What is absolute temperature?

The concept and reality of absolute temperature is amazing. It tells us something fundamental about the universe, including physical limits as to what is possible. The existence of absolute temperature is intimately connected with the existence of entropy as a thermodynamic state function. It also hints at the underlying quantum nature of reality.

Aside: Unfortunately, the Wikipedia page on this topic is mediocre and garbled. For example, it continues the myth that temperature is related to kinetic energy.

The zeroth law of thermodynamics allows the definition of empirical temperature. It is an equilibrium state variable that indicates whether a thermodynamic system will remain in the same state upon being brought into thermal contact with another system. Thermometers are systems with a single state variable.

Absolute temperature is a specific temperature scale that is central to thermodynamics and statistical mechanics. 

There are several equivalent definitions of absolute temperature. They start at different points. Except for the first one, the others show that the existence of absolute temperature is intimately connected to the second law and to entropy being an extensive quantity.

This is nicely discussed by Zemansky in chapter 8 of his text Heat and Thermodynamics, Fifth Edition (1968). [This was the text for my second year undergrad thermo course at ANU in 1980. At the time, I did not fully appreciate how profound some of it is. I just enjoyed all the multivariable calculus.] 

1. Ideal gas thermometers.

Consider a fixed mass of ideal gas whose volume is fixed. An ideal gas is defined as any gas at a temperature and pressure much larger than the critical temperature and pressure for the gas-liquid transition. Suppose the system is cooled and heated, and the pressure is measured as a function of the temperature measured by a separate thermometer calibrated by the Celsius scale. The pressure versus temperature curve is a straight line. If this line is extrapolated to zero pressure, this occurs at -273.15 degrees Celsius. The straight line has different slopes for different gases, but they all intercept the x-axis at the same point. Alternatively, one can take the pressure as fixed and measure the volume of the gas versus temperature. Extrapolation to zero volume also occurs at -273.15 degrees. 

This suggests that something special is happening at -273.15 degrees Celsius. One can define a special temperature scale where this temperature is zero. Historically, this was the beginning of the concept of absolute temperature.

However, we should be cautious about this approach. This is just an extrapolation and does not allow for the fact that ideal gases are rather special or that some very different physics might kick in below the critical temperature of helium.

2. The efficiency of Carnot cycles. 

This follows Zemansky (page 208). Consider a Carnot cycle abcda, where b to c and d to a are isothermal processes, between the same two reversible adiabatic surfaces, and involve heat transfers Q and Q_3, respectively. The absolute temperature scale T is defined by 

T/T_3 = Q/Q_3

with T_3 = 273.16, when the process d to a occurs at the triple point of water.

3. Integrating factor for heat

Heat is not a state property. It depends on processes. The first law says Delta Q = Delta U + P Delta V. If we consider a quasi-static process and integrate the heat transfer along the path taken (in state space), the result may depend on the path taken. On the other hand, if one integrates dQ/T, one finds that the result is independent of the path. This can then be used to define a new state variable, the entropy. 

The brief discussion above misses some subtle and profound features that only became clear in the 1960s following the work of Pippard, Turner, Landsberg, and Sears, which was inspired by an axiomatic approach to thermodynamics developed by Caratheodory.

Zemansky states

It is an extraordinary circumstance that not only does an integrating factor exist for the dQ of any system, but this integrating factor is a function of temperature only and is the same function for all systems! This universal character enables us to define an absolute temperature.

4. Applying the second law to a composite system

This treatment follows Schroeder, Thermal Physics (Section 3.1)

Schroeder defines entropy in terms of a multiplicity of states. However, I prefer to define entropy as the state function which tells us whether or not two states are accessible from one another by an adiabatic process. There are multiple possible versions of this empirical entropy state function, but let's choose one that is extensive, i.e., scales with the mass and volume of the system.

Consider an adiabatically isolated system containing an internal partition through which the conduction of heat can occur. Denote the two parts of the system by A and B. The entropy of each part can be written as a function U of its internal energy. 

The total entropy of the system can be written 

S = S_A (U_A) + S_B (U_B)

If the system is in thermal equilibrium, by the second law, the entropy of the whole system must be a minimum as a function of U_A and U_B.

Now, dU_A = - dU_B as the composite system is adiabatically isolated. Hence, we have.

The left-hand (right-hand) side of the equation only depends on the properties of system A (B). Thus, it is an intensive state variable which determines whether the system will be in equilibrium with another system. Hence, by the zeroth law, it defines a temperature scale.

T is the absolute temperature.

What is temperature?

Temperature is NOT the average kinetic energy.

