of matter and metrology

Two characteristics of states of matter are associated with them being referred to as quantum. One characteristic is the importance of quantum statistics of particles, i.e., that the system is composed of particles that obey Fermi-Dirac or Bose-Einstein statistics. The second characteristic is that a macroscopic property is quantized with values determined by Planck’s constant. I now discuss each of these with respect to emergence.

Quantum statistics. 

For a system of non-interacting  fermions and bosons at high temperatures the properties of the system are those of a classical ideal gas. As the temperature decreases there is a smooth crossover to low-temperature properties that are qualitatively different for fermions, bosons, and classical particles. This crossover occurs around a temperature, known as the degeneracy temperature, that is dependent on the particle density and Planck’s constant. 

Many of the properties resulting from quantum statistics also occur in systems of strongly interacting particles and this is central to the concept of Landau’s Fermi liquid and viewing liquid 4He as a boson liquid. If liquid 3He and the electron liquid in elemental metals are viewed as a gas of non-interacting fermions, the degeneracy temperature is about 1 K and 1000 K, respectively. Thermodynamic properties are qualitatively different above and below the degeneracy temperature. Low-temperature properties can have values that differ by orders of magnitude from classical values and have a different temperature dependence. In contrast to a classical ideal gas, a fermion gas has a non-zero pressure at zero temperature and its magnitude is determined by Planck’s constant. This degeneracy pressure is responsible for the gravitational stability of white dwarf and neutron stars.  

These properties of systems of particles can be viewed as emergent properties, in the sense of novelty, as they are qualitatively different from high-temperature properties. However, they involve a crossover as a function of temperature and so are not associated with discontinuity. They also are not associated with unpredictability as they are straightforward to calculate from a knowledge of microscopic properties.

Quantised macroscopic properties.

These provide a more dramatic illustration of emergence. Here I consider four specific systems: superconducting cylinders, rotating superfluids, Josephson junctions, and the integer Quantum Hall effect. All of these systems have a macroscopic property that is observed to have the following features.

i. As an external parameter is varied the quantity varies in a step-like manner with discrete values on the steps. This is contrast to the smooth linear variation seen when the material is not condensed into the quantum state of matter.

ii. The value on the steps is an integer multiple of some specific parameter.

iii. This parameter (unit of quantisation) only depends on Planck’s constant h and other fundamental constants. 

iv. The unit of quantisation does not depend on details of the material, such as chemical composition, or details of the device, such as its geometrical dimensions.

v. The quantisation has been observed in diverse materials and devices.

vi. Explanation of the quantisation involves topology.

Superconducting cylinders. A hollow cylinder of a metal is placed in a magnetic field parallel to the axis of the cylinder. In the metallic state the magnetic flux enclosed by the cylinder increases linearly with the magnitude of the external magnetic field. In the superconducting state, the flux is quantized in units of the magnetic flux quantum, Φ0 = h/2e where e is the charge on an electron. It is also found that in a type II superconductor the vortices that occur in the presence of an external magnetic field enclose a magnetic flux equal to Φ0.  

Rotating superfluids. When a cylinder containing a normal fluid is rotated about an axis passing down the centre of the cylinder the fluid rotates with a circulation proportional to the speed of rotation and the diameter of the cylinder. In contrast, in a superfluid, as the speed of rotation is varied the circulation is quantised in units of h/M where M is the mass of one atom in the fluid. This quantity is also the circulation around a single vortex in the superfluid. 

Josephson junctions. In the metallic state the current passing through a junction increases linearly with the voltage applied across the junction. In the superconducting state the AC Josephson effect occurs. If a beam of microwaves of constant frequency is incident on the junction, jumps occur in the current when the voltage is an integer multiple of h/2e. The quantisation is observed to better than one part in a million (ppm).

