The emergence of hadronic matter from interacting quarks and gluons

 A characteristic of emergent phenomena is how novel and complex properties can emerge from apparently simple laws. Quantum ChromoDynamics (QCD) describes the interaction of quarks and gluons. The classical Lagrangian is remarkably simple.

It has an SU(3) local gauge symmetry and the A_v^mu are the associated gauge fields (gluons).

The quarks are fermions with fractional electrical charge. The only parameters in the theory are the coupling constant g_s, which describes the self-interaction of the gluons, and the bare masses of the quarks, m_f. The gluons are massless. In the limit where the bare mass of quarks vanishes, the Lagrangian has chiral symmetry, which transforms quarks with left-handed symmetry into right-handed.

Frank Wilczek states that QCD "is conceptually simple. Its realisation in nature, however, is usually very complex. But not always."

Hadronic matter (nucleons and mesons) has properties that are qualitatively different from its components (interacting quarks and gluons). In other words, it is emergent. In hadronic matter, there are no particles with fractional electrical charge or massless bosons. Chiral symmetry is broken. Consequently, hadrons do not come in pairs with opposite parity and equal energy. Quarks are confined and this is associated with a string tension. The order parameter associated with confinement is the Polyakov (or Wilson) loop. The connection between chiral symmetry breaking and confinement is subtle. For a long time they were thought to be intimately connected but now that is not the case.

Although the underlying Lagrangian is simple, the spectrum of hadrons and their interactions is complex. There is a "zoo" of particles. This all comes from a single coupling constant!

On the one hand, this complexity is surprising. On the other hand, it is similar to how there is a simple coupling constant (the electronic charge) in the Hamiltonian that describes most of chemistry and condensed matter physics. Most of the particles are unstable and can be viewed as quasiparticles as they have a finite lifetime, even in the absence of electroweak interactions. 

The emergent state of hadronic matter only exists at "low" temperatures and densities. It "melts" at the high temperatures associated with the Big Bang, relativistic heavy ion colliders, or the high densities associated with neutron stars. But that and the associated phase diagram of QCD is another story...

Update. I revised this post due to some helpful clarifications from Chris Allton, who was visiting UQ this week.

Critical points in condensed matter illuminate universality

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

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

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

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

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

Condensed matter physics in flatland

Adventures in Flatland

In everyday life we think of most objects as having three dimensions. But what would life be like in a two-dimensional world? For one thing, it would be harder to move around. We could no longer step over things but would have to move around them. In 1884 Edwin Abbott published Flatland: A Romance of Many Dimensions, under the pseudonym, A. Square, a satirical novella about social life in Victorian England. People are represented by geometrical objects. Men are represented by shapes such as triangles and hexagons. Women are represented by lines. The social status of men increases with the number sides that their shape has and how many of the sides are of the same length. Abbott’s book created limited interest and was largely forgotten by the 1920s. Interest revived when theoretical physicists started to think about worlds in different dimensions. This interest was stimulated by Albert Einstein’s theories of relativity, that proposed that we live in a four-dimensional world, not a three-dimensional one. Time is the fourth dimension, and there is an intimate and concrete connection between time and space. Attempts to unify gravity with other fundamental forces has led to physicists proposing and studying theories with more than four dimensions.

Changing the number of spatial dimensions leads to different physics because it changes what is mathematically possible. In three dimensions, there were only five highly symmetrical shapes known as Platonic solids (tetrahedron, cube, octahedron, icosahedron, and dodecahedron). In contrast, in two dimensions it is possible to make an infinite number of symmetrical shapes, known as regular polygons, shapes made of straight lines of equal length such as squares or hexagons. Similarly, the number of Bravais lattices differ in two and three dimensions. Changing the number of spatial dimensions changes both what is mathematically possible and what is physically possible.

What would condensed matter physics be like in Flatland? This question received limited attention before the 1970s. Occasionally, theoretical physicists would investigate mathematical models of crystals or magnets in one or two dimensions just because the mathematics was simpler and more tractable than in three dimensions. The goal was to obtain insight into physics in three dimensions. We will consider a famous example, the Ising model. 

In the 1970s, several surprising developments led to significant interest in condensed matter physics in spatial dimensions different from the usual three. First, it became possible to make a wide range of material systems that were two-dimensional. Secondly, theoretical work showed that states of matter, and phase transitions between them, can be qualitatively different in one, two, and three spatial dimensions. And thirdly, considering different numbers of spatial dimensions turned out to be very fruitful for theory, particularly for understanding phase transitions near critical points. 

An extract from "Adventures in Flatland," chapter 5 in Condensed Matter Physics: A Very Short Introduction

Condensed matter physics is about how order emerges from disorder

 The order of things

Life and the world around us sometimes appears chaotic and random. We may feel this way about traffic, weather, economics, social change, politics, or our personal relationships. Perhaps that is why many yearn for regularity, predictability, order, and stability. Science is a search for patterns and order in the natural world. Condensed matter physics is about how order emerges from disorder.

This chapter explores how different states of matter are associated with different types of ordering of the atoms in the material. The symmetry of the state reflects the type of ordering, i.e., the patterns associated with the state. There is also a rigidity associated with the ordering and the rigidity determines the nature of the deviations from perfect ordering and results in entities such as vortices that are central to the physical properties of the state of matter.

