What is a quasiparticle?

 An example of emergent entities in condensed matter physics are quasiparticles. The concept can be described with the following analogue. When a horse gallops through the desert it stirs up a dust cloud that travels with it. The motion of the horse cannot be separated from the accompanying dust cloud. They act as one entity. Similarly, in a system consisting of many interacting particles, when one particle moves it carries with it a “cloud” of other particles. This composite entity is referred to as a quasiparticle. It turns out to be easiest to understand the whole system of particles in terms of the quasiparticles rather than in terms of the individual particles.

Quasiparticles are composite objects. Like the constituent particles in the system, quasiparticles each have properties such as charge, mass, and spin. However, these properties of a single quasiparticle may be different from those of the individual particles of which it is constituted. An example is holes in semiconductors; the many electrons in a crystal act collectively to produce a hole (the absence of a single electron), a quasiparticle with the opposite charge to that of a single electron. A more striking example is for the fractional quantum Hall states; the charge of the quasiparticles can be a fraction of the charge on a single electron.

Different musical instruments produce distinct sounds because they are made of different materials, and they vibrate in different ways in response to different stimuli. In general, the vibrations of a medium reflect something about the medium itself. Chapter 3 discussed how in a crystal the number of distinct ways that sound can travel through a crystal reflects the symmetry and ordering of the atoms in the crystal.

When the skin on a drum is hit by a drumstick the skin vibrates at particular frequencies. Similarly, a state of matter responds to external stimuli such as light, sound or heat, by oscillating at particular frequencies. These vibrations travel through the matter as waves. The properties of these waves reflect the particular order present in the state of matter. Here is a specific example. When a neutron with a particular energy and momentum is absorbed by a ferromagnetic crystal the interaction of the magnetism of the neutron with that of the atoms in the crystal produces a collective oscillation of the magnetic state of the crystal in time and space. Known as a spin wave, this oscillation has a particular frequency and wavelength. In quantum theory, waves and particles are equivalent to one another. The energy and momentum of a particle are related to the waves’ frequency and wavelength, respectively. Particles equivalent to light waves are known as photons; particulate equivalents of sound waves are known as phonons. And similarly, the particle equivalent of a spin wave is known as a magnon. These collective excitations are quasiparticles. Whereas the particles in a system may interact strongly with one another, the quasiparticles may interact weakly with one another. This makes analysis and understanding of the relevant theories more tractable.

The quasiparticle concept is a powerful theoretical tool in condensed matter physics. It is the basis for the construction of models that enable emergent phenomena to be understood in terms of the effective interactions between components such as quasiparticles, rather than in terms of the actual constituent particles and their interactions. This approach requires profound physical insight in order to discern what the truly essential components of a system are. Lev Landau was one of the first theoretical physicists to take this approach, introducing the idea of quasiparticles in his theories of superfluidity in 4He and of liquid 3He. This approach was also central to the BCS theory of superconductivity. Phil Anderson was also a master of the approach, using intuition to propose models that were simple enough for analysis and yet complex enough to capture the essential physics associated with a particular state of matter. In 1977 he was awarded the Nobel Prize for work using this approach to understand two specific systems: magnetic atoms in metals and the motion of electrons in materials that are not crystals and are dirty in the sense of containing many impurities.

An extract from Chapter 9, "Emergence: More is Different", in Condensed Matter Physics, A Very Short Introduction

A more detailed and technical discussion is in Section 8.2 of my review article on emergence.

Critical points in condensed matter illuminate universality

Every person is unique. No two people are identical. We differ in physical appearance, personality, fingerprints, heartbeat, gait, and DNA. Such differences are used to identify criminals and in video surveillance of citizens by nation states. Yet in other ways all humans are the same. We all have brains, hearts, and lungs. All our bodies use the same biochemistry to stay alive: whether to breathe oxygen, digest food, or fight infections. On some level we have common aspirations: to survive, to be loved, to be happy, and to find meaning and purpose. Yet these aspirations find many expressions. Humans have certain universal qualities and properties, yet at a finer level of detail there is a particularity of each of these properties. They are at one level the same but are not the same at another level. 