Integer Quantum Hall effect. In a normal conductor the Hall resistance increases linearly with the external magnetic field for small magnetic fields. In contrast, in a two-dimensional conductor at high magnetic fields the Hall resistance is quantized in units of h/2e^2. The quantisation is observed to better than one part in ten million. Reflecting universality, the observed value of the Hall resistance for each of the plateaus is independent of many details, including the temperature, the amount of disorder in the material, the chemical composition of system (silicon versus gallium arsenide), or whether the charge carriers are electrons or holes.

Other examples of macroscopic quantum effects are seen in SQUIDs (Superconducting Quantum Interference Devices). They exhibit quantum interference phenomena analogous to the double-slit experiment. The electrical current passing through the SQUID has a periodicity defined by the ratio of the magnetic flux inside the current loop of the SQUID and the quantum of magnetic flux.

The precision of the quantisation provides a means to accurately determine fundamental constants. Indeed, the title of the paper announcing the discovery of the integer quantum Hall effect was, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance.” It is astonishing that a macroscopic measurement of a property of a macroscopic system, such as the electrical resistance, can determine fundamental constants that are normally associated with the microscale and properties of atomic systems. 

Laughlin and Pines claimed that the quantisation phenomena described above reflect organizing principles associated with emergent phenomena, and their universality supports their claim of the unpredictability of emergent properties. 

Quantum states of matter and metrology

The universality of these macroscopic quantum effects has practical applications in metrology, the study of measurement and the associated units and standards. In 1990 new international standards were defined for the units of voltage and electrical resistance, based on the quantum Hall effect and the AC Josephson effect, respectively.

Prior to 1990 the standard used to define one volt was based on a particular type of electrical battery, known as a Weston cell. The new standard using the AC Josephson effect allowed voltages to be defined with a precision of better than one part per billion. This change was motivated not only by improved precision, but also improved portability, reproducibility, and flexibility. The old voltage standard involved a specific material and device and required making duplicate copies of the standard Weston cell. In contrast, the Josephson voltage standard is independent of the specific materials used and the details of the device. 

Prior to 1990 the international standard for the ohm was defined by the electrical resistance of a column of liquid mercury with constant cross-sectional area, 106.3 cm long, a mass of 14.4521 grams and a temperature 0 °C. Like the Josephson voltage standard, the quantum Hall resistance standard has the advantage of precision, portability, reliability, reproducibility, and independence of platform. The independence of the new voltage and resistance standards from the platform used reflects the fact that the Josephson and quantum Hall effects have the universality characteristic of emergent phenomena.

This post is an adaptation of material in Condensed Matter Physics: A Very Short Introduction

Quantifying the obscurity of academic writing

 Occasionally The Economist publishes nice graphs capturing social and economic trends. Here is one.

It is part of a nice article

Academic writing is getting harder to read—the humanities most of all

The downward trend in the humanities and social sciences is dramatic and perhaps not surprising.

I was surprised that in the natural sciences the trend wasn't worse. The Flesch reading ease score goes from about 26 around 1960 to about 18 today. 

However, this metric misses a lot. It only measures the number of words per sentence and the number of syllables per word. Thus, it is not sensitive to the amount of technical jargon or to overall clarity.

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.

From Leo Szilard to the Tasmanian wilderness

Richard Flanagan is an esteemed Australian writer. My son recently gave our family a copy of Flanagan's recent book, Question 7. It is a personal memoir that masterfully weaves together a dizzying array of topics, from nuclear physics to the Tasmanian wilderness. I mention it on this blog because of its endearing and fascinating portrayal of Leo Szilard, arguably one of the twentieth century's most creative, unconventional, and eccentric physicists.

The paragraph below gives an overview of the narrative that is used to weave together all the disparate topics.