The association of a state of matter with a specific type of ordering is illustrated in Figure 15 by an analogue with the dodgem bumper cars at an amusement park. A quiet day at the park is not much fun as collisions between cars are rare. In other words, there is little correlation between the relative locations and speeds of the cars. In comparison, on a busy day at the park the spatial separation of the cars is small, and their positions and speeds are more correlated with one another than on a quiet day. But, in both cases, there is no ordered arrangement of the cars. In contrast, after the park closes the cars are parked and arranged in an orderly manner. There is a rigidity associated with their spatial arrangement. One car cannot be moved without moving others. These three states of the dodgem cars are an analogue of three states of matter: gas, liquid, and crystal. 

Figure 15. A dodgem car analogue for the three states of matter: crystal, liquid, and gas. The only ordered arrangement is for the crystal (car park after hours) and this is associated with a specific symmetry and rigidity. The liquid and gas (busy and quiet day) only differ in density and the amount of correlation between the positions of the different atoms (dodgem cars).

In the dodgem car analogue, there are other possible types of ordering. In some amusement parks there is a track, and the cars are meant to all go in the same direction. The symmetry between clockwise and anti-clockwise of the track is then broken.  In the car park, Figure 15 shows cars that are symmetrical with respect to front and back. However, real cars have a front and back, and so can be parked either front first or back first. Hence, several types of ordering are possible: all cars park back first, all cars park front first, cars are front first or back first at random, alternating patterns of front first and back first as one goes along a row, alternating rows of front first and back first, and so on. These different types of ordering in the car park all have analogues in different solid states of matter.

Liquid crystals involve unique types of ordering. These materials are composed of elongated organic molecules, such as those shown in Figure 16. At high temperatures the material is in a liquid state and the orientations and positions of the molecules are random. The liquid has both continuous translational and rotational symmetry. At low temperatures the molecules form a solid crystal without the continuous translational and rotational symmetry of the liquid state. As the crystal is heated the temperature increases and there is a phase transition to the liquid crystal state, in which all the molecules point in the same direction, but their positions are random. Hence, the liquid crystal state has the continuous translational symmetry of the liquid, but not its continuous rotational symmetry, like the crystal. As the temperature increases further there is a transition to the liquid state (Figure 16). In terms of the dodgem car analogue the liquid crystal state is similar to when cars park in a field all pointing in the same direction but there are no grid lines, and their positions are then random.

The existence of a state in between a liquid and crystal was first proposed in 1888 by botanist and chemist Friedrich Reinitzer who was doing research on cholesterol at the Institute for Plant Physiology in Prague. He performed a heating experiment similar to that described in Figure 4. Instead of one melting transition he observed transitions at two distinct temperatures. 

Figure 16. Liquid crystals. (a) An example of the type of elongate organic molecule found in these materials. Each molecule can be represented by an oval shape. (b) In the nematic liquid crystal state, the molecules tend to point in the same direction, but their positions are random. 

There are multiple alternative orderings for liquid crystals with names such as nematic, smectic, chiral nematic, discotic, and chlorestic. In the smectic phase molecules form layers of oriented molecules. The character of the liquid crystal state can be detected by shining polarised light on the material. Liquid crystal displays (LCDs) in electronic devices use the property that an electric field can orient the molecules, and this changes the interaction of the material with polarised light.

For solid crystals the nature of the ordering and the symmetry associated with a specific crystal structure is clear once the spatial arrangements of the atoms in the crystal are determined, such as by X-ray diffraction. For other states of matter, such as superconductors, superfluids, and antiferromagnets, the nature of the ordering and the symmetry is often not apparent and has only been determined with significant scientific insight. 

An extract from "The order of things," chapter 4 in Condensed Matter Physics: A Very Short Introduction.

Symmetry matters in condensed matter physics

 Snowflakes form incredibly diverse structures, seen when they condense onto a plate of glass. Every snowflake is different. On the other hand, every snowflake is the same. They are all composed of ice, a solid state of water. Every snowflake is composed of units that have a six-fold symmetry (Figure 8). Every snowflake is composed solely of water molecules. This paradox of the particular and the universal is at the heart of condensed matter physics. Although diversity prevails anything is not possible. No snowflake has five-fold symmetry. Snowflakes have enchanted scientists for a long time. The astronomer Johannes Kepler studied them and in 1611 wrote a small book about them as a gift for his patron. Kepler suggested snowflakes provided clues to deeper questions about the composition of matter. Today, Kenneth Libbrecht, a physicist at Caltech, has spent most of his career studying snowflakes and has produced beautiful volumes of photographs of them.

Figure 8. A snowflake shows a six-fold symmetry, just like a hexagon. The snowflake appears identical when it is rotated by an angle of sixty degrees about an axis passing through its centre and perpendicular to the page.

Condensed matter physicists ask several questions about snowflakes. What is the reason for the six-fold symmetry of the snowflake? What is the connection between the macroscopic properties of snowflakes and the properties of the underlying microscopic constituents, molecules of H2O? How is the diversity of snowflake shapes possible? Is there a phase diagram that defines the external conditions under which the different shapes form?

There is a long history in art, architecture, philosophy, and science, of associating symmetry with beauty and perfection. The ancient Greek philosopher Plato was a proponent of this view. He studied a particular class of solid shapes: cube, tetrahedron, octahedron, icosahedron, and dodecahedron. Plato identified the first four shapes with the four “elements”: earth, wind, fire, and water, respectively, and the fifth with the heavens. Each of these solid shapes is highly symmetric. Every face of a Platonic solid is the same shape (square, triangle, pentagon,...) and each of those shapes has edges of equal length. 

