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.

The two-state model for spin crossover in organometallics

Previously, I discussed how spin-crossover is a misnomer for organometallic compounds and proposed that an effective Hamiltonian to describe the rich states and phase transitions is an Ising model in "magnetic field".

I introduce the two-state model that defines the model without the Ising interactions. To save me time on formatting in HTML, here is a pdf file that describes the model and what comparisons with experimental data (such as that below) tells us.

Future posts will consider how elastic interactions produce the Ising interaction and how frustrated interactions can produce multi-step transitions.

My review article on emergence

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

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

The abstract is below.

I welcome feedback. 

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

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.

Science job openings in sunny Brisbane, Australia

Bribie Island, just north of Brisbane.

The University of Queensland has just advertised several jobs that may be of interest to readers of this blog, particularly those seeking to flee the USA.

There is a junior faculty position for a theorist working at the interface of condensed matter, quantum chemistry, and quantum computing.

There is also a postdoc to work on the theory of strongly correlated electron systems with my colleagues Ben Powell and Carla Verdi.

There is a postdoc in experimental condensed matter, to work on scanning probe methods, such as STM, with my colleague Peter Jacobson.

Glasshouse Mountains. Just north of Brisbane.

Reviewing emergent computational abilities in Large Language Models

Two years ago, I wrote a post about a paper by Wei et al, Emergent Abilities of Large Language Models

Then last year, I posted about a paper Are Emergent Abilities of Large Language Models a Mirage? that criticised the first paper.

There is more to the story. The first paper has now been cited over 3,600 times. There is a helpful review of the state of the field.

Emergent Abilities in Large Language Models: A Survey

Leonardo Berti, Flavio Giorgi, Gjergji Kasneci

It begins with a discussion of what emergence is, quoting from Phil Anderson's More is Different article [which emphasised how new properties may appear when a system becomes large] and John Hopfield's Neural networks and physical systems with emergent collective computational abilities, which was the basis of his recent Nobel Prize. Hopfield stated

"Computational properties of use to biological organisms or the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons)."

Berti et al. observe, "Fast forward to the LLM era, notice how Hopfield's observations encompass all the computational tasks that LLMs can perform."

They discuss emergent abilities as in-context learning, defined as the "capability to generalise from a few examples to new tasks and concepts on which they have not been directly trained."

Here, I put this review in the broader context of the role of emergence in other areas of science.

Scales. 

Simple scales that describe how large an LLM is include the amount of computation, the number of model parameters, and the size of the training dataset. More complicated measures of scale include the number of layers in a deep neural network and the complexity of the training tasks.

Berti et al. note that the emergence of new computational abilities does not just follow from increases in the simple scales but can be tied to the training process. I note that this subtlety is consistent with experience in biology. Simple scales would be the length of an amino acid chain in a protein or base pairs in a DNA molecule, the number of proteins in a cell or the number of cells in an organism. More subtle scales include the number of protein interactions in a proteome or gene networks in a cell. Deducing what the relevant scales are is non-trivial. Furthermore, as emphasised by Denis Noble and Robert Bishop, context matters, e.g., a protein may only have a specific function if it is located in a specific cell.

Novelty. 

When they become sufficiently "large", LLMs have computational abilities that they were not explicitly designed for and that "small" versions do not have. 

The emergent abilities range "from advanced reasoning and in-context learning to coding and problem-solving."

The original paper by Wei et al. listed 137 emergent abilities in an Appendix!

Berti et al. give another example.

"Chen et al. [15] introduced a novel framework called AgentVerse, designed to enable and study collaboration among multiple AI agents. Through these interactions, the framework reveals emergent behaviors such as spontaneous cooperation, competition, negotiation, and the development of innovative strategies that were not explicitly programmed."

An alternative to defining novelty in terms of a comparison of the whole to the parts is to compare properties of the whole to those of a random configuration of the system. The performance of some LLMs is near-random (e.g., random guessing) until a critical threshold is reached (e.g., in size) when the emergent ability appears.

Discontinuities.

Are there quantitative objective measures that can be used to identify the emergence of a new computational ability? Researchers are struggling to find agreed-upon metrics that show clear discontinuities. That was the essential point of Are Emergent Abilities of Large Language Models a Mirage? 