“Without Rebecca West’s kiss H. G. Wells would not have run off to Switzerland to write a book in which everything burns, and without H. G. Wells’s book [The World Set Free] Leo Szilard would never have conceived of a nuclear chain reaction and without conceiving of a nuclear chain reaction he would never have grown terrified and without growing terrified Leo Szilard would never have persuaded Einstein to lobby Roosevelt and without Einstein lobbying Roosevelt there would have been no Manhattan Project and without the Manhattan Project there is no lever at 8.15 am on 6 August 1945 for Thomas Ferebee to release 31,000 feet over Hiroshima, there is no bomb on Hiroshima and no bomb on Nagasaki and 100,000 people or 160,000 people or 200,000 people live and my father dies. Poetry may make nothing happen, but a novel destroyed Hiroshima and without Hiroshima there is no me and these words erase themselves and me with them.”

You can read an extract here and a review in The Guardian here.

Are gravity and space-time emergent?

Attempts to develop a quantum theory of gravity continue to falter and stagnate. Given this, it is worth considering approaches that start with what we know about gravity at the macroscale and investigate whether it provides any hints about some underlying more microscopic theory. One such approach was taken by Thanu Padmanabhan and is elegantly described and summarised in a book chapter.

Gravity and Spacetime: An Emergent Perspective

Insights about microphysics from macrophysics 

Padmanabhan emphasises Boltzmann's insight: "matter can only store and transfer heat because of internal degrees of freedom". In other words, if something has a temperature and entropy then it must have a microstructure.

The approach of trying to surmise something about microphysics from macrophysics has a long and fruitful history, albeit probably with many false starts that we do not hear about. Kepler's snowflakes may have been the first example. Before people were completely convinced about the existence of atoms, the study of crystal facets and of Brownian motion provided hints of the atomic structure of matter. Planck deduced the existence of the quantum from the thermodynamics of black-body radiation.

Arguably, the first definitive determination of Avogadro's number was from Perrin's experiments on Brownian motion which involved macroscopic measurements.

Comparing classical statistical mechanics to bulk thermodynamic properties gave hints of an underlying quantum structure to reality. The Sackur-Tetrode equation for the entropy of an ideal gas hints at the quantisation of phase space. The Gibbs paradox hints that fundamental particles are indistinguishable. The third law of thermodynamics hints at the idea of quantum degeneracy.

Puzzles in classical General Relativity

Padmanabhan reviews aspects of the theory that he considers some consider to be "algebraic accidents" but he suggests that they may be hints to something deeper. These include the role of boundary terms in variational principles and he suggests hint at a classical holography (bulk behaviour is determined by the boundary). He also argues that the metric of space-time should not be viewed as a field, contrary to most attempts to develop a quantum field theory for gravity.

Thermodynamics of horizons

The key idea that is exploited to find the microstructure is that can define a temperature and an entropy for null surfaces (event horizons). These have been calculated for specific systems (metrics) including the following:

For accelerating frames of reference (Rindler) there is an event horizon which exhibits Unruh radiation with a temperature that was calculated by Fulling, Davies and Unruh.

The black hole horizon in the Schwarschild metric has the temperature of Hawking radiation.

The cosmological horizon in deSitter space is associated with a temperature proportional to the Hubble constant H. [This was discussed in detail by Gibbons and Hawking in 1977].

Estimating Avogadro's number for space-time

Consider the number of degrees of freedom on the boundary, N_s, and in the bulk, N_b. 

On the boundary surface, there is one degree of freedom associated with every Planck area (L_p^2) where L_p is the Planck length, i.e,  N_s = A/ L_p^2, where A is the surface area, which is related to the entropy of the horizon (cf. Bekenstein and Hawking).

In the bulk equipartition of energy is assumed so the bulk energy E = N_b k T/2 where

An alternative perspective on cosmology 

He presents a novel derivation of the dynamic equations for the scale factor R(t) in the Friedmann-Robertson-Walker metric of the universe in General Relativity. His starting point is a simple argument leading to 

V is the Hubble volume, 4pi/3H^3, where H is the Hubble constant, and L_P is the Planck length.

The right-hand side is zero for the deSitter universe, which is predicted to be the asymptotic state of our current universe.

Possible insights about the cosmological constant

One of the biggest problems in theoretical physics is to explain why the cosmological constant has the value that it does.

