"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 an associated quantum transition (at a wavelength of 1.2 cm) is the basis of the maser, the forerunner of the laser. This example of ammonia was discussed by Anderson at the beginning of his seminal More is Different article to illustrate how symmetry breaking leads to well-defined molecular structures in large molecules. 

Figure is taken from here.

Born-Oppenheimer approximation 

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

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

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

Qualitative difference with increase in molecular size

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

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

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