Like Plato, Kepler believed that “God is a geometer” and that God’s creation should reflect the perfection of God. These convictions led Kepler to propose in 1597 that the orbits of the planets around the Sun were circular and that the Platonic solids determined the relative size of the orbits. Later this model for the solar system was shown not to be true. In fact, Kepler himself became famous because he showed that the planets moved in elliptical, not circular orbits. Nevertheless, Kepler’s model was the beginning of a long history of successfully relating physical laws to symmetry and geometry.

A key discovery in physics from the past century is that symmetry is central to understanding a wide range of physical phenomena, whether colliding billiard balls, the allowed energies of an atom, the fundamental forces of nature, or different states of matter. Symmetries determine what is physically possible. For example, that energy cannot be created or destroyed is a consequence of the fact that physical laws do not change with time.

In this Chapter I explore three key ideas. First, transitions between different states of matter are associated with changes in symmetry. Thus, symmetry provides a criterion for specifying the qualitative difference between distinct states of matter. Second, for a specific state of matter the relevant symmetry constrains what is physically possible. Third, symmetry is central to making connections between the macroscopic and microscopic properties of a state of matter. The next chapter will explore how symmetry is associated with the type of ordering that occurs in a state of matter.

An extract from Chapter 3, Condensed Matter Physics: A Very Short Introduction

How many states of matter are there?

Diamond and graphite are distinct solid states of carbon. They have qualitatively different physical properties, at both the microscopic and the macroscopic scale. Condensed matter physics is all about states of matter. In science classes at school, you were probably taught that there are only three states of matter: solid, liquid, and gas. Like other things you were told in school, this is incorrect. There are endless, unlimited, distinct states of matter. 

Consider the “liquid crystals” that are the basis of LCDs (Liquid Crystal Displays) in the screens of televisions, computers, and smartphones. How can something be both a liquid and a crystal? A liquid crystal is a distinct state of matter. Solids can be found in many different states. We have already seen that there are two different solid states of carbon: graphite and diamond. In everyday life ice means simply solid water. But there are in fact eighteen different solid states of water, depending on the temperature of the water and the pressure that is applied to the ice. In each of these eighteen states there is a unique spatial arrangement of the water molecules and there are qualitative differences in the physical properties of the different solid states. Welcome to the world of condensed matter...

Extract from Chapter 1, Condensed Matter Physics: A Very Short Introduction

Classifying objects, people, and societies requires making qualitative distinctions. One book is easy to understand, and another is hard. One person is kind, and another is mean. One society is egalitarian, and another is not. Justifying such qualitative distinctions is hard. Not everyone will agree. Are there definitive criteria to justify a particular quality? Some claim they can quantify qualities such as these but that is contentious. In contrast, in condensed matter physics it is possible to give objective criteria that distinguish different states of matter. A state can only exist under specific external conditions, including defined ranges of parameters such as temperature and pressure. This chapter describes the clear signatures of transitions between different states that are observed as these parameters are varied. Some of the many known states of matter will be introduced including superconductors, superfluids, and magnets. On the way we will learn about “dry ice”, how to convert graphite into diamond, and how freeze-dried food is made.

Abrupt changes in properties

If you put some ice cubes in one empty glass and water in another, the ice does not change its shape, whereas water takes the shape of the glass. Solids are rigid and liquids are not. The distinct change from one state to another can be detected by observing an abrupt change or discontinuity in physical properties. For example, ice (solid water) has a different density to liquid water. This is evident because ice floats. The solid state of water has a lower density than the liquid state. To put it another way, water expands when it freezes. That’s why water pipes can burst if they freeze in cold weather.

A transition between two distinct states of matter is an example of a tipping point: a small change in a system variable can produce large changes in the system. For example, changing the temperature of water from +1 °C to -1 °C can produce a qualitative change in the system's properties. The water changes from liquid to solid. Tipping points occur in a wide range of physical, biological, and social systems. Examples include a stock market crash, the outbreak of an epidemic, and the operation of a room thermostat. Tipping points show that quantitative differences can become qualitative differences.

Extract from Chapter 2, Condensed Matter Physics: A Very Short Introduction

Information theoretic measures for emergence and causality

The relationship between emergence and causation is contentious, with a long history. Most discussions are qualitative. Presented with a new system, how does one identify the microscopic and macroscopic scales that may be most useful for understanding and describing the system? Can Judea Pearl’s seminal ideas about causality be implemented practically for understanding emergence?

Broadly speaking, a weakness of discussions of emergence and causality is that it is hard to define these concepts in a rigorous and quantitative manner that makes them amenable to empirical testing, with respect to theoretical models and to experimental data. 

Fortunately, in the past decade, there have been some specific proposals to address this issue, mostly using information theory. A helpful recent review is by Yuan et al. 

“Two primary challenges take precedence in understanding emergence from a causal perspective. The first is establishing a quantitative definition of emergence, whereas the second involves identifying emergent behaviors or phenomena through data analysis.

To address the first challenge, two prominent quantitative theories of emergence have emerged in the past decade. The first is Erik Hoel et al.’s theory of causal emergence [19] whereas the second is Fernando E. Rosas et al.’s theory of emergence based on partial information decomposition [24].

Hoel et al.’s theory of causal emergence specifically addresses complex systems that are modeled using Markov chains. It employs the concept of effective information (EI) to quantify the extent of causal influence within Markov chains and enables comparisons of EI values across different scales [19,25]. Causal emergence is defined by the difference in the EI values between the macro-level and micro-level."

