Signatures of soft matter

What is soft matter? 

Soft Matter: A Very Short Introduction by Tom McLeish has just been published.

McLeish identifies six characteristics of soft matter.

1. Thermal motion 

They exhibit large local spatial rearrangements of their microscopic constituents under thermal agitation. In contrast,  "hard" materials experience only small distortions due to thermal motion.

2. Structure on intermediate length scales

There are basic units ("fundamental" structures), typically involving a very large number (hundreds to thousands) of atoms, that are key to understand soft matter behaviour. These basic units are neither macroscopic nor microscopic (in the atomic sense), but rather mesoscopic (meso from the Greek word for middle). The relevant scales range from several nanometres up to a micrometer. An example of these length scales is those associated with (topological) defects in liquid crystals, such as those shown below.

Image is from here.

3. Slow dynamics

The mesoscopic length scales and complex structures lead to phenomena occurring on time scales of the order of seconds or minutes.

4. Universality

The same physical properties can arise from materials with quite different underlying chemistries.
This characteristic is of significant practical relevance. Solving a problem for one specific material can also solve it for whole families of materials. This universality is also of deep conceptual significance as understanding a general phenomenon is usually more powerful than just a specific example. 

5. Common experimental techniques

The dominant tools are microscopy, scattering (light, x-rays, neutrons), and rheometers which measure mechanical properties such as viscosity (rheology).

6. Multi-disciplinarity

Soft matter is studied by physicists, chemists, engineers, and biologists. 

The chapter titles in the book are 

Milkiness, muddiness, and inkiness [Colloids]

Sliminess and stickiness [Polymers]

Gelification and soapiness [Foams and Self-assembly]

Pearliness [Liquid crystals]

Liveliness [Active matter]

I highly recommend the book. Hopefully, later I will write a review.

Magnetic field induced (thermodynamic) phase transitions in graphite

One of the most common and reliable indicators of a phase transition into a new state of matter is anomalies (e.g. discontinuities or singularities) in thermodynamic properties such as specific heat capacity. This is how the superfluid phases of helium 4 and helium 3 were both discovered. Further transport experiments were required to show that the new states of matter were actually superfluids. This point was highlighted at the end of my last post.

In 2014, I wrote about the puzzling magnetoresistance of graphite and some experiments that were interpreted as evidence of a metal-insulator transition when the electrons are in the lowest Landau level of the graphene layers. This interpretation was partly motivated by theoretical predictions of charge density wave (CDW) transitions in this regime. I expressed some caution and skepticism about this interpretation, highlighting problems in other systems where magnetoresistance anomalies were given such interpretations.  I suggested that thermodynamic measurements should be performed.

In 2015, I highlighted similar issues, suggesting there is no metal-insulator transition in extremely large magnetoresistance materials, contrary to claims in luxury journals. Within two months my claim was shown to be correct.

I was delighted to recently learn from Benoit Fauque that he and his collaborators have now performed measurements of the specific heat of graphite in high magnetic fields.

Wide critical fluctuations of the field-induced phase transition in graphite 

Christophe Marcenat, Thierry Klein, David LeBoeuf, Alexandre Jaoui, Gabriel Seyfarth, Jozef Kačmarčík, Yoshimitsu Kohama, Hervé Cercellier, Hervé Aubin, Kamran Behnia, Benoît Fauqué

The figure below shows the ratio of the specific heat to temperature versus temperature for different values of the magnetic field. In an elemental metal, this would be the temperature and field-independent and equal to the specific heat coefficient gamma. The temperature dependence and peak at a particular temperature are reminiscent of the behaviour for a BCS superconductor or quasi-one-dimensional CDW transition.

I want to highlight a couple of nice things about this data and the analysis in the paper. First, the value of the specific heat coefficient at small fields is several orders of magnitude smaller than in an elemental metal (due to the low density and effective mass of charge carriers) and has a value consistent with band structure calculations. I presume that measuring such small values is a significant experimental achievement. 

