Phase diagrams elucidate emergence

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

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

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

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

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

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

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

Lamenting the disintegration of elite USA universities

Elite universities in the USA have nurtured and enhanced my whole academic life. In 1983, I moved to the USA as an international student, commenced a Ph.D. at Princeton, and then worked at Northwestern and Ohio State. After I returned to Australia in 1994, I visited the USA every year for several weeks for conferences, collaborations, and university visits. Much of my research was shaped by ideas I got from those trips. This blog started through the influence of I2CAM, a wonderful institution funded by the NSF. My movement into chemical physics was facilitated by attending workshops at the Telluride Science Center. I deeply appreciate my colleagues (and their institutions) for their stimulation, support, interest, encouragement, and hospitality. 

My trips to the USA only ended with COVID-19, retirement, family health issues, and my new general aversion to international travel. Currently I would be too scared to travel to the USA, based on what I read in the Travel Section of The Sydney Morning Herald.

Most importantly, what I have learned and done has been built largely on intellectual foundations laid by people in these elite universities.  Other parts of the world have played a role too, but my focus here is the USA due to current political events leading to the impending disintegration of these universities.

I readily acknowledge that these universities have flaws and need reform. On this blog, I occasionally discussed issues, such as the obsession with money, metrics, management, and marketing. Teaching undergraduates and robust scholarship has sometimes become subsidiary. I have critiqued some of the flaky science published in luxury journals by groups from these universities.

Nevertheless, if something is broken you do not fix it by smashing it. Consider a beautiful ancient vase with a large crack. You do not restore the vase by smashing it and hiring your teenage cousin to make a new one.

Reading about what is happening raises multiple questions. What is really happening? Why is it happening? How significant is it? What might it lead to? How should individuals and institutions respond? 

Today when I was on the UQ campus it was serene and the challenges my colleagues are facing, as formidable and important as they are, seem trifling compared to what I imagine is happening on Ivy campuses right now. In passing, I mention that Australia is not completely immune to what is happening in the USA. Universities here that receive some research grant funding from the USA government have had it paused or cancelled.

I can't imagine what it would be like to be an international student at Princeton right now.

On the one hand, I do not feel qualified to comment on what is happening as I am so distant. On the other hand, I do want to try and express some solidarity with and appreciation of institutions and colleagues that have blessed me and the world. I make a few general observations. This is my advice, for what it is worth, to my younger self.

Protect your mental health. You and your colleagues and your institutional are encountering an existential crisis, perhaps like none encountered before. Don't live in denial. But also don't let this consume you and destroy you as a person or a community. Limit your intake of news and how much you think about it and discuss it. Practise the basics: exercise; eat, drink, and sleep well; get help sooner than later; limit screen time; rest.

Expect the unexpected. Expect more surprises, pain, uncertainty, instability, intra-institutional conflict, and disappointments. 

Get the big picture. This is about a lot more than federal funding for universities. There are broader issues about what a university is actually for. What do you want to preserve and protect? What are you willing to compromise on? Beyond the university, many significant issues are at stake concerning politics, democracy, economics, pluralism, culture, and the law. This is an opportunity, albeit a scary one, to think about and learn about these issues.

Make the effort to have conversations across the divides. Try to  have civil and respectful discussions with people with different perspectives on how individuals and institutions should respond to the current situation. Talk to colleagues in the humanities and social sciences. Talk to those with different political perspectives, both inside and outside the university.

Read widely. History is instructive but not determinative. I recommend two very short books that I think are relevant and helpful.

On Tyranny: Twenty Lessons from the Twentieth Century by Timothy Snyder.

The Power of the Powerless, by Vaclav Havel, first published in 1978 in the context of living in communist totalitarian Czechoslovakia. I have a Penguin Vintage edition which includes a beautiful introduction by Timothy Snyder, written in 2018, for a 40th Anniversary edition. 

I thank Charles Ringma for bringing both books to my attention.

What do you think? I would love to hear from people in US universities who are living through this.

An authoritarian government takes over universities: one case history

Adventures of a Bystander, by Peter Drucker, contains the following account. Drucker was a faculty member at Frankfurt University in 1933.

“[S]everal weeks after the Nazis had come to power, was the first Nazi-led faculty meeting at the University. Frankfurt was the first university the Nazis tackled, precisely because it was the most self-confidently liberal of major German universities, with a faculty that prided itself on its allegiance to scholarship, freedom of conscience, and democracy. The Nazis knew that control of Frankfurt University would mean control of German academia altogether. So did everyone at the University. 

