Why is condensed matter in flatland so interesting?

I am working on a chapter on condensed matter physics in dimensions different from three for Condensed Matter Physics: A Very Short Introduction. 
This is a rich subject since it is associated with high-Tc superconductors, quantum Hall effects, Haldane spin chains, Kosterlitz-Thouless transition, critical phenomena in 4 - epsilon dimensions, .....
Obviously, I cannot only give the flavour of things.

I would like to get your perspective on a few questions. For some, I have my own answers but want to hear others. Bear in mind the answers have to be accessible to a non-expert audience.

1. What is the central idea or concept?

2. What is an analogy to explain how dimensionality changes things?

3. What is an example of cross-fertilisation with another field of physics or science?

4. What is a significant technological application where low-dimensionality is central?
[High mobility MOSFETs are not an example because the devices are not really using a property that only occurs in two dimensions].

Simple model Hamiltonians can describe complexity

An important idea in condensed matter physics, both soft and hard, is that the rich phenomena seen in materials that are chemically and/or structurally complex can often be described by relatively simple model Hamiltonians that involve only a few parameters. This is particularly true when the model and system have competing interactions. This often leads to two inter-related phenomena, that I have previously described for strongly interacting quantum many-body systems.

• the emergence of new energy scales that are much smaller than the energy parameters in the Hamiltonian
• subtle competition between different ground states that involve distinct types of ordering (or not)

These phenomena also occur in classical systems. A nice example is described in this 1993 paper.

hcp Ising model in the cluster-variation approximation 
R. McCormack, M. Asta, D. de Fontaine, G. Garbulsky, and G. Ceder

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.
Rich phase diagrams result by varying the relative concentration of the atoms A and B (e.g. gold and silver), or equivalently the difference in the chemical potential between A and B, or the relative size of the interatomic interactions, or the temperature. The phase diagram can contain many competing phases with well-defined stoichiometry: A, B, AB, A2B, A3B, A2B3, A3B5, ...
Furthermore, even for a single stoichiometry, there can be multiple possible distinct orderings (and crystal structures).

The hcp lattice can be viewed as layers of two-dimensional hexagonal lattices where each layer is displaced relative to others. A unit cell is shown below on the left, where V1, V2, and V3, denote nearest-neighbour (nn), next-nearest neighbour (nnn), and nnnn interactions.
For the case of perfect packing of hard spheres V1=V2.
Note, that even when only nn interactions are present, and they are antiferromagnetic, that the system is frustrated, and for a single layer the Ising model does not order at finite temperature and has a massively degenerate ground state (i.e. non-zero entropy).
The figure on the right shows a way to represent this unit cell and the interactions in terms of two hexagonal lattices superimposed on top of each other.

 The authors show that there is a plethora (menagerie) of possible ground states and stoichiometric orderings.

We predict 32 physically realizable ground states with stoichiornetries A, AB, A2B, A3B, A, B, and A4B3. Of these structures, six are stabilized by NN pairs and eight by NNN pairs; the remaining 18 structures require multiplet interactions for their stability. 

This is a nice example of how a simple model can describe complex and rich behaviour. It is also a nice example of emergence in that many of the details don't matter such as the identity of the atoms or the form of the interaction between them.

The commercial applications gap

These days too many seminars, papers, and grant applications begin with great claims about the potential commercial applications of the research being discussed.
We should be skeptical about any hype concerning technological applications of basic research in materials science.
There is a big gap between a commercial device/material and what you can do in the lab with millions of dollars worth of equipment on a milligram of a material or a single electronic device.

It does not matter whether it is a photovoltaic cell, a catalyst, or a superconducting wire. All of the following demanding criteria must be met. Furthermore, it must be better than any existing technology and any competitor on most of these counts.

Cheap to manufacture.
Scaleable to mass production.
Durable. Often on the scale of years or decades.
Reliable and reproducible. Devices, whether batteries or computer memories, must work all the time.
Healthy. Not expose the user or manufacturer to toxic materials.
Use materials available in abundance (silicon, water, ...) rather than scarce ones, such as some rare earth elements.
Environmentally friendly.

Thanks to Tanglaw Roman for emphasizing these issues to me.

This post was partly stimulated by re-reading the front page of The New York Times from March 20, 1987, which features an article Discoveries bring a `Woodstock' for physics. The article describes the famous session on cuprate superconductors at the 1987 March meeting of the American Physical Society. It is worth reading to see how so little of what was promised then has not happened (unfortunately).

Can you think of other criteria that new technologies must meet to be commercially viable?

The best book I read in 2019

My son is very good at giving good gifts. His all-time winner was Priya, the dog shown on my profile photo. For father's day this year, he gave me the book Why Nations Fail: The Origins of Power, Prosperity, and Poverty, by Daron Acemoglu and James A. Robinson. It could be viewed as an engaging political and economic history of the whole world, according to a leading economist and political scientist. The book is the fruit and popular presentation of a long research collaboration by the authors, which produced papers such as a mathematical model for political transitions. Examples discussed ranging from the Spanish conquest of the Aztecs to the Mugabe regime in Zimbabwe to China in the twentieth century to the southern USA after the civil war.