Secondly, the linear increase in gamma with the magnetic field and the rate of increase are also consistent with band structure. These agreements increase confidence in the reliability of the measurements and their identification with electronic contributions to the specific heat. For context, the field of strongly correlated electron materials is littered with dubious identifications. An example concerns claims of spinons in an organic charge-transfer salt.

Most importantly the data above suggests that there is a thermodynamic phase transition and that the transition temperature increases with the magnetic field. The corresponding phase diagram can be compared to that suggested by the magnetoresistance measurements from 2014. This is done in the figure below.

The fact that the transition temperature deduced from the specific heat is tracking the anomalies in the earlier magnetoresistance measurements suggests the identification of the latter with a metal-insulator transition back in 2014 was correct. I am happy to be have been proven wrong! That's good science!

Kapitsa, Landau, and quasi-particles

Earlier I suggested that the founders of condensed matter physics were Onnes, Landau, Bardeen, Anderson, and Wilson. I might also add Brian Josephson. But, as pointed out by Ben Powell, this list is theory-centric and so I am thinking more about experimentalists. I think my first addition would be Pyotr Kapitsa. He received a Nobel Prize for "his basic inventions and discoveries in the areas of low-temperature physics" and he managed to save Landau from the Soviet gulag. However, there is a lot more to Kapitsa. Two particular experimental achievements were finding ways to produce large quantities of liquid helium and the production of high magnetic fields. Both of these were key for revealing the details of the Fermi surface of metals through quantum magnetic oscillations and ultimately for finding new states of matter (such as superfluidity) and mapping out phase diagrams.

I was wondering how influential Kapitsa was in influencing Landau's scientific thinking. Biographical Memoirs of Fellows of the Royal Society has obituaries of Landau written by Kapitsa and by Evgeny Lifshitz. Kapitsa's is fairly boring, almost reading like something written by a Soviet bureaucrat, noting "The only interruption in his work at the Institute occurred between 1938 and 1939". No mention is made is that this was because he was in prison for mocking Stalin! Although the following is worth noting: 

 To what extent Landau valued ... connexion with experiment is revealed by the following. His theoretical department at the Institute was small (there were no more than ten research workers and aspirants). Although I suggested that the Academy might set up a special Institute of Theoretical Physics on as large a scale as he wished, Landau not only declined, but even refused to discuss the matter. He said that size was not important and he was extremely happy to be classed as a staff member of the experi mental institute.

It is also interesting that Landau never read any scientific literature himself, and never wrote anything!

Lifshitz's obituary is more detailed and focuses on Landau's science, rather than just reciting his CV. The following shows just how important Kapitsa was for Landau scientifically. Lifshitz states

But Landau’s greatest contribution to physics was the theory of quantum liquids. Its significance continues to increase and undoubtedly during recent decades it has also had a revolutionary effect on other fields of physics— solid state and even nuclear physics. 

The theory of superfluidity was stated by Landau in 1940-41 soon after the discovery in 1937 by P. L. Kapitza of this basic property of helium-II....

The discovery and explanation of superfluidity is also remarkable for its truly constructive interaction between experiment and theory. The research of Kapitza and Landau was carried out in close scientific co-operation and there is no doubt that results of the wide experimental research into processes of heat transfer in liquid helium carried out by Kapitza in 1939-41, had a stimulating effect on theoretical constructions. For his part, Landau formulated his theories while these experiments were still in progress, which made it possible to interpret the results of new experiments immediately. 

The basis of Landau’s theory is the notion of ‘quasi-particles’ (elementary excitations) which compose the energy spectrum of liquid helium. Landau was the first to put the question of the energy spectrum of a macroscopic body in this most general form, and he also found the character of the spectrum for a quantum liquid of the type to which liquid helium (the 4He isotope) belongs;

The concept of quasi-particles is arguably one of the most important in quantum many-body theory and condensed matter.