Above all, Frankfurt had a science faculty distinguished both by its scholarship and by its liberal convictions; and outstanding among the Frankfurt scientists was a biochemist of Nobel Prize caliber and impeccable liberal credentials. When the appointment of a Nazi commissar for Frankfurt was announced around February 25 of that year and when not only every teacher but also every graduate assistant at the University was summoned to a faculty meeting to hear his new master, everybody knew that a trial of strength was at hand. … 

The new Nazi commissar wasted no time on the amenities…. [He] pointed his finger at one department chairman after another and said: ‘You either do what I tell you or we’ll put you into a concentration camp.’ 

There was dead silence when he finished; everybody waited for the distinguished biochemist. The great liberal got up, cleared his throat, and said: ‘Very interesting, Mr. Commissar, and in some respects very illuminating. But one point I didn’t get too clearly. Will there be more money for research in physiology?’ The meeting broke up shortly thereafter with the commissar assuring the scholars that indeed there would be plenty of money for ‘racially pure science’.”

I became aware of this chilling story through Peter Woit's blog who got it from a blog post by Adam Przeworski

Superconductivity: a poster child for emergence

Superconductivity beautifully illustrates the characteristics of emergent properties.

Novelty. 

Distinct properties of the superconducting state include zero resistivity, the Meissner effect, and the Josephson effect. The normal metallic state does not exhibit these properties.

At low temperatures, solid tin exhibits the property of superconductivity. However, a single atom of tin is not a superconductor. A small number of tin atoms has an energy gap due to pairing interactions, but not bulk superconductivity.

There is more than one superconducting state of matter. The order parameter may have the same symmetry as a non-trivial representation of the crystal symmetry and it can have spin singlet or triplet symmetry. Type II superconductors in a magnetic field have an Abrikosov vortex lattice, another distinct state of matter.

Unpredictability. 

Even though the underlying laws describing the interactions between electrons in a crystal have been known for one hundred years, the discovery of superconductivity in many specific materials was not predicted. Even after the BCS theory was worked out in 1957 the discovery of superconductivity in intermetallic compounds, cuprates, organic charge transfer salts, fullerenes, and heavy fermion compounds was not predicted.71

Order and structure. 

In the superconducting state, the electrons become ordered in a particular way. The motion of the electrons relative to one another is not independent but correlated. Long-range order is reflected in the generalised rigidity, which is responsible for the zero resistivity. Properties of individual atoms (e.g., NMR chemical shifts) are different in vacuum, metallic state, and superconducting state.

Universality. 

Properties of superconductivity such as zero electrical resistance, the expulsion of magnetic fields, quantisation of magnetic flux, and the Josephson effects are universal. The existence and description of these properties are independent of the chemical and structural details of the material in which the superconductivity is observed. This is why the Ginzburg-Landau theory works so well. In BCS theory, the temperature dependences of thermodynamic and transport properties are given by universal functions of T/Tc where Tc is the transition temperature. Experimental data is consistent with this for a wide range of superconducting materials, particularly elemental metals for which the electron-phonon coupling is weak.

Modularity at the mesoscale. 

Emergent entities include Cooper pairs and vortices. There are two associated emergent length scales, typically much larger than the microscopic scales defined by the interatomic spacing or the Fermi wavelength of electrons. The coherence length is associated with the energy cost of spatial variations in the order parameter. It defines the extent of the proximity effect where the surface of a non-superconducting metal can become superconducting when it is in electrical contact with a superconductor. The coherence length turns out to be of the order of the size of Cooper pairs in BCS theory.  The second length scale is the magnetic penetration depth (also known as the London length) which determines the extent that an external magnetic field can penetrate the surface of a superconductor. It is determined by the superfluid density. The relative size of the coherence length and the penetration depth determines whether  the formation of an Abrikosov vortex lattice is stable in a large enough magnetic field.

Quasiparticles. 

The elementary excitations are Bogoliubov quasiparticles that are qualitatively different to particle and hole excitations in a normal metal. They are a coherent superposition of a particle and hole excitation (relative to the Fermi sea), have zero charge and only exist above the energy gap. The mixed particle-hole character of the quasiparticles is reflected in the phenomenom of Andreev reflection.

Singularities. 

Superconductivity is a non-perturbative phenomenon. In BCS theory the transition temperature, Tc, and the excitation energy gap are a non-analytic function of the electron-phonon coupling constant lambda, Tc \sim exp(-1/lambda).