One perspective on causal emergence is that it occurs when the dynamics of a system at the macro-level is described more efficiently by macro-variables than by the dynamics of variables from the micro-level.

Klein et al. used Hoel’s information-theoretic measures of causal emergence to analyse protein interaction networks (interactomes) in over 1800 species, containing more than eight million protein–protein interactions, across different scales. They showed the emergence of ‘macroscales’ that are associated with lower noise and uncertainty. The nodes in the macroscale description of the network are more resilient than those in less coarse-grained descriptions. Greater causal emergence (i.e., a stronger macroscale description) was generally seen in multicellular organisms compared to single-cell organisms. The authors quantified causal emergence in terms of mutual information (between large and small scales) and effective information (a measure of the certainty in the connectivity of a network). Philip Ball (2023) (pages 218-220) gives an account of this work in terms of the emergence of multicellularity in biological evolution. He introduced the term causal spreading (pages 225-7), arguing that over the history of evolution the locus of causation has changed.

Yuan et al. continue

"However, in Hoel’s theory of causal emergence, it is essential to establish a coarse-graining strategy beforehand. Alternatively, the strategy can be derived by maximizing the effective information (EI) [19]. However, this task becomes challenging for large-scale systems due to the computational complexity involved. To address these problems, Rosas et al. introduced a new quantitative definition of causal emergence [24] that does not depend on coarse-graining methods, drawing from partial information decomposition (PID)-related theory. PID is an approach developed by Williams et al., which seeks to decompose the mutual information between a target and source variables into non-overlapping information atoms: unique, redundant, and synergistic information [29]…"

The Figure below is taken from Rosas et al. Xt^j (j=1,…,n) are microscopic variables that define a Markov chain. Vt is a macroscopic variable that is completely determined by the microscopic variables.

“Diagram of causally emergent relationships. Causally emergent features have predictive power beyond individual components. Downward causation takes place when that predictive power refers to individual elements; causal decoupling when it refers to itself or other high-order features.”

Rosas et al. applied the method to specific systems, including Conway’s Game of Life, Reynolds’ flocking model, and neural activity as measured by electrocorticography. More recently, it was used to describe emergence in computer science, including the identification of modular structures. Calculations were performed for specific examples, including Ehrenfest’s urn model for diffusion, the Ising model with Glauber dynamics, a Hopfield neural network model for associative memory.

Yuan et al. also state the following:

"The second challenge pertains to the identification of emergence from data. In an effort to address this issue, Rosas et al. derived a numerical method [24]. However, it is important to acknowledge that this method offers only a sufficient condition for emergence and is an approximate approach. Another limitation is that a coarse-grained macro-state variable should be given beforehand to apply this method."

Sas et al. recently stated

“Empirical applications of this framework to study emergence … including the study of gene regulatory networks [22], the dynamics of the human brain [23], the internal dynamics of reservoir computing [24], and the formation of useful internal representations in machine learning [25].”

Yuan et al. also discuss two significant connections between causal emergence and machine learning. First, machine learning can be used to improve calculations of causal emergence. Second, causal emergence measures can be used to better understand how machine learning works and improve it.

The work described above built on earlier work by Crutchfield, who claimed that the identification of emergence and hierarchies could be made operational, stating that “different scales are delineated by a succession of divergences in statistical complexity at lower levels.” More recently, Rupe and Crutchfield have reported progress towards identifying emergent self-organisation in a system.

Although this work on quantitative measures of emergence based on information theory represents significant progress, there are many open problems. Examples include the extension to non-Markovian systems and the development of computationally feasible methods for large systems. The latter is particularly important in physical systems where spontaneous symmetry breaking occurs, as this only happens in the thermodynamic limit of an infinite system.

There is an unrecognised similarity between the work described above and techniques recently developed to characterise phase transitions in statistical mechanics models such as the Ising model and classical dimer models. Coarse-graining (CG) is optimised by maximising the Real-Space Mutual Information (RSMI) between a spatial block and its distant environment. 

In general, maximising mutual information is notoriously hard but can be done using state-of-the-art machine learning algorithms. Gokmen et al. have developed an algorithm that they claim “can, unsupervised, construct order parameters, locate phase transitions, and identify spatial correlations and symmetries for complex and large-dimensional real-space data.” Furthermore, the optimal CG explicitly identifies the scaling operators associated with the critical point. 

The classical dimer model provides a stringent test as “the relevant low-energy degrees of freedom are profoundly different from the microscopic building blocks of the theory and change qualitatively throughout the phase diagram.” In other words, the emergent entities (quasiparticles such as vortices associated with the height field, which is described by a sine-Gordon field theory) are different from the dimers.

It is encouraging to see that two different scientific communities have developed similar ideas to address this challenging problem of making discussions about emergence and causality more concrete and quantitative.

Undergraduates need to learn about the Ising model

A typical undergraduate course on statistical mechanics is arguably misleading because (unintentionally) it does not tell students several important things (related to one another).

Statistical mechanics is not just about how to calculate thermodynamic properties of a collection of non-interacting particles.

A hundred years ago, many physicists did not believe that statistical mechanics could describe phase transitions. Arguably, this lingering doubt only ended fifty years ago with Wilson's development of renormalisation group theory.

It is about emergence: how microscopic properties are related to macroscopic properties.

Leo Kadanoff commented, "Starting around 1925, a change occurred: With the work of Ising, statistical mechanics began to be used to describe the behaviour of many particles at once."

