Condensed matter physics in flatland

Adventures in Flatland

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

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

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

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

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

Information theoretic measures for emergence and causality

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

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

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

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

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

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

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

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

Yuan et al. continue

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

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

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

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

Yuan et al. also state the following:

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

Sas et al. recently stated

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

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

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

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

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

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

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

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

My best blog posts of 2025?

 Best wishes for the New Year!

Here is a list of the posts that I wrote last year that I hope get the most interest.

My review article on emergence. I wrote posts about emergence in a range of systems: thermodynamics, quantum gravity, economics,... They were drafts of sections for my review article. It may be best to just read the article.

Why is the state of universities such an emotional issue for me?

Undergraduates need to learn about the Ising model

I wrote a series of posts on so-called "spin-crossover" compounds. Here are two: Spin crossover is a misnomer, and Elastic interactions and complex patterns in binary systems

2025 Nobel Prize in Physics: Macroscopic quantum effects

As always, I welcome comments, feedback, and suggestions for new posts.

Elastic interactions and complex patterns in binary systems

One of the many beauties of condensed matter physics is that it can reveal and illuminate how two systems or phenomena that at first appear to be quite different actually involve similar physics. This is an example of universality: for emergent phenomena, many details don't really matter. One example is the similarities between superconductivity and superfluidity. A consequence of universality is that the same concepts, techniques, toy models, and effective theories can be used to describe a wide range of systems.

The complex organometallic molecules, known by the misnomer "spin crossover" compounds, exhibit a rich range of phase transitions and types of spatial order. Key aspects of the physics are the following.

• Each transition metal ion can be in one of two possible states: low-spin or high-spin. 
• The size of each molecular complex depends on the spin state.
• Consequently, the molecules interact with their neighbours via elastic interactions.

A toy model that can describe this is expanding balls connected by springs. Various versions of this type of model are reviewed here. The simplest version is the chain model below.

It turns out there are other classes of systems described by similar models. As far as I am aware, this was first pointed out in Consequences of Lattice Mismatch for Phase Equilibrium in Heterostructured Solids Layne B. Frechette, Christoph Dellago, Phillip L. Geissler

That paper is motivated by experiments on the growth of semiconductor quantum dots, by ion exchange, such as when CdSe is bathed in an Ag-rich solution and Ag2Se is produced with heterostructures (i.e., patterns of Ag and Se ions) that are different from the bulk crystal.

They consider the balls and springs model above on a triangular lattice.

They also point out how similar physics is relevant to binary metal alloys, e.g, AgCu, citing 

Ising model for phase separation in alloys with anisotropic elastic interaction—I. Theory, P. Fratzl and O. Penrose

Those authors consider a square lattice with elastic interactions associated with bond stretching along the edges and diagonals of the squares and bending of the square angles.

Frechette et al. also mention experiments on thin films of  DNA modified metallic nanoparticles. Compared to atomic systems these can tolerate larger lattice-mismatch before the formation of defects due to lattice strain.

Other systems (not mentioned) described by similar Ising models are metal-hydrogen systems, where the Ising pseudospin signifies whether a hydrogen atom is present at a particular site in the metallic crystal.

Frechette et al. start with the ball and springs model and "integrate out" the springs to obtain an effective Hamiltonian, which is an Ising model.

The spatial range of the interaction between Ising spins is shown in the colour-shaded plot below.

The interaction has two components.

One is an infinite range "ferromagnetic" part, seen as the light blue below.

The second is a short-range interaction which is mostly "antiferromagnetic" (i.e., red), but extends over several lattice sites. (Note, this interaction will be frustrated on the triangular lattice).

Using this toy model, Frechette et al. can obtain complex patterns (heterostructures) similar to those seen in quantum dots grown by ion exchange.

There is some subtle (and confusing) physics associated with deriving the Ising model from the ball and springs model. 

Due to the long-range nature of elastic interactions, the boundary conditions matter. 

The infinite range part of the Ising interaction arises from dealing with the lattice constant for the crystal, depending on the net "magnetisation" of the "spins". But that is a story for another day.

Rodney Baxter (1940-2025): Mathematical Physicist

I recently learnt that Rodney Baxter died earlier this year. He was adept at finding exact solutions to two-dimensional lattice models in statistical mechanics. He had a remarkably low public profile. But, during my lifetime, he was one of the Australian-based researchers who made the most significant and unique contributions to physics, broadly defined. Evidence of this is the list of international awards he received.