A singular structure is also evident in the properties of the current-current correlation function. Interchange of the limits of zero wavevector and zero frequency do not commute, this being intimately connected with the non-zero superfluid density.

Effective theories.

These are illustrated in the Figure below. The many-particle Schrodinger equation describes electrons and atomic nuclei interacting with one another. Many-body theory can be used to justify considering the electrons as a jellium liquid of non-interacting fermions interacting with phonons. Bardeen, Pines, and Frohlich showed that for that system there is an effective interaction between fermions that is attractive. The BCS theory includes a truncated version of this attractive interaction. Gorkov showed that Ginzburg-Landau theory could be derived from BCS theory. The London equations can be derived from Ginzburg-Landau theory. The Josephson equations only include the phase of order parameter to describe a pair of coupled superconductors.

The historical of the development of theories mostly went downwards. London preceded Ginzburg-Landau which preceded BCS theory. Today for specific materials where superconductivity is known to be due to electron-phonon coupling and the electron gas is weakly correlated one can now work upwards using computational methods such as Density Functional Theory (DFT) for Superconductors or the Eliashberg theory with input parameters calculated from DFT-based methods. However, in reality this has debatable success. The superconducting transition temperatures calculated typically vary with the approximations used in the DFT such as the choice of functional and basis set, and often differ from experimental results by the order of 50 percent. This illustrates how hard prediction is for emergent phenomena.

Potential and pitfalls of mean-field theory. 

Mean-field approximations and theories can provide a useful guide as what emergent properties are possible and as a starting point to map out properties such as phase diagrams. For some systems and properties, they work incredibly well and for others they fail spectacularly and are misleading. 

Ginzburg-Landau theory and BCS theory are both mean-field theories. For three-dimensional superconductors they work extremely well. However, in two dimensions as long-range order and breaking of a continuous symmetry cannot occur and the physics associated with the Berezinskii-Kosterlitz-Thouless transition occurs. Nevertheless, the Ginzburg-Landau theory provides the background to understand the justification for the XY model and the presence of vortices to proceed. Similarly, the BCS theory fails for strongly correlated electron systems, but a version of the BCS theory does give a surprisingly good description of the superconducting state.

Cross-fertilisation of fields. 

Concepts and methods developed for the theory of superconductivity bore fruit in other sub-fields of physics including nuclear physics, elementary particles, and astrophysics. Considering the matter fields (associated with the electrons) coupled to electromagnetic fields (a U(1) gauge theory) the matter fields can be integrated out to give a theory in which the photon has mass. This is a perspective on the Meissner effect in which the magnitude of an external magnetic field decays exponentially as it penetrates a superconductor. This idea of a massless gauge field acquiring a mass due to spontaneous symmetry breaking was central to steps towards the Standard Model made by Nambu and by Weinberg. 

Topological defects determine the strength and growth rate of crystals

 The quantum theory of solids developed in the 1920s provided a theoretical estimate of the ideal strength of crystals. The problem was that this estimate was a thousand times greater than the measured strength of metals. This paradox was resolved in 1934, when Egon Orowan, Michael Polanyi and G. I. Taylor, independently proposed that plastic deformation could be explained in terms of the theory of dislocations. Aside: this is an example of how macroscopic properties can be determined by structures at the mesoscale rather than microscopic properties.

By 1940 the accepted theory of crystal growth was that it occurred by nucleation of successive close-packed layers of the crystal and this provided algebraic expressions for growth rates that were consistent with experiment. However, around 1950 Keith Burton estimated the parameters in the theory and pointed out that it predicted a growth rate that was smaller than observed growth rates by a factor 10^1000, i.e., 1000 orders of magnitude!

This quantitative discrepancy was resolved by Burton, Nicolas Cabrera and Charles Frank in 1951 who showed the central role played by screw dislocations. A crystal does not grow by the independent nucleation of separate layers. Rather it grows from just one layer that heloicoidally overlapping itself. A signature of this growth mode is the presence of spiral steps on crystal surfaces and they were subsequently observed.

This history is beautifully recounted in the introduction to a review article on Snow Crystals published by Charles Frank in 1982. It was reprinted in 2009 with an introduction by Andrew Fisher.

Following the introduction Frank discusses how snow is an important example of crystal growth that is not attributable to the presence of screw dislocations.

In 2015, D.P. Woodruff wrote a commentary on the classic 1951 paper by Burton, Cabrera, and Frank.

Weather, chaos, and emergence

Weather involves many scales of distance, time, and energy. Describing weather means making decisions about what range of scales to focus on. 