When I came to UQ 25 years ago, I taught PHYS3020 Statistical Mechanics a couple of times. To my shame, I never discussed the Ising model. There is a nice section on it in the course textbook, Thermal Physics: An Introduction, by Daniel Schroeder. I guess I did not think there was time to "fit it in" and back then, I did not appreciate how important the Ising model is. This was a mistake.

Things have changed for the better due to my colleagues Peter Jacobson and Karen Kheruntsyan. They now include one lecture on the model, and students complete a computational assignment in which they write a Monte Carlo code to simulate the model.

This year, I am giving the lecture on the model. Here are my slides  and what I will write on the whiteboard or document viewer in the lecture.

Multi-step spin-state transitions in organometallics and frustrated antiferromagnetic Ising models

In previous posts, I discussed how "spin-crossover" material is a misnomer because many of these materials do not undergo crossovers but phase transitions due to collective effects. Furthermore, they exhibit rich behaviours, including hysteresis, incomplete transitions, and multiple-step transitions. Ising models can capture some of these effects.

Here, I discuss how an antiferromagnetic Ising model with frustrated interactions can give multi-step transitions. This has been studied previously by Paez-Espejo, Sy and Boukheddaden, and my UQ colleagues Jace Cruddas and Ben Powell. In their case, they start with a lattice "balls and spring" model and derive Ising models with an infinite-range ferromagnetic interaction and short-range antiferromagnetic interactions. They show that when the range of these interactions (and thus the frustration) is increased, more and more steps are observed.

Here, I do something simpler to illustrate some key physics and some subtleties and cautions.

fcc lattice

Consider the antiferromagnetic Ising model on the face-centred-cubic lattice in a magnetic field. 

[Historical trivia: the model was studied by William Shockley back in 1938, in the context of understanding alloys of gold and copper.]

The picture below shows a tetrahedron of four nearest neighbours in the fcc lattice.

Even with just nearest-neighbour interactions, the lattice is frustrated. On a tetrahedron, you cannot satisfy all six AFM interactions. Four bonds are satisfied, and two are unsatisfied.

The phase diagram of the model was studied using Monte Carlo by Kammerer et al. in 1996. It is shown above as a function of temperature and field. All the transition lines are (weakly) first-order.

The AB phase has AFM order within the [100] planes. It has an equal number of up and down spins.

The A3B phase has alternating FM and AFM order between neighbouring planes. Thus, 3/4 of the spins have the same direction as the magnetic field.

The stability of these ordered states is subtle. At zero temperature, both the AB and A3B states are massively degenerate. For a system of 4 x L^3 spins, there are 3 x 2^2L AB states, and 6 x 2^L   A3B states. At finite temperature, the system exhibits “order by disorder”.

On the phase diagram, I have shown three straight lines (blue, red, and dashed-black) representing a temperature sweep for three different spin-crossover systems. The "field" is given by h=1/2(Delta H - T Delta S). In the lower panel, I have shown the temperature dependence of the High Spin (HS) population for the three different systems. For clarity, I have not shown the effects of the hysteresis associated with the first-order transitions.

If Delta H is smaller than the values shown in the figure, then at low temperatures, the spin-crossover system will never reach the complete low-spin state.

Main points.

Multiple steps are possible even in a simple model. This is because frustration stabilises new phases in a magnetic field. Similar phenomena occur in other frustrated models, such as the triangular lattice, the J1-J2 model on a chain or a square lattice.

The number of steps may change depending on Delta S. This is because a temperature sweep traverses the field-temperature phase diagram asymmetrically.

Caution.

Fluctuations matter.

The mean-field theory phase diagram was studied by Beath and Ryan. Their phase diagram is below. Clearly, there are significant qualitative differences, particularly in the stability of the A3B phase.

The transition temperature at zero field is 3.5 J, compared to the value of 1.4J from Monte Carlo.

Monte Carlo simulations may be fraught.

Because of the many competing ordered states associated with frustration, Kammerer et al. note that “in a Monte Carlo simulation one needs unusually large systems in order observe the correct asymptotic behaviour, and that the effect gets worse with decreasing temperature because of the proximity of the phase transition to the less ordered phase at T=0”. 

Open questions.

The example above hints at what the essential physics may be how frustrated Ising models may capture it. However, to definitively establish the connection with real materials, several issues need to be resolved.

1. Show definitively how elastic interactions can produce the necessary Ising interactions. In particular, derive a formula for the interactions in terms of elastic properties of the high-spin and low-spin states. How do their structural differences, and the associated bond stretches or compressions, affect the elastic energy? What is the magnitude, range, and direction of the interactions?

[n.b. Different authors have different expressions for the Ising interactions for a range of toy models, using a range of approximations. It also needs to be done for a general atomic "force field".]

2. For specific materials, calculate the Ising interactions from a DFT-based method. Then show that the relevant Ising model does produce the steps and hysteresis observed experimentally.

"Ferromagnetic" Ising models for spin-state transitions in organometallics

In recent posts, I discussed how "spin crossover" is a misnomer for the plethora of organometallic compounds that undergo spin-state phase transitions (abrupt, first-order, hysteretic, multi-step,...)

In theory development, it is best to start with the simplest possible model and then gradually add new features to the model until (hopefully) arriving at a minimal model that can describe (almost) everything. Hence, I described how the two-state model can describe spin crossover. An Ising "spin" has values of +1 or -1, corresponding to high spin (HS) and low spin (LS) states. The "magnetic" field is half of the difference in Gibbs free energy between the two states. 