In condensed matter physics, the emergence of a new state of matter is (usually) associated with symmetry breaking and an order parameter. Figuring out what the relevant broken symmetry and the order parameter often requires brilliant insight and may even lead to a Nobel Prize (Neel, Josephson, Ginzburg, Leggett,...) A similar argument can be made with respect to the development of the Standard Model of elementary particles and gauge fields. Furthermore, the discontinuities only exist in the thermodynamic limit (i.e., in the limit of an infinite system), and there are many subtleties associated with how the data from finite-size computer simulations should be plotted to show that the system really does exhibit a phase transition.

Unpredictability.

The observation of new computational abilities in LLMs was unanticipated and surprised many people, including the designers of the specific LLMs involved. This is similar to what happens in condensed matter physics, where new states of matter have mostly been discovered by serendipity.

Some authors seem surprised that it is difficult to predict emergent abilities. "While early scaling laws provided some insight, they often fail to anticipate discontinuous leaps in performance."

Given the largely "black box" nature of LLMs, I don't find it the unpredictability surprising. It is hard for condensed matter systems, and they are much better characterised and understood.

Modular structures at the mesoscale.

Modularity is a common characteristic of emergence. In a wide range of systems, from physics to biology to economics, a key step in the development of the theory of a specific emergent phenomenon has been the identification of a mesoscale (intermediate between the micro- and macro-scales) at which modular structures emerge. These modules interact weakly with one another, and the whole system can be understood in these terms. Identification of these structures and the effective theories describing them has usually required brilliant insight. An example is the concepts of quasiparticles in quantum many-body physics, pioneered by Landau.

Berti et al. do not mention the importance of this issue. However, they do mention that "functional modules emerge naturally during training" [Ref. 7,43,81,84] and that "specialised circuits activate at certain scaling thresholds [24]".

Modularity may be related to an earlier post, Why do deep learning algorithms work so well? In the training process, a neural network rids noisy input data of extraneous details...There is a connection between the deep learning algorithm, known as the "deep belief net" of Geoffrey Hinton, and renormalisation group methods (which can be key to identifying modularity and effective interactions).

Is emergence good or bad?

Undesirable and dangerous capabilities can emerge. Those observed include deception, manipulation, exploitation, and sycophancy.

These concerns parallel discussions in economics. Libertarians, the Austrian school, and Federich Hayek tend to see the emergence as only producing socially desirable outcomes, such as the efficiency of free markets [the invisible hand of Adam Smith]. However, emergence also produces bubbles and crashes and recessions.

Resistance to control

A holy grail is the design, manipulation, and control of emergent properties. This ambitious goal is promoted in materials science, medicine, engineering, economics, public policy, business management, and social activism. However, it largely remains elusive, arguably due to the complexity and unpredictability of the systems of interest. Emergent properties of LLMs may turn out to offer similar hopes, frustrations, and disappointments. We should try, but have realistic expectations.

Toy models.

This is not discussed in the review. As I have argued before, a key to understanding a specific emergent phenomenon is the development of toy models that illustrate the phenomenon and the possible essential ingredients for it to occur. The following paper may be a step in that direction.

An exactly solvable model for emergence and scaling laws in the multitask sparse parity problem

Yoonsoo Nam, Nayara Fonseca, Seok Hyeong Lee, Chris Mingard, Ard A. Louis

In a similar vein, another possibly relevant paper is the review

Statistical Mechanics of Deep Learning

Yasaman Bahri, Jonathan Kadmon, Jeffrey Pennington1, Sam S. Schoenholz, Jascha Sohl-Dickstein and Surya Ganguli

They considered a toy model for the error landscape for a neural network, and show that the error function for a deep neural net of depth D corresponds to the energy function for a D-spin spherical spin glass. [Section 3.2 in their paper].

Emergence in Chemistry

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

Scales

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

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

Novelty

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

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

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

Diversity

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

Molecular structure

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

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

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

Figure is taken from here.

Born-Oppenheimer approximation 

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

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

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

Qualitative difference with increase in molecular size

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

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

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

What Americans might want to know about getting a job in an Australian university

Universities and scientific research in the USA are facing a dire future. Understandably, some scientists are considering leaving the USA. I have had a few enquiries about Australia. This makes sense, as Australia is a stable English-speaking country with similarities in education, culture, democracy, and economics. At least compared to most other possible destinations. Nevertheless, there are important differences between Australia and the USA to be aware of, particularly when it comes down to how universities function (and dis-function!) and how they hire people. 