All academic disciplines search for universals; they develop categories, concepts, and theories that overarch particularities. Biologists classify species of plants and animals and types of cells and viruses. All biological systems use the same molecules (DNA, RNA, and proteins) and chemical reactions. The same genetic code uses the information encoded in a piece of DNA to make proteins with specific functions. Anthropologists study the immense diversity of human cultures and societies. This diversity can be described in terms of universal concepts such as kinship, family, ritual, community, economics, law, and morality. Linguists study the common structures and grammars of the thousands of different human languages.  Although the world we live in is diverse, disciplines have each discovered some universals.

Condensed matter physicists study diverse states of matter and the transitions between them. A surprising discovery is that there is much more universality than might be expected, particularly given the chemical and structural diversity of materials. In this chapter, I will discuss the nature of this universality, how it emerges, and the length scales associated with transitions between different states of matter. Landau’s great insight was that many of the chemical and structural details of materials are irrelevant to understanding phase transitions. Furthermore, a precise classification of different types of phase transitions, into what are called universality classes, can be made. For example, superconducting, superfluid, and a subset of magnetic transitions are in the same class. The determinants of the universality classes are the symmetry of the state and the spatial dimensionality of the system. None of the other details matter.

Many phase diagrams (such as the Figure above) include a critical point, located at the end of a boundary between two different states of matter. A common example is the critical point that occurs at a specific temperature and pressure for a transition between a liquid and a gas. Understanding the physical properties of a material close to its critical point was a great challenge for theoretical physics, lasting a hundred years, and was only solved in the 1970s. The powerful theoretical ideas and techniques that were developed provide a quantitative way to relate the properties of a system at one length scale to properties at a different length scale. These techniques also have application to a wide range of other problems and fields including elementary particle physics, chaos theory, fractals, polymers, and machine learning. New insights were gained into universality and emergent phenomena.

An extract from "The Critical Point," chapter 6, Condensed Matter Physics: A Very Short Introduction

Will there be big new discoveries in condensed matter physics?

 There are two aspects to this question concerning the future of condensed matter physics. First, are there big things to be discovered? If yes, will they be discovered?

I believe the first answer is yes for two reasons. First, the past hundred years have given us a continual stream of discoveries, many of them unexpected. Every time that things get a little boring, pretty soon there is something exciting and new. Second, condensed matter physics is all about emergent phenomena in materials. Emergent phenomena are extremely hard to anticipate or predict. Because of the combinatorics of chemistry, the list of possible materials to study is endless. CMP presents an endless frontier to explore. However, just because such a frontier exists does not mean that it will be explored. Successful explorers require courage, creativity, resources, time, and freedom.

I am concerned that the wild frontiers of condensed matter may not be explored. It is worth reflecting on who were some of the pioneers of CMP and the character of their institutional environments.  Consider Kammerlingh Onnes, Landau, Kapitsa, Anderson, de Gennes, and Leggett. Some common elements of the context (institutional, historical, political) in which they made their discoveries were time, stability, job security, mental space, and intellectual freedom. For example, Anderson spent almost three decades at Bell Labs in its heyday. Thanks to the monopoly of Bell in providing telephone services in the USA, the parent company had a very secure and stable income, providing it the ability to provide substantial financial and institutional support for basic research.

These pioneers played a long game. They had the freedom to fail, to choose research topics, and to change directions. They did not follow fashion and were fiercely independent thinkers. Andrew Zangwill highlights this about Anderson in his biography. They largely had the resources they needed and did not have to worry or fight for funding. Their daily life was very different from that of a researcher today. Their mental space was not filled with an endless stream of distractions such as emails, grant proposals, conferences, reporting, reviewing, committees, metrics, ... Most of their time and mental energy was simply focused on curiosity-driven research. 

Today, there is intense competition for funding, institutional status, and career benefits associated with obtaining it, and a pressure to produce in the short term "outputs" (papers) and "impact" (citations) and "national benefit" (technological, commercial, security, and social). This naturally leads to researchers working on "safe" projects in fashionable areas that they are confident will produce results in the short term.

I hope that I am wrong. But, I fear that great discoveries may be missed.

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.

What is condensed matter physics? (revised)

I have just rewritten chapter 1 of Condensed Matter Physics: A Very Short Introduction.  I obtained very helpful feedback on my first version from a freelance editor. This has given me fresh eyes for the whole manuscript, which I am now rewriting. She suggested moving some strong material from later chapters into the first chapter, particularly the fact that CMP is all about emergence. I have also benefited from other readers and blog commenters. For example, David Sholl's asked about how CMP is different from other approaches to materials science.