On Baxter's scientific achievements, see the obituary from the ANU, and earlier testimonials from Barry McCoy in 2000, and by Vladimir Bahzanov, on the award of the Henri Poincaré Prize to Baxter in 2021.

Exact solutions of "toy models" are important in understanding emergent phenomena. Before Onsager found an exact solution to the two-dimensional Ising model in 1944, there was debate about whether statistical mechanics could describe phase transitions and the associated discontinuities and singularities in thermodynamic quantities. 

Exact solutions provide benchmarks for approximation schemes and computational methods. They have also guided and elucidated key developments such as scaling, universality, the renormalisation group and conformal field theory.

Exact solutions guided Haldane's development of the Luttinger liquid and our understanding of the Kondo problem.

I mention the specific significance of a few of Baxter's solutions. His Exact solution of the eight-vertex model in 1972 gave continuously varying critical exponents that depended on the interaction strength in the model. This surprised many because it seemed to be against the hypothesis of the universality of critical exponents. This was later reconciled in terms of connections to the Berezinskii-Kosterlitz-Thouless transition (BKT) phase transition, which was discovered at the same time. I am not sure who explicitly resolved this.

It might be argued that Baxter independently discovered the BKT transition. For example, consider the abstract of a 1973 paper, Spontaneous staggered polarization of the F-model

"The “order parameter” of the two-dimensional F-model, namely the spontaneous staggered polarization P0, is derived exactly. At the critical temperature P0 has an essential singularity, both P0 and all its derivatives with respect to temperature vanishing."

Following earlier work by Lieb, Baxter explored the connection of two-dimensional classical models with one-dimensional quantum lattice models. For example, the solution of the XYZ quantum spin chain is related to the Eight-vertex model. Central to this is the Yang-Baxter equation. Alexander B. Zamolodchikov connected this to integrable quantum field theories in 1+1 dimensions. [Aside: the Yang is C.N. Yang, of Yang-Mills and Yang-Lee fame, who died last week.]

Baxter's work had completely unanticipated consequences beyond physics. Mathematicians discovered profound connections between his exact solutions and the theory of knots, number theory, and elliptic functions. It also stimulated the development of quantum groups.

I give two personal anecdotes on my own interactions with Baxter. I was an undergraduate at the ANU from 1979 to 1982. This meant I was completely separated from the half of the university known as the Institute for Advanced Studies (IAS), where Baxter worked. Faculty in the IAS there did no teaching, did not have to apply for external grants, and had considerable academic freedom. Most Ph.D. students were in the IAS. By today's standards, the IAS was a cushy deal, particularly if faculty did not get involved in internal politics. As an undergraduate, I really enjoyed my courses on thermodynamics, statistical mechanics, and pure mathematics. My honours supervisor, Hans Buchdahl, suggested that I talk to Baxter about possibly doing a Ph.D. with him. I found him quiet, unassuming, and unambitious. He had only supervised a few students. He wisely cautioned me that Ph.D. students might not be involved in finding exact solutions but might just be comparing exact results to series expansions.

In 1987, when I was a graduate student at Princeton, Baxter visited, hosted by Elliot Lieb, and gave a Mathematical Physics Seminar. This visit was just after he received the Dannie Heinemann Prize for Mathematical Physics from the American Physical Society. These seminars generally had a small audience, mostly people in the Mathematical Physics group. However, for Baxter, many string theorists (Witten, Callen, Gross, Harvey, ...) attended. They had a lot of questions for Baxter. But, from my vague recollection, he struggled to answer them, partly because he wasn't familiar with the language of quantum field theory. 

I was told that he got nice job offers from the USA. He could have earned more money and achieved a higher status. For personal reasons, he turned down the offer of a Royal Society Research Professorship at Cambridge.  But he seemed content puttering away in Australia. He just loved solving models and enjoyed family life down under.

Baxter wrote a short autobiography, An Accidental Academic. He began his career and made his big discoveries in a different era in Australian universities. The ANU had generous and guaranteed funding. Staff had the freedom to pursue curiosity-driven research on difficult problems that might take years to solve. There was little concern with the obsessions of today: money, metrics, management, and marketing. It is wonderful that Baxter was able to do what he did. It is striking that he says he retired early so he would not have to start making grant applications!