There are two aspects to the problem.
1. The measured value is so small, 120 orders of magnitude smaller than what one estimates based on the quantum vacuum energy!

2. The measured value seems to be finely tuned (to 120 significant figures!) to the value of the mass energy.

He presents an argument that the cosmological constant is related to the Planck length 

where mu is of order unity.

Details of his proposed solution are also discussed here.

I am not technically qualified to comment on the possible validity or usefulness of Padmanabhan's perspective and results. However, I think it provides a nice example of a modest and conventional scientific alternative to radical approaches, such as the multiverse, or ideas that seem to be going nowhere such as AdS/CFT that are too often invoked or clung onto to address these big questions. 

Aside. In the same book, there is also a short and helpful chapter, Quantum Spacetime on loop quantum gravity by Carlo Rovelli. He explicitly identifies the "atoms" of space-time as the elements of "spin foam".

Emergent gauge fields in spin ices

Spin ices are magnetic materials in which geometrically frustrated magnetic interactions between the spins prevent long-range magnetic order and lead to a residual entropy similar to in ice (solid water).

Spin ices provide a beautiful example of many aspects of emergence, including how surprising new entities can emerge at the mesoscale. I think the combined experimental and theoretical work on spin ice was one of the major achievements of condensed matter physics in the first decade of this century.

Novelty

Spin ices are composed of individual spins on a lattice. The system exhibits properties that the individual spins and the high-temperature state do not have. The novel properties can be understood in terms of an emergent gauge field. Novel entities include spin defects reminiscent of magnetic monopoles and Dirac strings.

State of matter

Spin ices exhibit a novel state of matter, the magnetic Coulomb phase. There is no long-range spin order, but there are power-law (dipolar) correlations that fall off as the inverse cube of distance.

Toy models

Classical models such as the Ising or Heisenberg models with antiferromagnetic nearest-neighbour interactions on the pyrochlore lattice exhibit the emergent physics associated with spin ices: absence of long-range order, residual entropy, ice type rules for local order, and long-range dipolar spin correlations exhibiting pinch points. These toy models can be used to derive the gauge theories that describe emergent properties such as monopoles and Dirac strings.

Actual materials that exhibit spin ice physics such as dysprosium titanate (Dy₂Ti₂O₇) and holmium titanate (Ho₂Ti₂O₇) are more complicated. They involve quantum spins, ferromagnetic interactions, spin-orbit coupling, crystal fields, complex crystal structure and dipolar magnetic interactions. Chris Henley says these materials

"are well approximated as having nothing but (long-ranged) dipolar spin interactions, rather than nearest-neighbor ones. Although this model is clearly related to the “Coulomb phase,” I feel it is largely an independent paradigm with its own concepts that are different from the (entropic) Coulomb phase..."

Effective theory

Gauge fields described by equations analogous to electrostatics and magnetostatics in Maxwell’s electromagnetism are emergent in coarse-grained descriptions of spin ices. 

Consider a bipartite lattice where on each site we locate a tetrahedron. The "ice rules" require that two spins on each tetrahedron point in and two out. We can define a field L(i) on each lattice site i which is the sum of all the spins on the tetrahedron. The magnetic field B(r) is a coarse-graining of the field L(i). The ice rules and local conservation of flux require that 

The classical ground state of this model is infinitely degenerate. The emergent “magnetic” field [which it should be stressed is not a physical magnetic field] allows the presence of monopoles [magnetic charges]. These correspond to defects that do not satisfy the local ice rules in the spin system.

It can be shown that the total free energy of the system is

K is the "stiffness" or "magnetic permeability" associated with the gauge field. It is entirely of entropic origin, just like the elasticity of rubber.

[Aside: I would be curious to see a calculation of K from a microscopic model and an estimate from experiment. I have not stumbled upon one yet. Do you know of one? Henley points out that in water ice the entropic elasticity makes a contribution to the dielectric constant and this "has been long known."]