Aside: I had forgotten this and tended to think quasi-particles were introduced by Landau in his Fermi liquid theory paper fifteen years later.

The comments above follow the common narrative of the discovery of superfluidity, which as Sebastien Balibar argues is debatable. This narrative exclusively focuses on Kapitsa and Landau. The new state of matter, Helium-II, associated with a singularity in the specific heat of liquid 4He at the lambda temperature, was discovered in 1927 by Willem Keesom in Leiden. Superfluidity was independently discovered in 1937 by Allen and Misener. Theories of superfluidity, including the two-fluid model, by Laszlo Tisza and Fritz London, were developed before Landau's.

Nevertheless, the main point remains clear. It is highly likely that Landau and Kapitsa had a significant influence on one another. Such synergy between experiment and theory is at the heart of condensed matter physics. Kapitsa was definitely following the integrated approach of Kammerlingh Onnes: development of experimental techniques, careful measurements, addressing fundamental questions, and interaction with theorists.

The Devil is not in the details

Condensed matter physics aims to understand different and describe states of matter. Each state (phase) is associated with a particular type of order and phase diagrams encode the external parameters (temperature, pressure, chemical composition, magnetic field,...) that are necessary for each of the possible orderings to be stable.

Phase diagrams of even the simplest systems, such as binary alloys, can be quite rich, with many competing phases. Nevertheless, in many cases, simple microscopic models, with just a few degrees of freedom and a few parameters can describe these rich diagrams. Often a key is for the model to involve competing interactions, which can arise from different forces or from geometric frustration. Earlier, I wrote about how an Ising model on a hexagonal close-packed lattice could describe the plethora of distinct orderings that are observed in binary alloys. There the orderings are defined by the ordering wave vector and the composition of the unit cell. Another example occurs in spin-crossover materials and is described in recent work by my UQ colleagues.

Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks, Jace Cruddas and Ben J. Powell

A 2016 chemistry paper was First Step Towards a Devil's Staircase in Spin‐Crossover Materials

Another rich example is where there are two different spatial scales associated with interactions between the components of the system. In a lattice system, this can lead to ordering wavevectors that are incommensurate with the lattice. About forty years Per Bak wrote a nice series of papers that explored this situation.

Ising model with solitons, phasons, and "the devil's staircase", Per Bak and J. von Boehm 

This is an elegant and clear study of a simple Ising model in three dimensions with a frustrating interaction J_2 in the vertical direction. This is an example of an ANNNI model.

A mean-field theory for a state with a sinusoidally varying magnetisation (with wavevector 2 pi q) gives the phase diagram on the right above. The point P is a Lifshitz point, a tricritical point where the incommensurate (modulated) phase becomes stable. 

[Aside: in fermion models, a Lifshitz point is quite different: where the volume of the Fermi surface vanishes].

However, there is much more to the story. The authors then construct a mean-field theory where the magnetisation is commensurate, allowing for large unit cells. This leads to the phase diagram below.

The fractions p/q correspond to states with wavevector 2 pi p/q.

For example, the 1/4 state is below.

What does this have to do with a "devil's staircase"?

If for fixed J_2/J_1 the wavevector is plotted as a function of temperature it has steps of varying size and width.

A paper by Bruinsma and Bak considers an AFM Ising chain with 1/n^2 interaction, in a magnetic field, at zero temperature.

Note: In the figure below q is NOT the wavevector but rather the ratio of up to down spins. 

The magnetisation vs field curve has a fractal structure.

Bak also wrote a Physics Today article (that compares the phenomena to frequency mode locking) and a general review that includes examples of experimental realisations, ranging from magnets to atoms on surfaces.

This illustrates an important point that is often made in complexity science. Simple rules (theoretical models) can produce complex behaviour. 

This also illustrates characteristics of emergent phenomena. A wide range of physical systems can exhibit the same phenomena. Many of the details do not matter.

The devil is not in the details.