Undergraduates need to learn about the Ising model

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

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

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

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

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

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

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

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

One hundred years of Ising

In 1925, Ising published his paper on the solution of the model in one dimension. An English translation is here.https://www.hs-augsburg.de/~harsch/anglica/Chronology/20thC/Ising/isi_fm00.html

Coincidentally, next week I am giving a lecture on the Ising model to an undergraduate class in statistical mechanics. To flesh out the significance and relevance of the model, here are some of the interesting articles I have been looking at:

The Ising model celebrates a century of interdisciplinary contributions, Michael W. Macy, Boleslaw K. Szymanski and Janusz A. Hołyst

This mostly discusses the relevance of the model to understanding basic problems in sociology, including its relation to the classic Schelling model for social segregation.

The Ising model: highlights and perspectives, Christof Külske

This mostly discusses how the model is central to some work in mathematical physics and probability theory.

The Fate of Ernst Ising and the Fate of his Model, Thomas Ising, Reinhard Folk, Ralph Kennac, Bertrand Berche, Yurij Holovatche.

This includes some nice memories of Ising from his son, Thomas.

Aside: I wanted a plot of the specific heat for the one-dimensional model. According to Google AI "In a 1D Ising model with no external magnetic field, the specific heat is zero at all temperatures."

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.

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.

Emergence, structuralism, realism, and quarks

"Structuralism as an influential intellectual movement of the twentieth century has been advocated by Bertrand Russell, Rudolf Carnap, Nicholas Bourbaki, Noam Chomsky, Talcott Parsons, Claude Levi-Strauss, Jean Piaget, Louis Althusser, and Bas van Fraassen, among many others, and developed in various disciplines such as linguistics, mathematics, psychology, anthropology, sociology, and philosophy." 

In different words, structuralism and post-structualism have been and are still a really big deal in the humanities and social sciences. Structuralism is central to the rise and fall of a multitude of academic fashions, careers and reputations.  

"As a method of enquiry, it takes a structure as a whole rather than its elements as the major or even the only legitimate subject for investigations. Here, a structure is defined either as a system of stable relations among a set of elements, or as a self-regulated whole under transformations, depending on the specific subject under consideration. The structuralist maintains that the character or even the reality of a whole is mainly determined by its structuring laws, and cannot be reduced to its parts; rather, the existence and essence of a part in the whole can only be defined through its place in the whole and its relations with other parts."

In a sense, structuralism favours emergence over reductionism. But, note some of the strong exclusivist language highlighted in bold in the quote above. Structuralism seems to be an overreaction to extreme reductionism. 

Condensed matter physics has something concrete to contribute to these debates. Consider the case of Ising models defined on a range of lattices, as I discussed in a previous post. We do not have an exclusive interest in the whole system or in the parts of the system. Rather, we want to know the relationship between macroscopic properties [different ordered states], mesoscopic properties [domains, long-range correlations, networks], and microscopic properties [the individual spins and their local interactions].

That is the main point of this post. But for more context, keep reading.

The quotations above are taken from a book by Tian Yu Cao. 

From Current Algebra to Quantum Chromodynamics: A Case for Structural Realism

Cao is interested in a broad range of philosophical questions related to QCD, such as "If quarks cannot be observed in isolation should they be considered to be real?"

He continues:

In the epistemically interesting cases involving unobservable entities, the structuralist usually argues that it is only the structure and the structural relations of its elements, rather than the elements themselves (properties or entities with properties) that are empirically accessible to us. It is obvious that such an anti-reductionist holistic stance has lent some support to phenomenalism. 

However, as an effort to combat compartmentalization, which urge is particularly strong in mathematics, linguistics, and anthropology, the structuralist also tries to uncover the unity among various appearances, in addition to invariance or stable correlation under transformations, which can help discover the deep reality embodied in deep structures. Furthermore, if we accept the attribution of reality to structures, then the antirealist implications of the underdetermination thesis [which claims that since evidence cannot uniquely determine (or, worse, can even support conflicting) theoretical claims about certain unobservable entities, no theoretical entities should be taken as representation of reality], is somewhat neutralized, because then we can talk about the realism of structures, or the reality of the structural features of unobservable entities exhibited in evidence, although we cannot directly talk about the reality of the entities themselves that are engaged in the structural relations. In fact, this realist implication of structuralism was one of the starting points of current interests in structural realism.