A few people have asked me for advice. Below are some comparisons. Why should you believe me? I spent eleven years in the US (1983-1994) and visited at least once a year until 2018. On the other hand, there are some reasons to take what I say with a grain of salt. I have never been a faculty member in a US university. I retired four years ago from a faculty position in Australia. I actually haven't sat on a committee for almost ten years :). Hopefully, this post will prompt other readers to weigh in with other perspectives.

There are discussions in Australia about trying to attract senior people from the USA to come here. Whether that will come to anything substantial remains to be seen.

The best place to look for advertised positions is on Seek. 

Postdocs

This is where the news is best. Young people in the USA can apply for regular postdoc positions. Most are attached to specific grants and so involve working on a specific project. 

Ph.D. students

Most of the positions go to Australian citizens who get there own scholarship (fellowship) from the government. These are not tied to a grant or a supervisor (advisor) There are a few positions for international students, but not many. Usually they go to applicants with a Masters degree and publications.

Ph.D's are funded for 3 to 3.5 years. There is no required course work. Australian students have done a 4-year undergraduate degree and no Masters. This means tackling highly technical projects in theory is not realistic, except for exceptional students.

Faculty hiring is adhoc 

There is no hiring cycle. Positions tend to be advertised at random times depending on local politics, whims and bureaucracy. Universities and Schools (departments) claim they have strategic plans, but given fluctuations in funding, management, and government policy positions appear and disappear at random. Typically, the Dean (and their lackies), not the department, control the selection process, particularly for senior appointments. The emphasis is on metrics. Letters of reference are sometimes not even called for before short listing. Some hiring is done purely from online interviews and seminars.

Bias towards insiders 

People already in the Australian system know how to navigate it best. They may also already have a grant from the Australian Research Council and have done some teaching and (positive) student evaluations. They are known quantities to the managers and so a safer bet than outsiders. If you want to get a junior faculty position here (a lectureship) your chances may be better if you first come as a postdoc. However, there are exceptions...

Current funding crunches

Unfortunately, I fear the faculty market may be quite cool for the next few years. Many universities are actually trying to sack (fire) people due to funding shortfalls. These budget crises are due to post-covid, mismanagement, and the government trying to reduce international student numbers (due to the politics of a housing and cost-of-living crisis).

Australian Research Council

This is pretty much the sole source of funding in physics and chemistry. This is quite different to the USA where there were (pre-Trump) numerous funding agencies (NSF, DOE, DOD, ...).  They are currently reviewing and redesigning all their programs and so we will have to wait to see how this may impact the prospects of scientific refugees from the USA. (They used to have quite good Fellowship schemes for all career stages that were an excellent avenue for foreigners to come here). Some of my colleagues recommend following ARC Tracker on social media to be informed about the latest at ARC.

Thirty years ago, I came back to Australia from the USA. I had a wonderful stint doing science, largely because of generous ARC funding. Unfortunately, the system has declined. But I am sure it is better than being the USA right now.

There are many more things I could write about. Some have featured in previous rants about metrics and managerialism. Things to be aware of before accepting a job include faculty having little voice or power, student absenteeism, corrupt governance, and there is no real tenure or sabbaticals.

Thermodynamics and emergence

Novelty. 

Temperature and entropy are emergent properties. Classically, they are defined by the zeroth and second laws of thermodynamics, respectively. The individual particles that make up a system in thermodynamic equilibrium do not have these properties. Kadanoff provided an example illustrating the qualitative difference between macro- and micro-perspectives. He pointed out how deterministic behaviour can emerge at the macroscale from stochastic behaviour at the microscale. The many individual molecules in a dilute gas can be viewed as undergoing stochastic motion. However, collectively they are described by an equation of state such as the ideal gas law.

 Primas gave a technical argument, involving C* algebras, that temperature is emergent: it belongs to an algebra of contextual observables but not to the algebra of intrinsic observables.44 Following this perspective, Bishop argued that temperature and the chemical potential are (contextually) emergent.

Intra-stratum closure. 

The laws of thermodynamics, the equations of thermodynamics (such as TdS = dU + pdV), and state functions such as S(U,V), provide a complete description of processes involving equilibrium states. A knowledge of microscopic details, such as the atomic constituents or forces of interaction, is not necessary for the description.

Irreducibility. 