The model predicts equal numbers of HS and LS at a temperature

The two-state model is modified by adding Ising-type interactions between the “spins” (molecules). The Hamiltonian is then of the form

 The temperature dependence in the field arises because this is an effective Hamiltonian.

The Ising-type interactions are due to elastic effects. The spin-state transition in the iron atom leads to changes in the Fe-N bond lengths (an increase of about 10 per cent in going from LS to HS), changing the size of the metal-ligand (ML6 ) complex. This affects the interactions (ionic, pi-pi, H-bond, van der Waals) between the complexes. The volume of the ML6 complex changes by about 30 per cent, but typically the volume of the crystal unit cell changes by only a few per cent. The associated relaxation energies are related to the J’s. Calculating them is non-trivial and will be discussed elsewhere. There are many competing and contradictory models for the elastic origin of the J’s.

In this post, I only consider nearest-neighbour ferromagnetic interactions. Later, I will consider antiferromagnetic interactions and further-neighbour interactions that lead to frustration. 

Slichter-Drickamer model

This model was introduced in 1972 is beloved by experimentalists, especially chemists, because it provides a simple analytic formula that can be fit to experimental data.

The system is assumed to be a thermodynamic mixture of HS and LS. x=n_HS(T) is the fraction of HS. The Gibbs free energy is given by

This is minimised as a function of x to give the temperature dependence of the HS population.

The model is a natural extension of the two-state model, by adding a single parameter, Gamma, which is sometimes referred to as the cooperativity parameter.

The model is equivalent to the mean-field treatment of a ferromagnetic Ising model, with Gamma=2zJ, where z is the number of nearest neighbours. Some chemists do not seem to be aware of this connection to Ising. The model is also identical to the theory of binary mixtures, such as discussed in Thermal Physics by Schroeder, Section 5.4.

Successes of the model.

good quantitative agreement with experiments on many materials.

a first-order transition with hysteresis for T_1/2 < Tc =z J.

a steep and continuous (abrupt) transition for T_1/2 slightly larger than Tc.

Values of Gamma are in the range 1-10 kJ/mol. Corresponding vaules of J are in the range 10-200 K, depending on what value of z is assumed.

Weaknesses of the model.

It cannot explain multi-step transitions.

Mean-field theory is quantitatively, and sometimes qualitatively, wrong, especially in one and two dimensions.

The description of hysteresis is an artefact of the mean-field theory, as discussed below.

Figure. Phase diagram of a ferromagnetic Ising model in a magnetic field. (Fig. 8.7.1, Chaikin and Lubensky). Vertical axis is the magnetic field, and the horizontal axis is temperature. Tc denotes the critical temperature, and the double-line denotes a first-order phase transition between paramagnetic phases where the magnetisation is parallel to the direction of the applied field.

Curves show the free energy as a function of the order parameter (magnetisation) in mean-field theory. The dashed lines are the lines of metastability deduced from these free-energy curves. Inside these lines, the free energy has two minima: the equilibrium one and a metastable one. The lines are sometimes referred to as spinodal curves.

The consequences of the metastability for a field sweep at constant temperature are shown in the Figure below, taken from Banerjee and Bar.

How does this relate to thermally induced spin-state transitions?

Consider the phase diagram shown above of a ferromagnetic Ising model in a magnetic field. The red and blue lines correspond to temperature scans for two SCO materials that have different values of the parameters Delta H and DeltaS.

The occurrence of qualitatively different behaviour is determined by where the lines intercept the temperature and field axes, i.e. the values of T_1/2 /J and Delta H/J. If the former is larger than Tc/J, as it is for the blue line, then no phase transition is observed. 

The parameter Delta H/J determines whether at low temperatures, the complete HS state is formed.

The figure below is a sketch of the temperature dependence of the population of HS for the red and blue cases.

Note that because of the non-zero slope of the red line, the temperature  T_1/2 is not the average of the temperatures at which the transition occurs on the up and down temperature sweeps.

Deconstructing hysteresis.

The physical picture above of metastability is an artefact (oversimplification) of mean-field theory. It predicts that an infinite system would take an infinite time to reach the equilibrium state from the metastable state.

(Aside: In the context of the corresponding discrete-choice models in economics, this has important and amusing consequences, as discussed by Bouchaud.)

In reality, the transition to the equilibrium state can occur via nucleation of finite domains or in some regimes via a perturbation with a non-zero wavevector. This is discussed in detail by Chaikin and Lubensky, chapter 4.

The consequence of this “metastability” for a first-order transition in an SCO system is that the width of the hysteresis region (in temperature) may depend on the rate at which the temperature is swept and whether the system is allowed to relax before the magnetisation (fraction of HS) is measured at any temperature. Emprically, this is observed and has been highlighted by Brooker, albeit without reference to the theoretical subtleties I am highlighting here. She points out that up to 2014, chemists seemed to have been oblivious to these issues and reported results without testing whether their observations depended on the sweep rate or whether they waited for relaxation.

(Aside. The dynamics are different for conserved and non-conserved order parameters. In a binary liquid mixture, the order parameter is conserved, i.e., the number of A and B atoms is fixed. In an SCO material, the number of HS and LS is not conserved.)

In the next post, I will discuss how an antiferromagnetic Ising model can give a two-step transition and models with frustrated interactions can give multi-step transitions.

