What is condensed matter physics?

 Every day we encounter a diversity of materials: liquids, glass, ceramics, metals, crystals, magnets, plastics, semiconductors, foams, … These materials look and feel different from one another. Their physical properties vary significantly: are they soft and squishy or hard and rigid? Shiny, black, or colourful? Do they absorb heat easily? Do they conduct electricity? The distinct physical properties of different materials are central to their use in technologies around us: smartphones, alloys, semiconductor chips, computer memories, cooking pots, magnets in MRI machines, LEDs in solid state lighting, and fibre optic cables. Consequently, the science of materials attracts researchers in a wide range of disciplines: physics, chemistry, biology, mathematics, and the varieties of engineering (electrical, chemical, mechanical, material…). But why do different materials have different physical properties? 

There are more than one hundred different types of atoms, or chemical elements, in the universe. Any material is composed of a specific collection of different atoms, and they are arranged in a particular spatial pattern within the material. A central question is: 

How are the physical properties of a material related to the properties of the atoms from which the material is made?

Extract from Chapter 1, Condensed Matter Physics: A Very Short Introduction

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.

Wading through AI hype about materials discovery

 Discovering new materials with functional properties is hard, very hard. We need all the tools we can from serendipity to high-performance computing to chemical intuition. 

At the end of last year, two back-to-back papers appeared in the luxury journal Nature.

Scaling deep learning for materials discovery

All the authors are at Google. They claim that they have discovered more than two million new materials with stable crystal structures using DFT-based methods and AI.

On Doug Natelson's blog there are several insightful comments on the paper about why to be skeptical about AI/DFT based "discovery".

Here are a few of the reasons my immediate response to this paper is one of skepticism.

It is published in Nature. Almost every "ground-breaking" paper I force myself to read is disappointing when you read the fine print.

It concerns a very "hot" topic that is full of hype in both the science and business communities.

It is a long way from discovering a stable crystal to finding that it has interesting and useful properties.

Calculating the correct relative stability of different crystal structures of complex materials can be incredibly difficult.

DFT-based methods fail spectacularly for the low-energy properties of quantum materials, such as cuprate superconductors. But, they do get the atomic structure and stability correct, which is the focus of this paper.

It is a big gap between discovering a material that has desirable technological properties to one that meets the demanding criteria for commercialisation.

The second paper combines AI-based predictions, similar to the paper above, with robots doing material synthesis and characterisation.

An autonomous laboratory for the accelerated synthesis of novel materials

[we] realized 41 novel compounds from a set of 58 targets including a variety of oxides and phosphates that were identified using large-scale ab initio phase-stability data from the Materials Project and Google DeepMind

These claims have already been undermined by a preprint from the chemistry departments at Princeton and UCL.

Challenges in high-throughput inorganic material prediction and autonomous synthesis

We discuss all 43 synthetic products and point out four common shortfalls in the analysis. These errors unfortunately lead to the conclusion that no new materials have been discovered in that work. We conclude that there are two important points of improvement that require future work from the community: 

(i) automated Rietveld analysis of powder x-ray diffraction data is not yet reliable. Future improvement of such, and the development of a reliable artificial intelligence-based tool for Rietveld fitting, would be very helpful, not only to autonomous materials discovery, but also the community in general.

(ii) We find that disorder in materials is often neglected in predictions. The predicted compounds investigated herein have all their elemental components located on distinct crystallographic positions, but in reality, elements can share crystallographic sites, resulting in higher symmetry space groups and - very often - known alloys or solid solutions. 

Life is messy. Chemistry is messy. DFT-based calculations are messy. AI is messy. 

Given most discoveries of interesting materials often involve serendipity or a lot of trial and error, it is worth trying to do what the authors of these papers are doing. However, the field will only advance in a meaningful way when it is not distracted and diluted by hype and authors, editors, and referees demand transparency about the limitations of their work.  

Shape memory alloys

Recently I bought a small wire of NiTinol to have fun with and use in demonstrations to kids. This video gives a spectacular demonstration and attempts to explain how it works. I did not know about their use in stents for heart surgery.

I am still struggling to understand exactly how shape-memory alloys work. According to Wikipedia

The shape memory effect occurs because a temperature-induced phase transformation reverses deformation...Typically the martensitic (low-temperature) phase is monoclinic or orthorhombic . Since these crystal structures do not have enough slip systems for easy dislocation motion, they deform by twinning—or rather, detwinning.

Martensite is thermodynamically favored at lower temperatures, while austenite (B2 cubic) is thermodynamically favored at higher temperatures. Since these structures have different lattice sizes and symmetry, cooling austenite into martensite introduces internal strain energy in the martensitic phase. To reduce this energy, the martensitic phase forms many twins—this is called "self-accommodating twinning" and is the twinning version of geometrically necessary dislocations. 