A common view is that thermodynamics can be derived from statistical mechanics. However, this is contentious. David Deutsch claimed that the second law of thermodynamics is an “emergent law”: it cannot be derived from microscopic laws, like the principle of testability.

Lieb and Yngvason stated that the derivation from statistical mechanics of the law of entropy increase “is a goal that has so far eluded the deepest thinkers.”  In contrast, Weinberg claimed that Maxwell, Boltzmann, and Gibbs “showed that the principles of thermodynamics could in fact be deduced mathematically, by an analysis of the probabilities of different configurations… Nevertheless, even though thermodynamics has been explained in terms of particles and forces, it continues to deal with emergent concepts like temperature and entropy that lose all meaning on the level of individual particles.” (Dreams of A Final Theory, pages 40-41)

I agree that thermodynamic properties (e.g., equations of state, the temperature dependence of heat capacity, and phase transitions) can be deduced from statistical mechanics. However, thermodynamic principles, such as the second law, are not thermodynamic properties. Furthermore, these thermodynamic principles are required to justify the equations of statistical mechanics, such as the partition function, that are used to calculate thermodynamic properties. 

Macro hints of microscopics.

The Sackur-Tetrode equation for the entropy of an ideal gas hinted at the quantisation of phase space. The Gibbs paradox hinted that fundamental particles are indistinguishable. The third law of thermodynamics hints at quantum degeneracy.

Lamenting the destruction of science in the USA

I continue to follow the situation in the USA concerning the future of science with concern. Here are some of the articles I found most informative (and alarming).

Trump Has Cut Science Funding to Its Lowest Level in Decades (New York Times). It has helpful graphics.

On the proposed massive cuts to the NSF budget, the table below [courtesy of Doug Natelson] is informative and disturbing. At the end of the day it is all about people [real live humans and investment in human capital]

APS News | US physics departments expect to shrink graduate programs [I was quite surprised the expect shrinking isn't greater].

From an Update on NSF Priorities

Are you still funding research on misinformation/disinformation?

Per the Presidential Action announced January 20, 2025, NSF will not prioritize research proposals that engage in or facilitate any conduct that would unconstitutionally abridge the free speech of any American citizen. NSF will not support research with the goal of combating "misinformation," "disinformation," and "malinformation" that could be used to infringe on the constitutionally protected speech rights of American citizens across the United States in a manner that advances a preferred narrative about significant matters of public debate.

The Economist had a series of articles in the May 24 issue [Science and Technology section] that put the situation concerning research and universities in a broader context. The associated editorial is MAGA’s assault on science is an act of grievous self-harm, featuring the graphic below.

I welcome comments and suggestions of other articles.

Pattern formation and emergence

Patterns in space and/or time form in fluid dynamics (Rayleigh-Bénard convection and Taylor-Couette flow), laser physics, materials science (dendrites in the formation of solids from liquid melts), biology (morphogenesis), and chemistry (Belousov-Zhabotinsky reactions). External constraints, such as temperature gradients, drive most of these systems out of equilibrium. 

Novelty. 

The parts of the system can be viewed as the molecular constituents or small uniform parts of the system. In either case, the whole system has a property (a pattern) that the parts do not have.

Discontinuity. 

When some parameter becomes larger than a critical value, the system transitions from a uniform state to a non-uniform state. 

Universality. 

Similar patterns, such as convection rolls in fluids, can be observed in diverse systems regardless of the microscopic details of the fluid. Often, there is a single parameter, such as the Reynolds number, which involves a combination of fluid properties, that determines the type of patterns that form. Cross and Hohenberg highlighted how the models and mechanisms of pattern formation across physics, chemistry, and biology have similarities. Turing’s model for pattern formation in biology associated it with concentration gradients of reacting and diffusing molecules. However, Gierer and Meinhardt showed that it is sufficient to have a network with competition between short-range positive feedback and long-range negative feedback. This could occur in a circuit of cellular signals.

Self-organisation. 

The formation of a particular pattern occurs spontaneously, resulting from the interaction of the many components of the system.

Effective theories. 

A crystal growing from a liquid melt can form shapes such as dendrites. This process involves instabilities of the shape of the crystal-liquid interface. The interface dynamics are completely described by a few partial differential equations that can be derived from macroscopic laws of thermodynamics and heat conduction. A helpful review is by Langer. 

Diversity. 