Spin crossover is a misnomer

There are hundreds of organometallic compounds that are classified as spin-crossover compounds. As the temperature is varied the average spin per molecule can undergo a transition between low-spin and high-spin states.

The figure below shows several classes of transitions that have been observed. The vertical axis represents the fraction of molecules in the high-spin state, and the horizontal axis represents temperature.

Taken from Gutlich, Garcia, and Spiering and reproduced by Nicolazzi and Bousseksou

a) A smooth crossover. At the temperature T_{1/2} there are equal numbers of high and low spins.

b) There is sharp transition with the curve having a very large slope at T_{1/2}.

c) There is a discontinuous change in the spin fraction at the transition temperature, the value of which depends on whether the temperature is increasing or decreasing, i.e., there is hysteresis. The discontinuity and hysteresis are characteristic of a first-order phase transition.

d) There is a step in the curve when the high-spin fraction is close to 0.5. This is known as a two-step transition.

e) Although a crossover occurs, the system never contains only low- or high-spins.

But, there is more. Over the past decade, multiple-step transitions have been observed. An example of a four-step transition is below.

The figure is taken from a paper by Clements, Price, Neville, and Kepert.

Hysteresis is present and is larger at lower temperatures.

In a few cases of multiple-step transitions on the down-temperature sweep, the first step is missing compared to the up-temperature step.

Given the diverse behaviour described above, including sharp transitions and first-order phase transitions, spin "crossover" is a misnomer.

More importantly, given the chemical and structural complexity materials involved, is there a simple model effective Hamiltonian that can capture all this diverse behaviour?

Yes. An Ising model in a field. A preliminary discussion is here. I hope to discuss this in future posts. But first I need to introduce the simple two-state model and show what it can and cannot explain.

Phase diagrams elucidate emergence

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

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

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

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

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

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

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

Emergence and continuous phase transitions in flatland

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

Novelty

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

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

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

Discontinuity

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

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

Toy model

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

Modularity at the mesoscale

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

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

Universality

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

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

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

Singularity

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

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

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

Two-dimensional crystals

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

Predictability

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

A very effective Hamiltonian in nuclear physics

Atomic nuclei are complex quantum many-body systems. Effective theories have helped provide a better understanding of them. The best-known are the shell model, the (Aage) Bohr-Mottelson theory of non-spherical nuclei, and the liquid drop model. Here I introduce the Interacting Boson Model (IBM), which provides somewhat of a microscopic basis for the Bohr-Mottelson theory. Other effective theories in nuclear physics are chiral perturbation theory, Weinberg's theory for nucleon-pion interactions, and Wigner's random matrix theory.

The shell model has similarities to microscopic models in atomic physics. A major achievement is it explains the origins of magic numbers, i.e., nuclei with atomic numbers 2, 8, 20, 28, 50, 82, and 126 are particularly stable because they have closed shells. Other nuclei can then be described theoretically as an inert closed shell plus valence nucleons that interact with a mean-field potential due to the core nuclei and then with one another via effective interactions.

For medium to heavy nuclei the Bohr-Mottelson model describes collective excitations including transitions in the shape of nuclei.

An example of the trends in the low-lying excitation spectrum  to explain is shown in the figure below. The left spectrum is for nucleus with close to a magic number of nuclei and the right one for an almost half-filled shell. R_4/2 is the ratio of the energies of the J=4+ state to that of the 2+ state, relative to the ground state. B(E2) is the strength of the quadrupole transition between the 2+ state and the ground state.

The Interacting Boson Model (IBM) is surprisingly simple and successful. It illustrates the importance of quasi-particles, builds on the stability of closed shells, and neglects many degrees of freedom. It describes even-even nuclei, i.e., nuclei with an even number of protons and an even number of neutrons. The basic entities in the theory are pairs of nucleons. These are taken to be either an s-wave state or a d-wave state. There are five d-wave states (corresponding to the 2J+1 possible states of total angular momentum with J=2). Each state is represented by a boson creation operator and so the Hilbert space is six-dimensional. If the states are degenerate [which they are not] the model has U(6) symmetry.

The IBM Hamiltonian is written in terms of the most general possible combinations of the boson operators. This has a surprisingly simple form.

Note that it involves only four parameters. For a given nucleus these parameters can be fixed from experiment, and in principle calculated from the shell model. The Hamiltonian can be written in a form that gives physical insight, connects to the Bohr-Mottelson model and is amenable to a group theoretical analysis that makes calculation and understanding of the energy spectrum relatively simple.

Central to the group theoretical analysis is considering subalgebra chains as shown below

 

An example of an energy spectrum is shown below.

The fuzzy figures are taken from a helpful Physics Today article by Casten and Feng from 1984 (Aside: the article discusses an extension of the IBM involving supersymmetry, but I don't think that has been particularly fruitful).

The figure below connects the different parameter regimes of the model to the different subalgebra chains.

The nucleotide chart below has entries that have colour shading corresponding to their parameter values for the IBM model according to the symmetry triangle above.

The different vertices of the triangle correspond to different nuclear geometries and allow a connection to Aage Bohr's model for the surface excitations. 

This is discussed in a nice review article, which includes the figure above.

Quantum phase transitions in shape of nuclei

Pavel Cejnar, Jan Jolie, and Richard F. Casten

Aside: one thing that is not clear to me from the article concerns questions that arise because the nucleus has a finite number of degrees of freedom. Are the symmetries actually broken or is there tunneling between degenerate ground states?   