When I taught thermodynamics to second year undergraduates one of the preconceived notions that was hard to dislodge from students was that temperature IS a measure of the average kinetic energy of the atoms or molecules in a system.

First, I will give the merits of this view and then explain why it is problematic.

A profound and important insight from Maxwell's kinetic theory of ideal gases was that the average kinetic energy of the atoms/molecules in the gas is related to the absolute temperature defined by Kelvin. This result was important because it provided a microscopic basis for Joule's discovery of the mechanical equivalence of heat.

The result does not just hold for an ideal gas. Classical statistical mechanics can be used to show that for any system of interacting particles, the average kinetic energy of each particle is 3/2 kT. The proof proceeds in the same manner as the equipartition theorem. In the partition function, the integral over momentum factorises and can be evaluated exactly as it is Gaussian integral.

However, this simple relationship between temperature and kinetic energy does not hold for quantum systems. Consider the case of a harmonic oscillator, with frequency omega. By the virial theorem, the average kinetic energy is equal to the average potential energy. Thus, the average kinetic energy is half of the internal energy U(T), which is a universal function f(T/omega). Thus, if we compare two oscillators with different frequencies, at the same temperature, they will have different kinetic energies.

This problem is not just some quantum exotica that is only relevant at extremely low temperatures. Most solids are "quantum" at room temperature because they have a Debye temperature in the range of 200-1000 K.

Temperature is a macroscopic variable, not a microscopic one. It should be defined in terms of the zeroth law of thermodynamics.

Temperature is a state variable associated with a system in thermal equilibrium. It tells us whether that system will be in thermal equilibrium with another system. Consider two separated systems with temperatures T1 and T2. If they are brought into thermal contact, their states will not change if and only if T1=T2.

A thermometer is a system with a single state variable. The value of that variable is an empirical temperature.

Aside. This view of temperature was used by Planck in his book, Treatise on Thermodynamics, first published in 1905.

I am thankful to my undergraduate mentor, Hans Buchdahl for teaching me that thermodynamics is conceptually coherent and beautiful. 

This discussion illustrates that temperature is an emergent property. It is a property of a macroscopic system that the parts of the system do not have. The temperature is independent of the microscopic composition of the system or its history. This universality is a characteristic of many emergent properties.

In another post, I hope to explain what the absolute temperature, first introduced by Kelvin, is.

10 key ideas about emergence

Consider a system comprised of many interacting components. 

1. Many different definitions of emergence have been given. I take the defining characteristic of an emergent property of the system is novelty, i.e, the individual components of the system do not have this property.

2. Many other characteristics have been associated with emergence, such as universality, unpredictability, irreducibility, diversity, self-organisation, discontinuities, and singularities. However, it has not been established whether these characteristics are necessary or sufficient for novelty.

3. Emergent properties are ubiquitous across scientific disciplines from physics to biology to sociology to computer science. Emergence is central to many of the biggest scientific challenges today and some of the greatest societal problems.

4. Reality is stratified. A key concept is that of strata or hierarchies. At each level or stratum,  there is a distinct ontology (properties, phenomena, processes, entities, and effective interactions) and epistemology (theories, concepts, models, and methods). This stratification of reality leads to semi-autonomous scientific disciplines and sub-disciplines.

5. A common challenge is understanding the relationship between emergent properties observed at the macroscopic scale (whether in societies or in solids) and what is known about the microscopic scale: the components (whether individual humans or atoms) and their interactions. Often a key (but profound) insight is identifying an emergent mesoscopic scale (i.e., a scale intermediate between the macro- and micro- scales) at which new entities emerge and interact with one another weakly.

6. A key theoretical method is the development and study of effective theories and toy models. Effective theories can describe phenomena at the mesoscopic scale and be used to bridge the microscopic and macroscopic scales. Toy models involve just a few degrees of freedom, interactions, and parameters. Toy models are amenable to analytical and computational analysis and may reveal the minimal requirements for an emergent property to occur. The Ising model is a toy model that elucidates critical phenomena and key characteristics of emergence.

7. Condensed matter physics elucidates many of the key features and challenges of emergence. Unlike brains and economies, condensed states of matter are simple enough to be amenable to detailed and definitive analysis but complex enough to exhibit rich and diverse emergent phenomena.

8. The ideas above about emergence matter for scientific strategy in terms of choosing methodologies, setting priorities, and allocating resources.

9. An emergent perspective that does not privilege the parts or the whole can address contentious issues and fashions in the humanities and social sciences, particularly around structuralism.

10. Emergence is also at the heart of issues in philosophy including the nature of consciousness, truth, reality, and the 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.