Diverse patterns are observed, particularly in biological systems. In toy models, such as the Turing model, with just a few parameters, a diverse range of patterns, both in time and space, can be produced by varying the parameters. Many repeated iterations can lead to a diversity of structures. This may result from a sensitive dependence on initial conditions and history. For example, every snowflake is different because, as it falls, it passes through a slightly different environment, with small variations in temperature and humidity, compared to others.

Toy models. 

Turing proposed a model for morphogenesis in 1952 that involved two coupled reaction-diffusion equations. Homogeneous concentrations of the two chemicals become unstable when the difference between the two diffusion constants becomes sufficiently large. A two-dimensional version of the model can produce diverse patterns, many resembling those found in animals. However, after more than seventy years of extensive study, many developmental biologists remain sceptical of the relevance of the model, partly because it is not clear whether it has a microscopic basis. Kicheva et al., argue that “pattern formation is an emergent behaviour that results from the coordination of events occurring across molecular, cellular, and tissue scales.” 

Other toy models include Diffusion Limited Aggregation, due to Witten and Sander, and Barnsley’s iterated function system for fractals that produces a pattern like a fern.

Here is a beautiful lecture on Pattern Formation in Biology by Vijaykumar Krishnamurthy

 

Remembering the student protestors who died 36 years ago

Memorial plaque in the Great Court of the University of Queensland today.

Emergence and quantum theories of gravity

Einstein’s theory of General Relativity successfully describes gravity and large scales of length and mass. In contrast, quantum theory describes small scales of length and mass. Emergence is central to most attempts to unify the two theories. Before considering specific examples, it is useful to make some distinctions.

First, a quantum theory of gravity is not necessarily the same as a theory to unify gravity with the three other forces described by the Standard Model. Whether the two problems are inextricable is unknown.

Second, there are two distinct possibilities on how classical gravity might emerge from a quantum theory. In Einstein’s theory of General Relativity, space-time and gravity are intertwined. Consequently, the two possibilities are as follows.

i. Space-time is not emergent. Classical General Relativity emerges from an underlying quantum field theory describing fields at small length scales, probably comparable to the Planck length.

ii. Space-time emerges from some underlying granular structure. In some limit, classical gravity emerges with the space-time continuum. 

Third, there are "bottom-up" and "top-down" approaches to discovering how classical gravity emerges from an underlying quantum theory, as was emphasised by Bei Lok Hu.

Finally, there is the possibility that quantum theory itself is emergent, as discussed in an earlier post about the quantum measurement problem. Some proposals of Emergent Quantum Mechanics (EQM) attempt to include gravity.

I now mention several different approaches to quantum gravity and for each point out how they fit into the distinctions above.

Gravitons and semi-classical theory

A simple bottom-up approach is to start with classical General Relativity and consider gravitational waves as the normal modes of oscillation of the space-time continuum. They have a linear dispersion relation and move with the speed of light. They are analogous to sound waves in an elastic medium. Semi-classical quantisation of gravitational waves leads to gravitons which are a massless spin-2 field. They are the analogue of phonons in a crystal or photons in the electromagnetic vacuum. However, this reveals nothing about an underlying quantum theory, just as phonons with a linear dispersion relation reveal nothing about the underlying crystal structure.

On the other hand, one can start with a massless spin-2 quantum field and consider how it scatters off massive particles. In the 1960s, Weinberg showed that gauge invariance of the scattering amplitudes implied the equivalence principle (inertial and gravitational mass are identical) and the Einstein field equations. In a sense, this is a top-down approach, as it is a derivation of General Relativity from an underlying quantum theory. In passing, I mention Weinberg used a similar approach to derive charge conservation and Maxwell’s equations of classical electromagnetism, and classical Yang-Mills theory for non-abelian gauge fields. 

Weinberg pointed out that this could go against his reductionist claim that in the hierarchy of the sciences, the arrows of the explanation always point down, saying “sometimes it isn't so clear which way the arrows of explanation point… Which is more fundamental, general relativity or the existence of particles of mass zero and spin two?”

More recently, Weinberg discussed General Relativity as an effective field theory

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

This is a bottom-up approach. Weinberg then went on to discuss a top-down  approach:

“it is usually assumed that in the quantum theory of gravitation, when Λ reaches some very high energy, of the order of 10^15 to 10^18 GeV, the appropriate degrees of freedom are no longer the metric and the Standard Model fields, but something very different, perhaps strings... But maybe not..."