The whole is qualitatively different from the parts: beer, birds, and brains

Pint of Science is an annual event in cities all around Australia. Local scientists give short talks about their research to general audiences. I am speaking tonight, along with my colleague Ben Powell. 

I found the tips to speakers very helpful. This led me to try and make the talk more of a personal story, reduce the amount of text on slides, and aim for engagement rather than focusing on scientific details or on technical details of your own research.

Here is the current version of my slides.

The introduction is based on this video and poem about emergence in economics.

This provides an example of how "free" economic markets can work well sometimes. But I will also point out that they can also fail spectacularly, another emergent phenomenon! 

The role of effective theories and toy models in understanding emergent properties

Two of the approaches to the theoretical description of systems with emergent properties that have been fruitful are effective theories and toy models. These leverage our limited knowledge of many details about a system with many interacting components.

Effective theories

An effective theory is valid at a particular range of scales. This exploits the fact that in complex systems there is often a hierarchy of scales (length, energy, time, or number). In physics, examples of effective theories include classical mechanics, general relativity, classical electromagnetism, and thermodynamics. The equations of an effective theory can be written down almost solely from consideration of symmetry and conservation laws. Examples include the Navier-Stokes equations for fluid dynamics and non-linear sigma models in elementary particle physics. Some effective theories can be derived by the “coarse-graining” of theories that are valid at a finer scale. For example, the equations of classical mechanics result from taking the limit of Planck’s constant going to zero in the equations of quantum mechanics. The Ginzburg-Landau theory for superconductivity can be derived from the BCS theory. The parameters in effective theories may be determined from more microscopic theories or from fitting experimental data to the predictions of the theory. For example, transport coefficients such as conductivities can be calculated from a microscopic theory using a Kubo formula.

Effective theories are useful and powerful because of the minimal assumptions and parameters used in their construction. For the theory to be useful it is not necessary to be able to derive the effective theory from a smaller scale theory, or even to have such a smaller scale theory. For example, even though there is no accepted quantum theory of gravity, general relativity can be used to describe phenomena in astrophysics and cosmology and is accepted to be valid on the macroscopic scale. Some physicists and philosophers may consider smaller-scale theories as more fundamental, but that is contested and so I will not use that language. There also are debates about how effective field theories fit into the philosophy of science.

Toy models

In his 2016 Nobel Lecture, Duncan Haldane said, “Looking back, … I am struck by how important the use of stripped down “toy models” has been in discovering new physics.” 

Here I am concerned with a class of theoretical models that includes the Ising, Hubbard, Agent-Based Models, NK, Schelling, and Sherrington-Kirkpatrick models. I refer to them as “toy” models because they aim to be as simple as possible, while still capturing the essential details of a particular emergent phenomenon. At the scale of interest, the model is an approximation, neglecting certain degrees of freedom and interactions. In contrast, at the relevant scale, effective theories are often considered to be exact because they are based on general principles.

Historical experience has shown that there is a strong justification for the proposal and study of toy models. They are concerned with a qualitative, rather than a quantitative, description of experimental data. A toy model is usually introduced to answer basic questions about what is possible. What are the essential ingredients that are sufficient for an emergent phenomena to occur? What details do matter? For example, the Ising model was introduced in 1920 to see if it was possible for statistical mechanics to describe the sharp phase transition associated with ferromagnetism.  

In his book The Model Thinker and online course Model Thinking, Scott Page has enumerated the value of simple models in the social sciences. An earlier argument for their value in biology was put by JBS Haldane in his seminal article about “bean bag” genetics. Simplicity makes toy models more tractable for mathematical analysis and/or computer simulation. The assumptions made in defining the model can be clearly stated. If the model is tractable then the pure logic associated with mathematical analysis leads to reliable conclusions. This contrasts with the qualitative arguments often used in the biological and social sciences to propose explanations. Such arguments can miss the counter-intuitive conclusions associated with emergent phenomena and the rigorous analysis of toy models. Such models can show what is possible, what are simple ingredients for a system sufficient to exhibit an emergent property, and how a quantitative change can lead to a qualitative change. In different words, what details do matter? 