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.

Ising model basics

The Ising model is a paradigm in both statistical mechanics and condensed matter physics. Today for most theorists it is so familiar that some of its historical and conceptual significance is lost.
Previously, I posted about what students can learn from computer simulations of the Ising model.

If you had to talk about the Ising model to an experimental chemist what would you say?
[Last week I had to do this].

The Ising model is the simplest effective model Hamiltonian that can describe a thermodynamic system that undergoes a first-order phase transition and has a phase diagram containing a critical point.

On each site i of a lattice one defines a spin sigma_i= +1 or -1, representing spin up or spin down.

The Hamiltonian H is

J_ij describes the interaction between spins on sites i and j. In the simplest version the interactions are only between nearest neighbours, and have the same value J.
h is the external magnetic field.

If J is positive, the ground state at h=0 is a ferromagnet.
If J is negative, the ground state at h=0 is an anti-ferromagnet for a bipartite lattice.

[Caution: just like for the Heisenberg model, some authors define the Hamiltonian with the opposite sign of J].

For h=0 there is a critical point at a finite temperature Tc, for lattices of dimension two and higher.

The spins sigma_i= +/- 1 defined at each lattice site i, were originally to represent the atomic magnetic moments in a ferromagnetic material. However, the sigma's can represent any two states of the site i. For example, the ``spin'' or pseudo-spin can represent the presence or absence of an atom or molecule in a ``lattice gas'', atom A or atom B in a binary alloy (mixture), or the low-spin and high-spin states in a spin-crossover material.

The mean-field theory of the Ising model is mathematically equivalent to the thermodynamic theory of binary mixtures with an entropy of an ideal mixture.
There is a nice discussion of such mixtures in Section 5.4 [and the associated problems] of Introduction to Thermal Physics by Schroeder.
[Here are the slides for a lecture I have given based on that text].
Chapter 15 of the text by Dill and Bromberg is also helpful as it has more detail.
Neither text makes an explicit connection to the Ising model. Following this paper on alloys, one has

This is shown in Section 8.1.2 of James Sethna's text, Statistical MechanicsEntropy, Order Parameters and Complexity.

When interactions beyond nearest-neighbours are included in the Ising model or when the lattice is frustrated (e.g. fcc or triangular) a richer phase diagram is possible. Examples include the ANNNI model and some models for spin-state ice considered by Jace Cruddas and Ben Powell.

Symmetry is the origin of all interactions

In Phil Anderson's review of Lucifer's Legacy: The Meaning of Asymmetry by Frank Close, Anderson makes the following profound and cryptic comment.

In a book focusing, as this does, on symmetry, it seems misleading not to explain the fundamental principle that all interaction follows from symmetry: the gauge principle of London and Weyl, modelled on and foreshadowed by Einstein's derivation of gravity from general relativity (Einstein seems to be at the root of everything). The beautiful idea that every continuous symmetry implies a conservation law, and an accompanying interaction between the conserved charges, determines the structure of all of the interactions of physics. It is not appropriate to try to approach advanced topics such as electroweak unification and supersymmetry without this foundation block.

To see how this plays out in electrodynamics see here.

What is condensed matter physics?

How would you answer this question if you were asked by a non-expert who likes to learn and understand new things?
For example, a smart high school student, your uncle who reads a lot of popular science, an academic colleague in sociology, an economics graduate, ...
A draft of my answer is here.
I welcome suggestions for improvements.
Feel free to try it out on people you know who might be interested.

Ten key ideas about condensed matter physics?

I am slowly working towards writing a Condensed Matter Physics: Very Short Introduction.
But first I am trying to clarify my audience and goals. Some earlier posts have helped me clarify this.

My intended audience is probably not you! Rather it is a person who wants to get the flavour of what CMP is actually about.  Examples might include a smart final year high school who wants to study science at university, or a first-year chemistry undergraduate, or an economics graduate, or a sociology professor, ...

My goal is to show that CMP is intellectually exciting, intellectually challenging, and intellectually important.

The VSI format is 8-10 chapters and 30-35 thousand words. It is meant to be written in the style of an engaging essay not a technical paper.

My plan is to basically have one clear and specific idea that I want to communicate in each chapter. I am thinking that in order to increase interest and comprehension that for each chapter I will aim to include.

An easily understandable analogy to illustrate the main idea.
A few relevant and illuminating figures.
An interesting historical anecdote.
An example of a technological application.
An example of cross-fertilisation to another field of science.

So here is the current version of my chapter headings and the main idea(s) I want each chapter to communicate.