BTW. Did you know that there is a cyclone heading for Brisbane right now!

Now, back to physics :)

Key physics involves thermal convection which reflects an interplay of gravity, thermal expansion, viscosity and thermal conduction. This can lead to Rayleigh-Bénard convection and convection cells.

The multiple scales are associated with multiple entities:

-the molecules that make up the fluid

-small volumes of fluid that are in local thermodynamic equilibrium with a well-defined temperature, density, and velocity

-individual convection cells (rolls)

-collections of cells.

At each scale, the corresponding entities can be viewed as emerging from the interacting entities at the next smallest scale. Hence, they are collective degrees of freedom.

In principle, a complete description, including the transition to turbulence, is given by the equations of fluid dynamics, including the Navier-Stokes equation. Despite the apparent simplicity of these equations, making definitive predictions from them remains elusive.

A famous toy model was studied by the meteorologist Edward Lorenz in 1963, in a seminal paper, "Deterministic Nonperiodic Flow." Under the restrictive conditions of considering the dynamics of a single convection roll the model can be derived from the full hydrodynamic equations.

Lorenz's study stimulated the field of chaos theory, and is beautifully described in James Gleick's book Chaos: The Making of a New Science.

Here, I discuss Lorenz's model in the context of emergence.

 

The model consists of (just) three coupled ODEs (ordinary differential equations):

The variables x(t), y(t), and z(t) describe, respectively, the amplitude of the velocity mode, the temperature mode, and the mode measuring the heat flux Nu, the Nusselt number. x and y characterize the roll pattern.

The model has three dimensionless parameters: r, sigma, and b.

r is the ratio of the temperature difference between the hot and cold plate, to its critical value for the onset of convection. It can also be viewed as the ratio of the Rayleigh number to its critical value.

sigma is the Prandtl number, the ratio of the kinematic viscosity to the thermal diffusivity. Sigma is about 0.7 in air and 7 in water. Lorenz used sigma = 10.

b is of order unity and conventionally taken to have the value 8/3. It arises from the nonlinear coupling of the fluid velocity and temperature gradient in the Boussinesq approximation.

The model is a toy model because for values of r larger than r_c (defined below) "the three mode approximation for the PDEs describing thermal convection... ceased to be physically

realistic, but mathematically the model now starts to show its most fascinating properties,.."

Novelty

The model has several distinct types of long-time dynamics: stable fixed points (no convection), limit cycles (convective rolls), and most strikingly a chaotic strange attractor (represented below). The chaos is reflected in the sensitive dependence on initial conditions.

Briefly, a strange attractor is a curve of infinite length that never crosses itself and is contained in a finite volume. This means it has a fractal structure and a non-trivial Hausdorff dimension [calculated in this paper to be 2.0627160].

Discontinuities

Quantitative changes lead to qualitative changes. For r < 1, no convection occurs. For r > 1, convective rolls develop, but these become unstable for 

and a strange attractor develops.

Phase diagram

Lorenz only considered one set of parameter values [r =28, sigma=10, and b=8/3]. This was rather fortunate, because then strange attractor was waiting to be discovered. 

The phase diagram maps out the qualitatively different behaviours that occur as a function of sigma (vertical axis) and r (horizontal axis). 

Different phases are the fixed points P± associated with convective rolls (black), orbits of period 2 (red), period 4 (green), period 8 (blue), and chaotic attractors (white).

EXTENDED PHASE DIAGRAM OF THE LORENZ MODEL

H. R. DULLIN, S. SCHMIDT, P. H. RICHTER, and S. K. GROSSMANN

Universality

The details of the molecular composition of the fluid and the intermolecular interactions are irrelevant beyond how they determine the three parameters in the model. Hence, qualitatively similar behaviour can occur in systems with a wide range of chemical compositions and physical properties.

Unpredictability

Although the system of three ODEs is simple, discovery of the strange attractor and the chaotic dynamics was unanticipated. Furthermore, the dynamics in the chaotic regime are unpredictable, given the sensitivity to initial conditions.

Top-down causation

The properties and behaviour of the system are not just determined by the properties of the molecules and their interactions. The external boundary conditions, the applied temperature gradient and the spatial separation L of the hot and cold plates, are just as important in determining the dynamics of the system, including motion as much smaller length scales.

Emergence and continuous phase transitions in flatland

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

Novelty

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

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

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

Discontinuity

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

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

Toy model

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

Modularity at the mesoscale

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

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

Universality

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

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

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

Singularity

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

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

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

Two-dimensional crystals

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

Predictability

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