String theory 

Versions of string theory from the 1980s aimed to unify all four forces. They were formulated in terms of nine spatial dimensions and a large internal symmetry group, such as SO(32), where supersymmetric strings were the fundamental units. In the low-energy limit, vibrations of the strings are identified with elementary particles in four-dimensional space-time. A particle with mass zero and spin two appears as an immediate consequence of the symmetries of the string theory. Hence, this was originally claimed to be a quantum theory of gravity. However, subsequent developments have found that there are many alternative string theories and it is not possible to formulate the theory in terms of a unique vacuum.

AdS-CFT correspondence

In the context of string theory, this correspondence conjectures a connection (a dual relation) between classical gravity in Anti-deSitter space-time (AdS) and quantum conformal field theories (CFTs), including some gauge theories. This connection could be interpreted in two different ways. One is that space-time emerges from the quantum theory. Alternatively, the quantum theory emerges from the classical gravity theory.   This ambiguity of interpretation has been highlighted by Alyssa Ney, a philosopher of physics. In other words, it is ambiguous which of the two sides of the duality is the more fundamental. Witten has argued that AdS-CFT suggests that gauge symmetries are emergent. However, I cannot follow his argument.

Seiberg reviewed different approaches, within the string theory community, that lead to spacetime as emergent. An example of a toy model is a matrix model for quantum mechanics [which can be viewed as a zero-dimensional field theory]. Perturbation expansions can be viewed as discretised two-dimensional surfaces. In a large N limit, two-dimensional space and general covariance (the starting point for general relativity) both emerge. Thus, this shows how both two-dimensional gravity and spacetime can be emergent. However, this type of emergence is distinct from how low-energy theories emerge. Seiberg also notes that there are no examples of toy models where time (which is associated with locality and causality) is emergent.

Loop quantum gravity 

This is a top-down approach where both space-time and gravity emerge together from a granular structure, sometimes referred to as "spin foam" or a “spin network”, and has been reviewed by Rovelli. The starting point is Ashtekar’s demonstration that General Relativity can be described using the phase space of an SU(2) Yang-Mills theory. A boundary in four-dimensional space-time can be decomposed into cells and this can be used to define a dual graph (lattice) Gamma. The gravitational field on this discretised boundary is represented by the Hilbert space of a lattice SU(2) Yang-Mills theory. The quantum numbers used to define a basis for this Hilbert space are the graph Gamma,  the “spin” [SU(2) quantum number] associated with the face of each cell, and the volumes of the cells. The Planck length limits the size of the cells. In the limit of the continuum and then of large spin, or vice versa, one obtains General Relativity.

Quantum thermodynamics of event horizons

A bottom-up approach was taken by Padmanabhan. He 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. He does this by considering the connection between event horizons in General Relativity and the temperature of the thermal radiation associated with them. He frames his research as attempting to estimate Avogadro’s number for space-time.

The temperature and entropy associated with event horizons has been calculated for the following specific space-times:

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

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

c. The cosmological horizon in deSitter space is associated with a temperature proportional to the Hubble constant H, as discussed in detail by Gibbons and Hawking.

Padmanabhan considers the number of degrees of freedom on the boundary of the event horizon, Ns, and in the bulk, Nb. He argues for the holographic principle that Ns = Nb. On the boundary surface, there is one degree of freedom associated with every Planck area, Ns = A/Lp2, where Lp is the Planck length and A is the surface area, which is related to the entropy of the horizon, as first discussed by Bekenstein and Hawking. In the bulk, classical equipartition of energy is assumed so the bulk energy E = Nb k T/2. 

Padmanabhan gives an alternative perspective on cosmology through 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, 4\pi/3H^3, where H is the Hubble constant, and Lp 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.

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

where mu is of order unity and gives a value consistent with observation.

The triumphs of lattice gauge theory

When first proposed by Ken Wilson in 1974, lattice gauge theory was arguably a toy model, i.e., an oversimplification. He treated space-time as a discrete lattice purely to make analysis more tractable. Borrowing insights and techniques from lattice models in statistical mechanics, Wilson could then argue for quark confinement, showing that the confining potential was linear with distance.

Earlier, in 1971, Wegner had proposed a Z2 gauge theory in the context of generalised Ising models in statistical mechanics to show how a phase transition was possible without a local order parameter, i.e., without symmetry breaking. Later, it was shown that the phase transition is similar to the confinement-deconfinement phase transition that occurs in QCD. [A nice review from 2014 by Wegner is here]. This work also provided a toy model to illustrate the possibility of a quantum spin liquid.