  A local spin flip produces a pair of oppositely charged monopoles. The monopoles are deconfined in that they can move freely through the lattice. They are joined together by a Dirac string.

This contrasts with real magnetism where there are no magnetic charges, only magnetic dipoles; one can view magnetic charges as confined within dipoles.

There is an effective interaction between the two monopoles [charges] that has the same form as Coulomb’s law.  There are only short-range (nearest neighbour) direct interactions between the spins. However, these act together to produce a long-range interaction between the monopoles (which are deviations from local spin order).

Universality

The novel properties of spin ice occur for both quantum and classical systems, Ising and Heisenberg spins, and for a range of lattices. The same physics occurs with water ice, magnetism, and charge order.

Modularity at the mesoscale

The system can be understood as a set of weakly interacting modular units. These include the tetrahedra of spins, the magnetic monopoles, and the Dirac strings. The measured temperature dependence of the specific heat of Dy₂Ti₂O₇  is consistent with that calculated from Debye-Huckel theory for deconfined charges interacting by Coulomb's law, and shown as the blue curve below. The figure is taken from here.

Pinch points.

The gauge theory predicts that the spin correlation function (in momentum space) has a particular singular form exhibiting pinch points [also known as bow ties], which are seen experimentally.

Unpredictability

Most new states of matter are not predicted theoretically. They are discovered by experimentalists, often by serendipity. Spin ice and the magnetic Coulomb phase seems to be an exception. Please correct me if I am wrong.

Sexy magnetic monopoles or boring old electrical charges?

I am hoping a reader than clarify this issue. What is wrong with the following point of view. In the discussion above the "magnetic field" B(r) could equally well be replaced with an "electric field" E(r). Then the spin defects are just analogous to electrical charges and the "Dirac strings" become like a polymer chain with opposite electrical charges at its two ends. This is not as sexy. 

Note that Chris Henley says Dirac strings are "a nebulous and not very helpful notion when applied to the Coulomb phase proper (with its smallish polarisation), for the string's path is not well defined... It is only in an ordered phase... that the Dirac string has a clear meaning."

Or is the emergent field actually "magnetic"? It describes spin defects and these are associated with a local magnetic moment. Furthermore, the long-range dipolar correlations (with associated pinch points) of the gauge field are detected by magnetic neutron scattering and so the gauge field should be viewed as "magnetic" and not "electric".

Emergent gauge fields in quantum many-body systems?

In spin ice, the emergent gauge field is classical and arises in a spin system that can be described classically. This does raise two questions that have been investigated extensively by Xiao-Gang Wen. First, he has shown how certain mean-field treatments of frustrated antiferromagnetic (with quantum spin liquid ground states) and doped Mott insulators lead to emergent gauge fields. As fascinating as his work is, it needs to be stressed that there is no definitive evidence for these emergent gauge fields. They just provide appealing theoretical descriptions. This is in contrast to the emergent gauge fields for spin ice.

Second, based on Wen's success at constructing these emergent gauge fields he has pushed provocative (and highly creative) ideas that the gauge fields and fermions that are considered "fundamental" in the standard model of particle physics may be emergent entities. This is the origin of the subtitle of his 2004 book, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons.

To prepare this post I found the articles below helpful.

Emergent particles and gauge fields in quantum matter

Ben J. Powell

Maxwell electromagnetism as an emergent phenomenon in condensed matter

J. Rehn and R. Moessner

The “Coulomb Phase” in Frustrated Systems

Chris Henley

Emergence and protein folding

Proteins are a distinct state of matter. Globular proteins are tightly packed with a density comparable to a crystal but without the spatial regularity found in crystals. The native state is thermodynamically stable, in contrast to the globule state of synthetic polymers which is often glassy and metastable, with a structure that depends on the preparation history.

For a given amino acid sequence the native folded state of the protein is emergent. It has a structure, properties, and function that the individual amino acids do not, nor does the unfolded polymer chain. For example, the enzyme catalase has an active site whose function is as a catalyst to make hydrogen peroxide (which is toxic) decay rapidly.