Toy models can guide what experimental data to gather and how to analyse it. Insight can be gained by considering multiple models as that approach can be used to rule out alternative hypotheses. Finally, there is value in the adage, “all models are wrong, but some are useful.”

Due to universality, sometimes toy models work better than expected, and can even give a quantitative description of experimental data. An example is the three-dimensional Ising model, which was eventually found to be consistent with data on the liquid-gas transition near the critical point. Although, not a magnetic system, the analogy was bolstered by the mapping of the Ising model onto the lattice gas model. This success led to a shift in the attitude of physicists towards the Ising model. According to Martin Niss, from 1920-1950, it was viewed as irrelevant to magnetism because it did not describe magnetic interactions quantum mechanically. This was replaced with the view that it was a model that could give insights into collective phenomena. From 1950-1965, the view diminished that the Ising model was irrelevant to describing critical phenomena because it oversimplified the microscopic interactions.

Physicists are particularly good and experienced at the proposal and analysis of toy models. I think this expertise is a niche that they could exploit more in contributing to other fields, from biology to the social sciences. They just need humility to listen to non-physicists about what the important questions and essential details are.

Emergence and the Ising model

The Ising model is emblematic of “toy models” that have been proposed and studied to understand and describe emergent phenomena. Although originally proposed to describe ferromagnetic phase transitions, variants of it have found application in other areas of physics, and in biology, economics, sociology, neuroscience, complexity theory, …  

Quanta magazine had a nice article marking the model's centenary.

In the general model there is a set of lattice points {i} with a “spin” {sigma_i = +/-1} and a Hamiltonian

where h is the strength of an external magnetic field and J_ij is the strength of the interaction between the spins on sites i and j. The simplest models are where the lattice is regular, and the interaction is uniform and only non-zero for nearest-neighbour sites.

The Ising model illustrates many key features of emergent phenomena. Given the relative simplicity of the model, exhaustive studies since its proposal in 1920, have given definitive answers to questions often debated about more complex systems. Below I enumerate some of these insights: novelty, quantitative change leads to qualitative change, spontaneous order, singularities, short-range interactions can produce long-range order, universality, three horizons/scales of interest, self-similarity, inseparable horizons, and simple models can describe complex behaviour.

Most of these properties can be illustrated with the case of the Ising model on a square lattice with only nearest-neighbour interactions (J_ij = J). Above the critical temperature (Tc = 2.25J), and in the absence of an external magnetic field the system has no net magnetisation. Below Tc, at net magnetisation occurs. For J > 0 (J < 0) this state is ferromagnetic (antiferromagnetic).

Novelty

The state of the system below Tc is qualitatively different than that at very high temperatures or the state of a set of non-interacting spins. Thus, the non-zero magnetisation is an emergent property, as defined in this post. This state is also associated with spontaneous symmetry breaking and more than one possible equilibrium state, i.e., the magnetisation can be positive or negative.

Quantitative change leads to qualitative change

The qualitative change associated with formation of the magnetic state can occur with a small quantitative change in the value of the ratio T/J, i.e., either by decreasing T or increasing J. Formation of the magnetic state is also associated with the quantitative change of increasing the number of spins from a large finite number to infinity. 

Singularities

For a finite number of spins all the thermodynamic properties of the system are an analytic function of the temperature and magnitude of an external field. However, in the thermodynamic limit, these properties become singular at T=Tc and h=0. This is the critical point in the phase diagram of h versus T. Some of the quantities, such as the specific heat capacity and the magnetic susceptibility, become infinite at the critical point. These singularities are characterised by critical exponents, most of which have non-integer values. Consequently, the free energy of the system is not an analytic function of T and h.

Spontaneous order

The magnetic state occurs spontaneously. The system self-organises. There is no external field causing the magnetic state to form. There is long-range order, i.e., the value of spins that are infinitely apart from one another are correlated. 

Short-range interactions can produce long-range order.

Although there is no direct long-range interaction between spins, long-range order can occur. Prior to Onsager’s exact solution of the two-dimensional model, many scientists were not convinced that this was possible.