Perhaps, what was not anticipated was that lattice QCD could be used to calculate accurately properties of elementary particles.

The discrete nature of lattice gauge theory means it is amenable to numerical simulation. It is not necessary to have the continuum limit of real spacetime because of universality. Due to increases in computational power over the past 50 years and innovations in algorithms, lattice QCD can be used to calculate properties of nucleons and mesons, such as mass and decay rates, with impressive accuracy. The figure below is taken from a 2008 article in Science. 

The mass of three mesons is typically used to fix the mass of the light and strange quarks and the length scale. The mass of nine other particles, including the nucleon, is calculated with an uncertainty of less than one per cent, and in agreement with experimental values.

An indication that this is a strong coupling problem is that about 95 per cent of the mass of nucleons comes from the interactions. Only about 5 per cent is from the rest mass of the constituent quarks.

For more background on computational lattice QCD, there is a helpful 2004 Physics Today article, which drew a critical response from Herbert Neuberger. A recent (somewhat) pedagogical review by Sasa Prelovsek just appeared on the arXiv.

Characteristics of static disorder can emerge from electron-phonon interactions

Electronic systems with large amounts of static disorder can exhibit distinct properties, including localisation of electronic states and sub-gap band tails in the density of states and electronic absorption. 

Eric Heller and collaborators have recently published a nice series of papers that show how these properties can also appear, at least on sufficiently long time scales, in the absence of disorder, due to the electron-phonon interaction. On a technical level, a coherent state representation for phonons is used. This provides a natural way of taking a classical limit, similar to what is done in quantum optics for photons. Details are set out in the following paper 

Coherent charge carrier dynamics in the presence of thermal lattice vibrations, Donghwan Kim, Alhun Aydin, Alvar Daza, Kobra N. Avanaki, Joonas Keski-Rahkonen, and Eric J. Heller

This work brought back memories from long ago when I was a postdoc with John Wilkins. I was puzzled by several related things about quasi-one-dimensional electronic systems, such as polyacetylene, that underwent a Peierls instability. First, the zero-point motion of the lattice was comparable to lattice dimerisation that produced an energy gap at the Fermi energy. Second, even in clean systems, there was a large subgap optical absorption. Third, there was no sign of the square-root singularity expected in the density of states, predicted by theories which treated the lattice classically, i.e., calculated electronic properties in the Born-Oppenheimer approximation.

I found that on the energy scales relevant to the sub-gap absorption, the phonons could be treated like static disorder and make use of known exact results for one-dimensional Dirac equations with random disorder. This explained the puzzles.

Effect of Lattice Zero-Point Motion on Electronic Properties of the Peierls-Fröhlich State

The disorder model can also be motivated by considering the Feynman diagrams for the electronic Green's function perturbation expansion in powers of the electron-phonon interaction. In the limit that the phonon frequency is small, all the diagrams become like those for a disordered system, where the strength of the static disorder is given by 

I then teamed up with another postdoc, Kihong Kim, who calculated the optical conductivity for this disorder model.

Universal subgap optical conductivity in quasi-one-dimensional Peierls systems

Two things were surprising about our results. First, the theory agreed well with experimental results for a range of materials, including the temperature dependence. Second,  the frequency dependence had a universal form. Wilkins was clever and persistent at extracting such forms, probably from his experience working on the Kondo problem.

Could quantum mechanics be emergent?

One of the biggest challenges in the foundations of physics is the quantum measurement problem. It is associated with a few key (distinct but related) questions.

i. How does a measurement convert a coherent state undergoing unitary dynamics to a "classical" mixed state for which we can talk about probabilities of outcomes?

ii. Why is the outcome of an individual measurement always definite for the "pointer states" of the measuring apparatus?

iii. Can one derive the Born rule, which gives the probability of a particular outcome?