1.     What is condensed matter physics?

CMP is concerned with studying and understanding material systems composed of large numbers of atoms. How do the properties of the system emerge from the properties of the constituent atoms and the interactions between them? It is a multi-faceted approach to studying materials and involves a unifying set of concepts. Quite abstract ideas and concepts can be quite powerful for understanding quite practical systems.

2.     A plethora of states of matter

Even for the simplest materials, there is a multitude of different phases, i.e. qualitatively different states of matter. Transitions between distinct phases are defined by discontinuities in properties. Phase diagrams encode what phase is stable under specific external conditions such as temperature, pressure, and magnetic field.

3.     Symmetry matters

Distinct phases are associated with distinct ordering of the system. A unifying concept to distinguish and classify different phases and their associated ordering is how they differ in the type of symmetry that they have. What different classes of symmetry are mathematically possible significantly constrains what is physically possible.

4.     The order of things

The type and quantity of order and the broken symmetry in a distinct state of matter can be described by a small set of numbers represented by the “order parameter’’.

5.     Adventures in flatland

Confining a material to one or two dimensions can lead to new states of matter. Furthermore, imagining a world of variable dimension can actually lead to a better understanding of materials in our three-dimensional world.

6.     The critical point: details do not matter

Under a very special set of external conditions, a phase transition is not associated with discontinuous properties. These conditions are represented by the critical point in the phase diagram.  Very different material systems can have the same properties close to the critical point. Understanding this universality requires looking at the system at many different length scales.

7.     Quantum matter

The weirdness of quantum theory is most commonly manifest at the level of single atoms and molecules. Surprisingly, quantum effects can also be seen “with the naked eye” in states of matter such as superconductors and superfluids.

8.     Topology matters

Abstract ideas about shapes help us understand spatially non-uniform broken symmetry states. They also lead to new states of quantum matter, that do not involve broken symmetry.

9.     Emergence matters

Condensed matter physics is all about emergence: the sum is greater than the parts. From a system composed of many interacting components new (often unanticipated) properties, concepts, and organising principles emerge. Reality is stratified.

10.  Future challenges

Almost all new states of matter are discovered by experiment and often by accident rather than being predicted theoretical. An open question and challenge is to what extent one can predict new states or to design materials with specific properties. There are significant open challenges in all facets of CMP: synthesis, characterisation, measurement, computation, and theory. Finally, given the great success of CMP at understanding emergence in complex systems a challenge is to adapt the approach and concepts to other complex systems, ranging from biology to sociology.

I welcome feedback.
But keep in mind the audience.
I do not want to add material, but perhaps even cut material (e.g. chapter 8).

What do you wish non-CMP people understood about CMP?

Generalised rigidity is a key concept

What are some of the most important concepts in condensed matter physics?
In a recent comment on this blog Gautam Menon suggested that one of them is that of generalised rigidity, i.e. the elasticity of order parameters associated with broken symmetry phases.

 A while ago I wrote a post trying to introduce Phil Anderson's discussion of the concept.
Thinking about this made me appreciate just how important and useful the concept is.

Basically, generalised rigidity quantifies how the free energy of a system varies when introduces spatial variations in the order parameter. These variations can result from boundary conditions, fluctuations, or topological defects.

Depending on the type of broken symmetry there are just a few parameters, maybe only one, involved in defining the rigidity. One is looking at "linear" response and so symmetry determines how many different terms one can write down that are second order in a gradient operator.

A concrete example is the Frank free energy density associated with non-chiral nematic crystals.

Here n is a unit vector (the order parameter) and there are just three parameters and K1, K2, and K3. The three terms represent pure splay, bend, and twist, respectively. Spatial uniformities are at the heart of liquid crystal displays.

Historical aside. It is impressive that Frank wrote this down in 1958, without any reference to Landau.

For s-wave superconductivity and superfluids such as 4He, there is just one parameter, known as the superfluid stiffness or superfluid density. This is the coefficient of the gradient term in the Ginzburg-Landau theory and determines the superconducting "coherence length".

The generalised rigidity is also important because it is central to the renormalisation group theory of critical phenomena, which start with Ginzburg-Landau-Wilson functionals (effective actions). For most cases, one discovers that higher order gradient terms are "irrelevant'' to long wavelength properties and the rigidities are renormalised by fluctuations.

The spin stiffness is the only parameter (energy scale) that appears in a non-linear sigma model treatment of ferromagnetism and antiferromagnetism. The model is sufficient to describe all of the long-wavelength and low-energy properties.

The XY model also involves just one parameter, the rigidity.  From this model one gets the Kosterlitz-Thouless transition.