Universality

The values of the critical exponents are independent of many details of the model, such as the value of J, the lattice constant and spatial anisotropy, and the presence of small interactions beyond nearest neighbour. Many details do not matter. This is why the model can give a quantitative description of experimental data near the critical temperature, even though the model Hamiltonian is a crude descriptions of the interactions in a real material. It can describe not only magnetic transitions but also transitions in liquid-gas, binary alloys, and binary liquid mixtures.

Three horizons/scales of interest

There are three important length scales associated with the model. Two are simple: the lattice constant, and the size of the whole lattice. These are the microscopic and macroscopic scale. The third scale is emergent and temperature dependent: the correlation length, i.e., the distance over which spins are correlated with one another. This can also be visualised as the size of magnetisation domains seen in Monte Carlo simulations. 

The left, centre, and right panels above show a snapshot of a likely configuration of the system at a temperature less than, equal to, and greater than the critical temperature, Tc, respectively.

Understanding the connection between the microscopic and macroscopic properties of the system requires studying the system at the intermediate scale of the correlation length. This scale also defines emergent entities [magnetic domains] that interact with one another weakly and via an effective interaction.

Self-similarity

At the critical temperature, the correlation length is infinite. Consequently, rescaling the size of the system, as in a renormalisation group transformation, the state of the systems does not change. The system is said to be scale-free or self-similar like a fractal pattern. This is an example of self-organised criticality.

Inseparable horizons

I now consider how things change when the topology or dimensionality of the lattice changes or when interactions beyond nearest neighbours are added. This can change the relationships between the parts and the whole. Some details of the parts matter. Changing from a two-dimensional rectangular lattice to a linear chain the ordered state disappears. Changing to a triangular lattice with antiferromagnetic nearest-neighbour interactions removes the ordering at finite temperature and there are an infinite number of ground states at zero temperature. Thus, some microscopic details do matter.

The main point of this example is that to understand a large complex system we have to keep both the parts and the whole in mind. It is not either/or but both/and. Furthermore, there may be an intermediate scale, at which new entities emerge.

Aside: I suspect heated debates about structuralism versus functionalism in social sciences, and the humanities are trying to defend intellectual positions (and fashions) that overlook the inseparable interplay of the microscopic and macroscopic that the Ising model captures.

Simple models can describe complex behaviour

Now consider an Ising model with competing interactions, i.e. the neighbouring spins of a particular spin compete with one another and with an external magnetic field to determine the sign of the spin. This can be illustrated with the an Ising model on a hexagonal close packed (hcp) lattice with nearest neighbour antiferromagnetic interactions and an external magnetic field. The lattice is frustrated and can be viewed as layers of hexagonal (triangular) lattices where each layer is displaced relative to one another.

This model has been studied by materials scientists as it can describe the many possible phases of binary alloys, AxB1-x, where A and B are different chemical elements (for example, silver and gold) and the Ising spins on site i has value +1 or -1, corresponding to the presence of atom A or B on that site. The magnetic field corresponds to the difference in the chemical potentials of A and B, and is related to their relative concentration.

The authors studied the Ising model on the hexagonal close-packed (hcp) lattice in a magnetic field. The authors are all from materials science departments and are motivated by the fact that the problem of binary alloys AxB1_x can be mapped onto an Ising model. A study of this model found rich phase diagrams including 32 stable ground states with stoichiometries, including A, AB, A2B, A3B, A5B, and A4B3. Even for a single stoichiometry, there can be multiple possible distinct orderings (and crystal structures). Of these structures, six are stabilized by purely nearest-neighbour interactions, eight by addition of next-nearest neighbour interactions. The remaining 18 structures require multiplet interactions for their stability. 

A second example is the Anisotropic Next-Nearest Neighbour Ising (ANNNI) model, which supports a plethora of ordered states, including a phase diagram with a fractal structure, known as the Devil’s staircase.

These two Ising models illustrate how relatively simple models, containing competing interactions (described by just a few parameters) can describe rich behaviour, particularly a diversity of ground states.

An emergentist perspective on public policy issues that divide

How is the whole related to the parts?

Which type of economy will produce the best outcomes: laissez-faire or regulated?

Can a government end an economic recession by "stimulus" spending?  

What is the relative importance of individual agency and social structures in causing social problems such as poverty and racism?

These questions are all related to the first one. Let's look at it from an emergentist perspective, with reference to physics. 