Here is the new version. 

I welcome comments. But, again you are probably not my intended audience. Rather, it is your family, undergraduates, or colleagues in biology or social sciences.

Mean-field theories: helpful or misleading? From Hubbard to COVID-19 models

Mean-field theory (self-consistent field theory) is incredibly valuable. It gives significant insights into what is possible with a particular model.
What kind of phases and broken symmetries may be possible?
How does the phase diagram depend on different parameters in a model?
Indeed, mean-field theory is the basis of the whole Landau paradigm for spontaneous symmetry breaking and phase transitions.
Implementations of Density Functional Theory (DFT) in computational materials science are basically mean-field theories. Most of computational quantum chemistry involves some sort of mean-field theory.

Mean-field theories do not take into account fluctuations, dynamic or spatial.
Basically, a many-body problem is reduced to a one-body problem.

A good mean-field theory can win you a Nobel Prize. That's what Anderson, BCS, Ginsberg, Abrikosov, and Leggett all did!
Can you think of others?

However, mean-field theory does have its limitations.
It is usually quantitatively wrong. It often gives unreliable values for transition temperatures. In spatial dimensions less than four, mean-field theory gives the wrong values for the critical exponents near a phase transition.

An even bigger problem is that mean-field can be qualitatively wrong.
For many models (e.g. the Ising model or Heisenberg model) mean-field theory always gives a transition from a disordered to an ordered phase at a non-zero temperature.
However, in one dimension the Ising model has no phase transition in one dimension. For a Heisenberg ferromagnet or antiferromagnet, there is no transition at finite temperature in two dimensions.
The Mermin-Wagner theorem states that in two dimensions a superconductor or superfluid never has long-range order at finite temperature. Instead, there is a Kosterlitz-Thouless transition, to a distinct state of matter, with power-law correlations.

Mean-field theory can also fail to predict the existence of states of matter. For example, for Hubbard models, mean-field theory can produce several states: a Fermi liquid metal, a ferromagnetic metal, an antiferromagnetic metal, and a spin-density-wave insulator. But it is quite possible the model also can have non-magnetic Mott insulating phases, superconductivity, non-Fermi liquid metals, and pseudogap states.

In the next post, I will discuss some issues that arise in mean-field theories used in modeling the COVID-19 epidemic.

Was Landau the first condensed matter theorist?

Expert readers: please note this post is written for the general audience of a Very Short Introduction. General comments welcome.

Condensed matter physics is not just defined by the objects it studies: condensed states of matter. Rather, the field is also defined by a particular approach. The focus is on finding unifying concepts and organizing principles to address fundamental questions concerning a wide range of phenomena in materials that are chemically and structurally diverse. This approach means looking at the different scales (length, time, and energy) associated with phenomena. In particular, CMP often looks at scales intermediate between the macroscopic and atomic scales. I argued before, that in this sense Kamerlingh Onnes was the first condensed matter experimentalist. In a similar sense, Lev Landau (1908 - 1968) is arguably the first condensed matter theorist, with three papers that he published in 1937, marking the beginning of theoretical CMP.

Landau lived in the Soviet Union and his 1937 papers were almost his last because in 1938 he was arrested for comparing Stalinism to Nazism. The Institute Director, Pyotr Kapitsa, personally wrote to Stalin to no avail. After a year Kapitsa wrote to Molotov (of cocktail fame!), then the nominal head of government, arguing that Landau was indispensable to ``clear up one of the most puzzling areas in modern physics."  In the year following his release Landau developed a theory to explain many of the experiment results on superfluid helium that Kapitsa had obtained. (A nice thank you present!) Landau made notable contributions in all areas of theoretical physics, not just condensed matter. Wikipedia lists more than twenty separate entries describing results, equations, or phenomenon that bear Landau's name. With his former student, Evgeny Lifshitz, Landau co-authored a classic nine-volume series, Course in Theoretical Physics, that is still a standard reference today. Landau also founded a School of Theoretical Physics that produced a plethora of distinguished theoretical physicists. Tragically, Landau’s scientific career ended after a terrible car accident when he was fifty-two years old. He died six years later from injuries associated with the accident. In 1962, Landau was awarded the Nobel Prize in Physics for his work on the theory of superfluidity.