Shape memory alloys

Recently I bought a small wire of NiTinol to have fun with and use in demonstrations to kids. This video gives a spectacular demonstration and attempts to explain how it works. I did not know about their use in stents for heart surgery.

I am still struggling to understand exactly how shape-memory alloys work. According to Wikipedia

The shape memory effect occurs because a temperature-induced phase transformation reverses deformation...Typically the martensitic (low-temperature) phase is monoclinic or orthorhombic . Since these crystal structures do not have enough slip systems for easy dislocation motion, they deform by twinning—or rather, detwinning.

Martensite is thermodynamically favored at lower temperatures, while austenite (B2 cubic) is thermodynamically favored at higher temperatures. Since these structures have different lattice sizes and symmetry, cooling austenite into martensite introduces internal strain energy in the martensitic phase. To reduce this energy, the martensitic phase forms many twins—this is called "self-accommodating twinning" and is the twinning version of geometrically necessary dislocations. 

In different words, I think the essential idea may be the following. In most metals large strains are accomodated by topological defects such as dislocations. These become entangled leading to work hardening and irreversible changes is macroscopic shapes. Shape memory alloys are different because of the low symmetry unit cell. The most natural defects are twinning domain walls and they are not topological and so their formation is reversible.

I am looking forward to reading the book chapter Shape memory alloys by Vladimir Buljak, Gianluca Ranzi

Another fascinating phenomena that is related to shape-memory is "superelasticity", which I discussed in an earlier post on organic molecular crystals, and has recently been reviewed.

I welcome clarification  of the essential physics.

Gravitational waves and ultra-condensed matter physics

In 2016, when I saw the first results from the LIGO gravitational wave interferometer my natural caution and skepticism kicked in. They had just observed one signal in an incredibly sensitive measurement. A lot of data analysis was required to extract the signal from the background noise. That signal was then fitted the results of numerical simulations of the solutions to Einstein's gravitational field equations describing the merger of two black holes. Depending on how you count about 15 parameters are required to specify the parameters of the binary system [distance from earth, masses, relative orientations of orbits, .... The detection events involve displacement of the mirrors in the interferometer by about 30 picometres!

What on earth could go wrong?!

After all, this was only two years after the BICEP2 fiasco which claimed to have detected anisotropies in the cosmic microwave background due to gravitational waves associated with cosmic inflation. The observed signal turned out to be just cosmic dust! It led to a book, by the cosmologist Brian Keating, Losing the Nobel Prize: A Story of Cosmology, Ambition, and the Perils of Science’s Highest Honor

Well, I am happy to be wrong, if it is good for science. Now almost one hundred gravitational wave events have been observed and one event GW170817 has been correlated with an x-ray observation.

But detecting some gravitational waves is quite a long way from gravitational wave astronomy, i.e, using gravity wave detectors as a telescope, in the same sense as the regular suite of optical, radio, X-ray, ... detectors. I was also skeptical about that. But it does not seem that gravity wave detectors are providing a new window into the universe.

A few weeks ago I heard a very nice UQ colloquium by Paul Lasky, What's next in gravitational wave astronomy?

Paul gave a nice overview of the state of the field, both past and future. 

A key summary figure is below. It shows different possible futures when two neutron stars merge.

The figure is taken from the helpful review

The evolution of binary neutron star post-merger remnants: a review, Nikhil Sarin and Paul D. Lasky

A few of the things that stood out to me.

1. One stunning piece of physics is that in the black hole mergers that have been observed the combined mass of the resulting black hole is three solar masses less than the total mass of the two separate black holes. The resulting loss of mass energy (E=mc^2) of three solar masses is converted into gravitational wave energy within seconds. During this time the peak radiant power was more than fifty times the power of all the stars in the observable universe combined!

I have fundamental questions about a clear physical description of this energy conversion process. First, defining "energy" in general relativity is a vexed and unresolved question with a long history. Second, is there any sense in which needs to describe this in terms of a quantum field theory: specifically conversion of neutron matter into gravitons?

2. Probing nuclear astrophysics in neutron stars. It may be possible to test the equation of state (relation between pressure and density) of nuclear matter. This determines the Tolman–Oppenheimer–Volkoff limit; the upper bound to the mass of cold, non-rotating neutron stars. According to Sarin and Lasky

The supramassive neutron star observations again provide a tantalising way of developing our understanding of the dynamics of the nascent neutron star and the equation of state of nuclear matter (e.g., [37,121,127–131]). The procedure is straight forward: if we understand the progenitor mass distribution (which we do not), as well as the dominant spin down mechanism (we do not understand that either), and the spin-down rate/braking index (not really), then we can rearrange the set of equations governing the system’s evolution to find that the time of collapse is a function of the unknown maximum neutron star mass, which we can therefore infer. This procedure has been performed a number of times in different works, each arriving at different answers depending on the underlying assumptions at each of the step. The vanilla assumptions of dipole vacuum spin down of hadronic stars does not well fit the data [37,127], leading some authors to infer that quark stars, rather than hadronic stars, best explain the data (e.g., [129,130]), while others infer that gravitational radiation dominates the star’s angular momentum loss rather than magnetic dipole radiation (e.g [121,127]).

As the authors say, this is a "tantalising prospect" but there are many unkowns. I appreciate their honesty. 

3. Probing the phase diagram of Quantum Chromodynamics (QCD)

This is one of my favourite phase diagrams and I used to love to show it to undergraduates.

Neutron stars are close to the first-order phase transition associated with quark deconfinement.

When the neutron stars merge it may be that the phase boundary is crossed.