Consider the Ising model in two or more dimensions. The presence of nearest neighbour interactions between spins leads to emergent properties: long-range ordering of the spins, spontaneous symmetry breaking below the critical temperature, and singularities in the temperature dependence of thermodynamic properties such as the specific heat and magnetic susceptibility. Individual uncoupled spins have neither property. Even a finite number of spins do not. (Although, a large number of spins do exhibit suggestive properties such as an enhancement of the magnetic susceptibility near the critical temperature). Thus, the whole system has properties that are qualitatively different from the parts. 

On the other hand, the properties of the parts, such as how strongly the spins couple to an external field and interact with their neighbours, influence the properties of the whole. Some details of the parts matter. Other details don't matter. Adding some interaction with spins beyond nearest neighbours does not change any of the qualitative properties, provided those longer-range interactions are not too large. On the other hand, changing from a two-dimensional rectangular lattice to a linear chain removes the ordered state. Changing to a triangular lattice with an antiferromagnetic nearest-neighbour interaction removes the ordering and there are multiple ground states. Thus, some microscopic details do matter.

For illustrative purposes, below I show a sketch of the temperature dependence of the magnetic susceptibility of the Ising model for three cases: non-interacting spins (J=0), two dimensions (d=2), and one dimension (d=1). This shows how interactions can significantly enhance/diminish the susceptibility depending on the parameter regime.

The main point of this example is to show that to understand a large complex system we have to keep both the parts and the whole in mind. In other words, we need both microscopic and macroscopic pictures. There are two horizons, the parts and the whole, the near and the far. There is a dialectic tension between these two horizons. It is not either/or but both/and.

I now illustrate how this type of tension matters in economics and sociology, and the implications for public policy. If you are (understandably) concerned about whether Ising models have anything to do with sociology and economics, see my earlier posts about these issues. The first post introduced discrete-choice models that are essentially Ising models. A second post discussed how these show how equilibrium may never be reached leading to the insight that local initiatives can "nucleate" desired outcomes. A third post, considered how heterogeneity can lead to qualitative changes including hysteresis so that the effectiveness of "nudges" can vary significantly.

A fundamental (and much debated) question in sociology is the relationship between individual agency and social structures. Which determines which? Do individuals make choices that then lead to particular social structures? Or do social structures constrain what choices individuals make. In sociology, this is referred to as the debate between voluntarism and determinism. A middle way, that does not preference agency or structure, is structuration, proposed by Anthony Giddens.

Social theorists who give primacy to social structures will naturally advocate solving social problems with large government schemes and policies that seek to change the structures. On the other side, those who give primacy to individual agency are sceptical of such approaches, and consider progress can only occur through individuals, and small units such as families and communities make better choices. The structure/agency divide naturally maps onto political divisions of left versus right, liberal versus conservative, and the extremes of communist and libertarian. An emergentist perspective is balanced, affirming the importance of both structure and agency.

Key concepts in economics are equilibrium, division of labour, price, and demand. These are the outcomes of many interacting agents (individuals, companies, institutions, and government). Economies tend to self-organise. This is the "invisible hand" of Adam Smith. Thus, emergence is one of the most important concepts in economics. 

A big question is how the equilibrium state and the values of the associated state variables (e.g., prices, demand, division of labour, and wealth distribution) emerge from the interactions of the agents. In other words, what is the relationship between microeconomics and macroeconomics?

What are the implications for public policy? What will lead to the best outcomes (usually assumed to be economic growth and prosperity for "all")? Central planning (or at least some government regulation) is pitted against laissez-faire. For reasons, similar to the Ising and sociology cases, an emergentist perspective is that the whole and the parts are inseparable. This is why there is no consensus on the answers to specific questions such as, can government stimulus spending move an economy out of a recession? Keynes claimed it could but the debate rages on.

An emergentist perspective tempers expectations about the impact of agency, both individuals and government. It is hard to predict how a complex system with emergent properties will respond to perturbations such as changes in government policy. This is the "law" of unintended consequences.

“The curious task of economics is to demonstrate to men how little they really know about what they imagine they can design.”

Friedrich A. Hayek, The Fatal Conceit: The Errors of Socialism

I think this cuts both ways. This is also reason to be skeptical about those (such as Hayek's disciples) who think they can "design" a better society by just letting the market run free.