Emergence of the classical world from the quantum world via decoherence

A quantum system always interacts to some extent with its environment. This interaction leads to decoherence, whereby quantum interference effects are washed out. Consequently, superposition states of the system decay into mixed states described by a diagonal density matrix. A major research goal of the past three decades has been understanding decoherence and the extent to which it does provide answers to the quantum measurement problem. One achievement is that decoherence theory seems to give a mechanism and time scale for the “collapse of the wavefunction” within the framework of unitary dynamics. However, this is not the case because decoherence is not the same as a projection (which is what a single quantum measurement is). Decoherence does not produce definite outcomes but rather statistical mixtures. Decoherence only resolves the issue if one identifies ensembles of measured states with ensembles of the decohered density matrix (the statistical interpretation of quantum mechanics). Thus, it seems decoherence only answers the first question above, but not the last two. On the other hand, Zurek has pushed the decoherence picture further and given a “derivation” of the Born rule within its framework. In other words, decoherence does not solve the quantum measurement problem: measurements always produce definite outcomes.

One approach to solving the problem is to view quantum theory as only an approximate theory. In particular, it could be an effective theory for some underlying theory valid at time and length scales much smaller than those for which quantum theory has been precisely tested by experiments. 

Emergence of quantum field theory from a “classical” statistical theory

Einstein did not accept the statistical nature of quantum theory and considered it should be derivable from a more “realistic” theory. In particular, he suggested “a complete physical description, the statistical quantum theory would …. take an approximately analogous position to the statistical mechanics within the framework of classical mechanics.”

Einstein's challenge was taken up in a concrete and impressive fashion by Stephen Adler in a book, “Quantum Theory as an Emergent Phenomenon: The Statistical Mechanics of Matrix Models as the Precursor of Quantum Field Theory”, published in 2004.  A helpful summary is given in a review by Pearle.

The starting point is "classical" dynamical variables qr and pr which are NxN matrices, where N is even. Half of these variables are bosonic, and the others are fermionic. They all obey Hamilton's equations of motion for an unspecified Hamiltonian H. Three quantities are conserved: H, the fermion number N, and (very importantly) the traceless anti-self-adjoint matrix, 

where the first term is the sum for all the bosonic variables of their commutator, and the second is the sum over anti-commutators for the fermionic variables.

Quantum theory is obtained by tracing over all the classical variables with respect to a canonical ensemble with three (matrix) Lagrange multipliers [analogues of temperature and chemical potential in conventional statistical mechanics] corresponding to the conserved quantities H, N, and C. The expectation values of the diagonal elements of C are assumed to all have the same value, hbar!

An analogy of the equipartition theorem in classical statistical mechanics (which looks like a Ward identity in quantum field theory) leads to dynamical equations (trace dynamics) for effective fields. To make these equations look like regular quantum field theory, an assumption is made about a hierarchy of length, energy, and "temperature" [Lagrange multiplier] scales, which cause the Trace dynamics to be dominated by C rather than H, the trace Hamiltonian. Adler suggests these scales may be Planck scales. Then, the usual quantum dynamical equations and the Dirac correspondence of Poisson brackets and commutators emerge. Most of the actual details of the trace Hamiltonian H do not matter; another case of universality, a common characteristic of emergent phenomena.

The “classical” field C fluctuates about its average value. These fluctuations can be identified with corrections to locality in quantum field theory and with the noise terms which appear in the modified Schrodinger equation of "physical collapse" models of quantum theory.

More recently, theorists including Gerard t’Hooft and John Preskill have investigated how quantum mechanics can emerge from other deterministic systems. This is sometimes known as the emergent quantum mechanics (EmQM) hypothesis.

Underlying deterministic systems considered include

Hamilton-Randers systems defined in co-tangent spaces of large-dimensional configuration spaces, 

neural networks,

cellular automata,

fast-moving classical variables, and the

 boundary of a local classical model with a length that is exponentially large in the number of qubits in the quantum system. 

In most of these versions of EmQM the length scale at which the underlying theory becomes relevant is conjectured to be of the order of the Planck length.

The fact that quantum theory can emerge from such a diverse range of underlying theories again illustrates universality.

The question of quantum physics emerging from an underlying classical theory is not just a question in the foundations of physics or in philosophy. Slagle points out that Emergent Quantum Mechanics may mean that the computational power of quantum computers is severely limited. He has proposed a specific experimental protocol to test for EmQM. A large number d of entangling gates (the circuit depth d) are applied to n qbits in the computational basis, followed by the inverse gates. This is followed by measurement in the computational basis. The fidelity should decay exponentially with d, whereas for EmQM will decay much faster above some critical d, for sufficiently large n.

Independent of experimental evidence, EmQM provides an alternative interpretation to quantum theory that avoids the thorny issues such as the many-worlds interpretation.