                                              Landau and Kapitsa in 1948.

Landau’s first 1937 paper was concerned with developing the simplest possible theory that could describe the properties of a material near a critical point in the phase diagram, such as associated with a liquid-gas transition or a ferromagnet. A key assumption was that most of the microscopic details, such as the chemical composition of the material, don’t matter much. Landau introduced an order parameter to quantify the amount of ordering present and the symmetry of the ordering. The order parameter varies with temperature and other external parameters such as pressure or magnetic field. It is only non-zero in the ordered state. Landau wrote down the simplest form for the (free) energy of the system as a function of the order parameter. It turns out that symmetry significantly constrains the possible forms for this function. Furthermore, the form is qualitatively different for temperatures above and below the critical temperature. From this simple theory, Landau obtained results for how the order parameter varies with temperature and how there should be a jump (discontinuity) in the specific heat at the critical temperature. What was particularly important was the idea of universality: that most of the microscopic details did not matter and that a wide range of materials and states of matter should have similar properties. Furthermore, the ideas in this paper were foundational for the important ideas about the critical point.

A significant achievement of Landau’s approach to phase transitions was that in 1950, together with Vitaly Ginsburg (1916 - 2009), Landau proposed a theory that could describe many of the properties of superconductors, including how they behaved in the presence of a magnetic field and in thin films. For this work, Ginsburg shared the Nobel Prize in Physics in 2003. Although the Ginzburg-Landau theory could explain a wide range of superconducting phenomena, it left many questions unanswered, including the actual nature and origin of the ordering associated with the superconducting state.

The Ginzburg-Landau theory suggested that the relevant symmetry was a particular symmetry associated with electromagnetism: gauge symmetry. This is a rather abstract concept, but one can give a simple example that may help. With regard to electricity we are familiar with voltage: for example, a 9-volt battery, or a 240-volt appliance. The voltage refers to the electric potential energy; the larger the voltage the stronger the electrical driving force. Voltages are all relative, i.e., they are defined relative to some reference. What is physical is differences in voltage. This is similar to how gravitational potential energy (or elevation) is always defined relative to some reference height, e.g. the floor of the room, sea level, the center of the earth. There is also a gauge symmetry associated with magnetic fields and quantum theory but these are both more complicated and abstract. Later I will discuss experimental manifestations of this breaking of gauge symmetry.

[I am mindful that there are many subtleties about what the ordering and the broken symmetry actually are. For example, this is a breaking of a global gauge symmetry not a local one (which is not allowed by Elitzur's theorem). However, such subtleties are beyond a general audience].

Do you agree that in the sense I discuss Landau was the first condensed matter theorist?
Perhaps it should be van der Waals?

Any corrections?

Any suggestions on how to make this more accessible to a general audience?

My tentative answers to some big questions about CMP

In my last post, I asked a number of questions about Condensed Matter Physics (CMP) that my son asked me. On reflection, my title ``basic questions" was a misnomer, because these are actually rather profound questions. Also, it should be acknowledged that the answers are quite personal and subjective. Here are my current answers.

1. What do you think is the coolest or most exciting thing that CMP has discovered? 

Superconductivity.

explained?

BCS theory of superconductivity.
Renormalisation group (RG) theory of critical exponents.

2. Scientific knowledge changes with time. Sometimes long-accepted ``facts''  and ``theories'' become overturned.  What ideas and results are you presenting that you are almost absolutely certain of? 

Phase diagrams of pure substances.
Crystallography.
Landau theory and symmetry breaking as a means to understand almost all phase transitions.
RG theory.
Bloch's theorem and band theory as a framework to understand the electronic properties of crystals.
Quantisation of vortices.
Quantum Hall effects.
Emergence.

What might be overturned?

I will be almost certain of everything I will write about in the Very Short Introduction. This is because it centers around concepts and theories that have been able to explain a very wide swathe of experiments on diverse materials and that have been independently reproduced by many different groups.
I am deliberately avoiding describing speculative theories and the following.
Ideas, results, and theories based on experiments that did not involve the actual material claimed, involved significant curve fitting, or large computer simulations.
Many things published in luxury journals during the last twenty years.

3. What are the most interesting historical anecdotes? 

These are so interesting and relevant to major discoveries that they are worth including in the VSI.
Graphene and sellotape.
Quasi-crystals.
Bardeen's conflict with Josephson.
Abrikosov leaving his vortex lattice theory in his desk drawer because Landau did not like it.