Protein folding is an example of self-organisation. A key question is how the order of the folded state arises from the disorder (random configuration) of the unfolded state.

There are hierarchies of structures, length scales, and time scales associated with the folding.

The hierarchy of structures are primary, secondary, tertiary, and ternary structures. The primary structure is the amino acid sequence in the heteropolymer. Secondary structures include alpha-helices and beta-sheets, shown in the figure above in orange and blue, respectively. The tertiary structure is the native folded state. An example of a ternary structure is in hemoglobin which consists of four myoglobin units in a particular geometric arrangement.

The hierarchy of time scales varies over more than fourteen orders of magnitude, including folding (msec to sec), helix-coil transitions (microsec), hinge motion (nanosec), and bond vibrations (10 fsecs).

Folding exhibits a hierarchy of processes, summarised in the figure below which is taken from

Cooperativity and modularity in protein folding

Masaki Sasai, George Chikenji, Tomoki P. Terada

Modularity 

"Protein foldons are segments of a protein that can fold into stable structures independently. They are a key part of the protein folding process, which is the stepwise assembly of a protein's native structure." (from Google AI)

See for example.

Discontinuities

The folding-unfolding transition [denaturation] is a sharp transition, similar to a first-order phase transition. This sharpness reflects the cooperative nature of the transition. There is a well-defined enthalpy and entropy change associated with this transition.

Universality

Proteins exhibit "mutational plasticity", i.e., native structures tolerant to many mutations (changes in individual amino acids). Aspects of the folding process such as its speed, reliability, reversibility, and modularity appear to be universal, i.e., hold for all proteins.

Diversity with limitations

On the one hand, there are a multitude of distinct native structures and associated biological functions. On the other hand, this diversity is much smaller than the configuration space, presumably because thermodynamic stability vasts reduces the options.

Effective interactions

These are subtle. Some of the weak ones matter as the stabilisation energy of the native state is of order 40 kJ per mole, which is quite small as there are about 1000 amino acids in the polymer chain. Important interactions include hydrogen bonding, hydrophobic, and volume exclusion. In the folded state monomers interact with other monomers that are far apart on the chain. The subtle interplay of these competing interactions produces complex structures with small energy differences, as is often the case with emergent phenomena.

Toy models

1. Wako-Saito-Munoz-Eaton model

This is an Ising-like model on a chain. A short and helpful review is

A simple theoretical model goes a long way in explaining complex behavior in protein folding

Victor Muñoz 

Note that the interactions are not pairwise but involves strings of "spins" between native contacts.

2. Dill's HP polymer on a lattice

This consists of a polymer which has only two types of monomer units and undergoes a self-avoiding walk on a lattice. H and P denote hydrophobic and polar amino acid units, denoted by red and blue circles, respectively, in the figure below. The relative simplicity of the model allows complete enumeration of all possible confirmations for short chains. The model is simpler in two dimensions, yet still captures essential features of the folding problem.  

As the H-H attraction increases the chain undergoes a relatively sharp transition to just a few conformations that are compact and have hydrophobic cores. The model exhibits much of the universality of protein folding. Although there are 20 different amino acids in real proteins, the model divides them into two classes and still captures much of the phenomena of folding, including mutational plasticity.

co-operativity - helical order-disorder transition is sharp

Organising principles

Certain novel concepts such as the rugged energy landscape and the folding funnel apply at a particular scale.

This post drew on several nice papers written by Ken Dill and collaborators including

The Protein Folding Problem, H.S. Chan and K.A. Dill, Physics Today, 1993

The Protein Folding Problem: The Role of Theory

Roy Nassar, Gregory L. Dignon, Rostam M. Razban, Ken A. Dill, Journal of Molecular Biology, 2021

Interestingly, in the 2021 article, Dill claims that the protein folding problem [which is not the prediction problem] has now essentially been solved.