In different words, I think the essential idea may be the following. In most metals large strains are accomodated by topological defects such as dislocations. These become entangled leading to work hardening and irreversible changes is macroscopic shapes. Shape memory alloys are different because of the low symmetry unit cell. The most natural defects are twinning domain walls and they are not topological and so their formation is reversible.

I am looking forward to reading the book chapter Shape memory alloys by Vladimir Buljak, Gianluca Ranzi

Another fascinating phenomena that is related to shape-memory is "superelasticity", which I discussed in an earlier post on organic molecular crystals, and has recently been reviewed.

I welcome clarification  of the essential physics.

Towards real materials applications

There is a chasm between finding a material that has a desirable property that is key to a technological application and producing a commercial product. In the hype about materials research, the width of this chasm is too often glossed over.

The Structure of Materials by Samuel M. Allen and Edwin L. Thomas (based on a course in Materials Science and Engineering at MIT) introduces the tetrahedron of

structure, properties, processing, and performance. In condensed matter physics the focus is largely on the relationship between structure and properties. But, for engineering, these are both also related to performance and processing (i.e. ability to make materials and devices).

 The book also emphasises the multiple length scales associated with the structure of "real" materials. The scales range from the atomic scale of Angstroms to the scale of micrometers associated with objects such as grain boundaries, topological defects, and domain walls. These longer length scales are also relevant in liquid crystals, glasses, and polymers.

Where is materials research heading?

One way to answer this question is to look at the reports prepared every decade by the National Academies in the USA. I have recently been looking through the 2019 report, Frontiers of Materials Research: A Decadal Survey.

There are several reasons why I like to look at these reports. A previous post mentioned a similar 2007 report prepared for the USA Department of Energy.

I can learn a lot about materials science and engineering. See, for example, the figure below.

The reports help put condensed matter physics in the broader context of research in materials science and engineering. 

[Previously, I have argued that CMP is a particular approach to materials research and is distinct from materials physics. Although there is a significant overlap in the materials studied and some of the methods used, the driving questions are distinctly different].

The reports provide choice quotes for grant applications. Here is one from pages 24-25.

Key Finding: Basic research in fundamental science directions, meaning work that neither anticipates nor seeks a specific outcome, is the deep well that both satisfies our need to understand our universe and feeds the technological advances that drive the modern world. It lays the groundwork for future advances in materials science as in other fields of science and technology. Discoveries without immediate obvious application often represent great technical challenges for further development (e.g., high-Tc superconductivity, carbon nanotubes) but can also lead to very important advances, often years in the future. 

Key Recommendation: It is critically important that fundamental research remains a central component of the funding portfolio of government agencies that support materials research. Paradigm-changing advances often come from unexpected lines of work.

Here is one from page 6.

Key Finding: Quantum materials science and engineering, which can include superconductors, semiconductors, magnets, and two-dimensional and topological materials, represents a vibrant area of fundamental research. New understanding and advances in materials science hold the promise of enabling transformational future applications, in computing, data storage, communications, sensing, and other emerging areas of technology. This includes new computing directions outside Moore’s law, such as quantum computing and neuromorphic computing, critical for low-energy alternatives to traditional processors. Two of NSF’s “10 big ideas” specifically identify support of quantum materials (see The Quantum Leap: Leading the Next Quantum Revolution and Midscale Research Infrastructure).

The reports are based on the consensus of a range of experts. Hence, they arguably more objective than survey articles written in luxury journals by individuals hyping their field.

But, right now the reason I am reading this report is that I am writing the last chapter of Condensed Matter Physics: A Very Short Introduction, and need to address the question of where CMP is heading. Some earlier preliminary thoughts are here.

Here are a few of my thoughts about this report. I would love to hear the perspectives of others. 

First, I should give some important caveats. I have only skimmed the report. It was written by people who know much more than I. Writing a report that is based on a diverse community of interests and perspectives is extremely difficult. The main audience for such reports is not scientists themselves but rather funding agencies and policymakers.

The Summary begins with "The past decade has seen extraordinary advances in materials research" (page 3). Chapter 2 describes "significant advances" from the past decade. There is no doubt there have been many advances. It is great to read about them. Section 2.4 concerns Quantum Materials and Strongly Correlated Systems. Most of the advances described there are incremental advances from discoveries made before 2010, such as topological insulators. This haunts me with a nagging concern that CMP has not seen a big discovery in the past decade. For quantum materials is superconductivity in twisted bilayer graphene the leading candidate? Other suggestions?

A lot of attention is given to the potential of computational materials science, including when combined with data science methods (e.g. machine learning), topological matter, and quantum information processing in solid-state devices. However, I remain skeptical about the hype associated with these subjects, particularly with regard to technological applications. Big data need big theory too.

Significant attention is given to the relevance of materials research to USA defense, national security, and economic competitiveness. I wonder if this is because the report is being pitched to a MAGA government. Although I agree on the relevance, for many of us that is not the motivation for our interest in materials.

Update. In a comment below, David Sholl pointed out that NSF is not happy with the report. The background given there is also worth reading.