What are the most significant historical events? 

Discovery of x-ray crystallography
Discovery of superconductivity.
Landau's 1937 paper.
BCS paper.
Wilson and Fisher.

Who were the major players?

They are so important that they are worthy of a short bio in the text.
Onnes.
Landau.
Bardeen.
Anderson.
Wilson.

4. What are the sexy questions that CMP might answer in the foreseeable future?

Is room-temperature superconductivity possible?

Metastability and first-order phase transitions

One of the simplest examples of a first-order phase transition is occurs in a ferromagnet at a temperature below the critical temperature and in an external magnetic field. The transition occurs when the field is varied so that it changes sign.

This can be described in terms of the following Landau free energy where H is the external field and r is negative.

One observes hysteresis as for non-zero H there is a metastable state.
The order parameter phi versus H is shown below

The boundaries of the region of metastability are defined by the field Hc given by

The above description is taken from a review article by Kurt Binder.
I have never seen this in a textbook.
Have you?
Any clear detailed presentations of this topic would be appreciated.

Lecture on Fermi liquid theory

Last week I gave a single lecture on Fermi liquids to the honours year undergraduate course PHYS4030 Condensed Matter Physics.
The slides do not include the expression for the quasi-particle lifetime or the rough argument  (based on phase space considerations) used to justify its form. This is because I do that on the whiteboard.
Perhaps such an important topic justifies more than one lecture...

Ranking Landau

I have decided I should start incorporating a little more history and biographies into my lectures. It is important that students learn something about the "giants" who have paved the way for us.
Since I just taught about "Landau levels" I had a one power point slide biography of Landau. Here is my ranking, for both originality and impact, of Landau's major achievements:

1.  Theory of continuous phase transitions 
2.  Fermi liquid theory
3.  Ginzburg-Landau theory of superconductivity
4.  Theory of superfluidity
5.  The Course in Theoretical Physics

I don't think Landau levels or Landau damping was on the scale as the above achievements. Other people would have worked these out fairly soon if he had not. The main thing that was impressive about "Landau levels" was that he did it when he was 22! and that he sort of "predicted" de Haas van Alphen oscillations in the same year they were observed.

1. has proven to be incredibly important not just in providing a unifying framework for condensed matter but also for broken symmetry in particle physics. It also led to the renormalisation group.

2. is the basis for understanding not just elemental metals but also nuclear physics. Furthermore, it showed the power of using quantum field theoretical techniques based on Green's functions to understand quantum many-body physics.

In preparing I learnt that Landau kept a list ranking system of other theoretical physicists. According to Wikipedia:

Landau kept a list of names of physicists which he ranked on a logarithmic scale of productivity ranging from 0 to 5. The highest ranking, 0.5, was assigned to Albert Einstein. A rank of 1 was awarded to "historical giants" Isaac Newton, Satyendra Nath Bose, Eugene Wigner, and the founding fathers of quantum mechanics, Niels Bohr, Werner Heisenberg, Paul Dirac and Erwin Schrödinger. Landau ranked himself as a 2.5 but later promoted himself to a 2. David Mermin, writing about Landau, referred to the scale, and ranked himself in the fourth division, in the article My Life with Landau: Homage of a 4.5 to a 2.

I must say I think Landau was being modest. Due to the significance of 1. and 2. above, I would rank Landau above Bose, Wigner, and Bohr. As I have blogged before as much as Bohr advanced our understanding of quantum theory I think he also retarded it in several respects.

                   Bohr and Landau in Moscow in 1961
How would you rank these "greats" and Landau's achievements?

Postscript. 2 October 2020. I recently learned that at Landau's 50th birthday party, his colleagues gave him two stone tablets engraved with Landau's "ten commandments", based on his ten most important papers.

Density matrix (1927); Landau diamagnetism (1930); dynamics of ferromagnets (1935, written with Evgenii Lifshitz); theory of phase transitions (1937); intermediate state of superconductors (1937); statistical theory of nuclei (1937); theory of superfluidity (1941); renormalization of electron charge in quantum electrodynamics (1954, with Alexei Abrikosov and Isaac Khalatnikov); theory of Fermi Liquid (1956); and two-component neutrino theory (1957).