To summarise the main ideas I think it is helpful to define the key terminology and concepts in the book. Some of the terminology used seems to me to be slightly different from how it is used by others. A more detailed summary of the book can also be found in these lecture slides.

Creative destruction is the process whereby technical innovations (e.g., steam engines or mobile phones) lead to new companies and new markets that create new wealth for some while also destroying the wealth of some and profitability of other companies. Countries that encourage, produce and embrace innovation grow economically. Those that discourage it, particularly to protect their own markets and monopolies, decline economically.

Extractive political institutions are governments (ranging from national down to counties) that concentrate power with a select few (an `elite') who work to maintain their stranglehold on power, and use their power to enrich themselves. There are few constraints on the power of political rulers. Dictatorship or one-party rule is an example, whether fascist or communist.

In contrast, inclusive political institutions, encourage the involvement of diverse groups in decision making and the formulation of laws and policies. The focus is on the common good. There are checks and balances (a free press, an independent judiciary, the rule of law) that constrain the power of politicians.

Extractive economic institutions are public utilities, banks, and companies, that are set up and used by a select few (an `elite') to extract wealth from others. This is done by corruption, rent-seeking, monopolies, crony capitalism, or opposing innovation.

In contrast, inclusive economic institutions are those that encourage free markets (especially labour markets), competition, and innovation. In particular, participation in economic activity by diverse parties is encouraged, from job opportunities to starting new companies.

There is an intimate connection between the character of economic and political institutions.
There are virtuous cycles whereby inclusive political institutions facilitate the development of inclusive economic institutions and vice versa. Acemoglu and Robinson argue that a key example is how the Glorious Revolution in England in 1688 created conditions that led to the Industrial Revolution, and the consequent economic growth, that made England far richer than most other countries.

In contrast, there are vicious cycles where extractive political institutions facilitate the development of extractive economic institutions and vice versa. For example, if a wealthy company has a monopoly that they wish to protect they will hire lobbyists and donate money to political candidates that will pass laws that reflect their own narrow interests and priorities rather than those of broader society.

The iron law of oligarchy is that if the leaders of an extractive political institution are replaced, particularly in a revolution, that the institution will remain extractive. The autocracy and wealth of the Russian Tsar were replaced with that of the Soviet elite. Arguably, this is also what has happened in the post-colonial era. Colonial governments that extracted wealth from colonies for white European `elites' were replaced with national governments controlled by local `elites' that extract wealth from their fellow nationals.

Watershed moments are key times where a transition in institutions from extractive to inclusive, or vice versa, can happen.

The book provides countless fascinating examples from history to make their case. The key message of the book is that we should be investing in developing and protecting inclusive institutions.  This can only be done by broad coalitions of different groups in society. Thus civil society is very important.

I might disagree a little with the book, perhaps they don't emphasise enough the key role that individuals can play, both for good and for bad, in shaping and reforming institutions. For example, contrast Stalin and Gorbachov, or Mao Zedong and Deng Xiaoping. Servant leaders whose focus is not on themselves but on the common good and who act with integrity can steer things in the right direction.

The authors discuss the amazing story of Botswana. It gained independence in 1966 and was one of the poorest countries in the world. Today, it has the highest Human Development Index (HDI) and GDP per person, and the lowest corruption index of African countries. Since 1966, it has had continuous democratic elections (the longest in Africa). A key role was played by the founding leaders Seretse Khama and Quett Masire.

A question that occurred to me, stimulated by the book, is to what extent have universities become extractive institutions (while marketing themselves as inclusive institutions)?

The central role of scales in condensed matter

An important concept in condensed matter is the role played by scales, i.e. how big or small physical quantities are. Length, time, energy, and temperature are all physical quantities.

For example, there are many different length scales associated with a piece of material, say a block of copper, ranging from centimetres to a fraction of a nanometer. This covers lengths varying by a factor of a trillion, i.e., twelve orders of magnitude. The piece of copper may have dimensions of a centimetre. But it may be composed of small metallic grains of micron (micrometer) dimensions, and that can only be seen with a microscope. On an even smaller scale is the size of the individual copper atoms that make up the material, with dimensions less than a nanometer. Using different experimental techniques a scientist can ``zoom in and out'' and examine the properties of a material at different length scales.

Similarily one can investigate properties of a material at different time scales. This is similar to how one may use a high-speed movie camera to observe something and then replay it in slow motion. In a metal there are different time scales associated with different phenomena: the vibration of an atom, the time between collisions of electrons with each other, the period of the collective oscillation of all of the electrons.

There are also different energy scales associated with a material. Examples include the energy required to move a single atom a particular distance, the energy required to remove a single electron from the crystal, the kinetic energy of an electron inside the material, and the energy required to compress the whole material by a certain amount.

In quantum theory, energy and time are related by a proportionality factor known as Planck's constant. Thus, the energy scale and time scale associated with a specific phenomenon are related to each other.

The magnitude or scale of the temperature is also important. Temperature is related to energy via heat. Using clever refrigeration techniques materials can be cooled down to temperatures of less than one-thousands of a degree above absolute zero. This means that the properties of a material can be studied over a temperature range varying by about a factor of one million (six orders of magnitude).

This wide range of length, time, energy, and temperature scales is central for condensed matter physics in several respects. Overall, it means that phenomena, experimental techniques, theories, and concepts are relevant to a particular scale.

Experimental techniques have to be designed to investigate and ``probe'' the relevant phenomena at the relevant scale. Theories are also constructed with a concern with the relevant scales. Perhaps this is obvious.

There are also three profound and unanticipated aspects of the role of scales in condensed matter. 

a. Whereas, the existence of the atomic and macroscopic scales is obvious, due to collective behaviour (emergence) there are intermediate scales of length and time associated with particular phenomena. Before, I have discussed examples of emergent energy scales and length scales.

b. In distinct systems, the same phenomena can occur at scales that differ by many orders of magnitude. A striking example is the occurrence of superfluidity in liquid 3He at temperatures below one-thousandth of a one degree Kelvin and in neutron stars at temperatures below one hundred thousand degrees.

c. Through a highly sophisticated theoretical method, known as the renormalisation group and scaling, it is possible to make concrete connections between the properties of a system at different scales.

It is worth considering whether this wide range of scales and the central role they play occurs in other academic disciplines. In biology, this is certainly true, with a hierarchy of scales from biomolecules to protein networks to cells to organs. In economics, one goes from individual consumers to microeconomics to macroeconomics. The size of personal incomes, businesses, and government debt can also range of many orders of magnitude. In sociology, there is also a range of scales. Indeed, emergence does shape many of the big questions of many disciplines.
Arguably, what is really unique about CMP is b. and c. above.

I thank my son for asking me to clarify this central role of scales in condensed matter.

Crowdsourcing answers to some science questions

Often when I write a post commenter's suggest some useful references. Answers to any of the following questions would be appreciated. The questions relate to things I am curious about, working on or subjects of possible blog posts.

1. Quasi-particles are a key concept in quantum many-body theory. Is there an analogous concept in classical many-body systems, e.g., dense liquids or plasmas?

2. Is there a simple physical argument, possibly accessible to non-experts, of why decreased dimensionality leads to increased fluctuations?
(An example is the Mermin-Wagner theorem). I understand how to mathematically show that fluctuations increase due to decreased phase space, but I am skeptical that I could make this argument comprehensible to a non-expert?

3. Why does increased CO2 in the atmosphere lead to an increased frequency of extreme weather events (cyclones, droughts, floods, ...)? What is the basic physics involved?
This is the scientific aspect of climate change that I understand the least. It also seems to be the aspect of climate change that could be the worst. I write this in the context of the current bushfires in Australia.

4. Who was the first person to write down the Landau theory for a superfluid transition, suggesting that the order parameter was a complex number?
Was it Ginzburg and Pitaevskii in 1958?

5. Who was the first person to fully appreciate that at a critical point the correlation length of the order parameter diverges and fluctuations in the order parameter become large?
[In 1914 Ornstein and Zernike solved this problem for a liquid-gas transition].

6. Are there philosophical problems or paradoxes associated with the principle of least action in quantum mechanics?
Consider a particle that moves from x at time t to x' at time t'. The path taken is that which is an extrema of the action (time integral of the Lagrangian) along that path relative to others. Superficially, that sounds like the particle ''considers" all the possible paths and then "chooses" the right one. Spooky action at a distance? This makes it sound like to understand classical mechanics you have to consider it as a limit of quantum mechanics and just perhaps embrace the many-worlds interpretation....

What was the greatest discovery of the past decade?

To my readers my best wishes for the New Year and the new decade!

It is worth reflecting on what has been achieved in condensed matter and chemical physics over the past decade. Which discovery or achievement would you rate as the most exciting, surprising, or significant?

To benchmark things, this is what I would say about previous decades, with regard to hard condensed matter, with a personal bias towards strongly correlated electron systems.

1970s: scaling and the renormalisation group

1980s: quantum Hall effects, cuprate superconductivity, heavy fermions, scanning tunneling microscopy (STM)

1990s: Dynamical Mean-Field Theory (DMFT), Kondo effect in quantum dots, superconducting qubits, (Angle-Resolved PhotoElectron Spectroscopy) ARPES advances, DMRG

2000s: iron-based superconductors, graphene, DMFT+DFT, topological insulators

2010s: ?

To be honest, I am worried that with each decade the discoveries are somewhat less exciting or significant. On the other hand, incremental advances, particularly steady ones over several decades should not be looked down on. An example is computational electronic structure methods and increases in the energy and momentum resolution associated with inelastic neutron scattering spectroscopy and ARPES.

What would you nominate for the past decade?