Note that for both the non-linear sigma model and XY model in one and two dimensions there is no long-range order (symmetry breaking) [Mermin-Wagner theorem] yet the rigidity still has meaning.

The rigidity also determines the emergent length scales in these systems, including the size of topological defects such as vortices, skyrmions, disinclinations,....

A nice detailed discussion of much of the above is in Chaikin and Lubensky, particularly chapter 6.

What is condensed matter physics?

What do condensed matter physicists study?

High school students are often taught there are three states of matter: solids, liquids, and gases. However, this is misleading as there are many more states of matter. Liquid crystals, superconductors, and ferromagnets are distinct states of matter that do not fit in the high school classification. Condensed matter physics (CMP) is concerned with practically any material system that involves a large number (say at least a million) of interacting atoms or molecules. We can consider this to be a complex system because there are many different ways of arranging the constituents (atoms or molecules) of the system.

What approaches and techniques do condensed matter physicists use to study and understand these systems?

CMP provides a coherent intellectual framework for a multi-faceted approach to investigate and understand complex material systems.
First, one can look at the material at many different scales ranging from the microscopic level (scale of individual atoms and molecules) to the mesoscopic (roughly thousands of atoms or molecules, micrometer scale) to the macroscopic (what can be seen with the naked eye). The different scales can be different system sizes, length scales, energy scales, and time scales.
At every scale one can use different tools and approaches, which fall into three broad categories: experimental, theoretical, and computational. All three are intellectually and technically challenging. All are important.

Experiment
There are several distinct parts to this.
Synthesis and fabrication: one has to make a sample of the material. This involves chemistry. Making large clean samples is an art in itself.
Characterisation: this concerns testing that one actually has a sample of chemical composition and purity desired.
Property measurement: this concerns determining what the physical properties (for example, crystal structure or electrical resistance) of the sample are. Often one varies external conditions such as temperature, magnetic field, and pressure, and determines how the properties of interest vary with these parameters. Some of the most interesting condensed matter physics happens under extreme conditions: low temperatures, high magnetic fields, or high pressures.

Theory and model building
The fundamental question that one is trying to answer is: How do the material properties emerge from the chemical composition and atomic structure of the material? In particular, what are the physical mechanisms responsible for the different states of matter found in the material? In CMP it is found that these questions are best understood in terms of deciding on the essential system components and  physical interactions between them that occur at different length and energy scales. Constructing (or dreaming up!) the simplest possible model for these interactions is a real art.

Computation
This has several aspects often requiring the use of state-of-the-art supercomputers and algorithms. One is broadly known as quantum chemistry and concerns starting with a knowledge of the basic chemical composition and calculating from quantum theory the properties of the system. In spite of massive advances in computational power and algorithms over the past 60 years, one is still confined to relatively small numbers of atoms and/or unreliable approximation schemes.
The second computational side is calculating properties of the theoretical models that can be compared to experiment. Even for "simple" models usually requires either massive computational power on small systems or unreliable approximation schemes.

Finally, an important challenge is that of intellectual synthesis and critical evaluation. Here, one tries to bring together the results of all these complementary investigations to gain a coherent picture of the material and its properties. Inevitably, there are inconsistencies, sometimes minor and sometimes major. Investigators then have to decide in which element the problem lies.

I think CMP is more complex, challenging, and full of surprises than other areas of physics, such as atomic physics, elementary particle physics, fluid mechanics, and optics. There is a lot more that is unknown in CMP and a lot more that can go wrong.

Why is condensed matter physics important and interesting?

I am trying to get some momentum in writing A Very Short Introduction to Condensed Matter Physics. The intended audience is the intellectually curious person with little background in science. My goal is to convince them that CMP is important and interesting. I can think of several reasons.

1. CMP is intimately connected with everyday technology ranging from liquid crystal displays to computer chips.
2. CMP comprises the majority of physics (employees, papers, conferences, ...) and has significant interaction with areas of science and engineering.
3. CMP is a rich source of creative ideas, concepts, and techniques that represent a significant intellectual achievement and are relevant to many other intellectual endeavors.
4. CMP is full of surprises. We keep discovering new unanticipated phases of matter.
5. CMP presents significant scientific challenges: theoretical, computational, and experimental (from characterisation to sample synthesis).

I am going to focus on 3.
However, it is interesting that the traditional route is 1. Furthermore, different people (including reviewers of the book proposal) are quite divided about 1. versus 3.
[More on that later following this article].

What are the big picture ideas of condensed matter, that are significant intellectual achievements in their own right and particularly relevant to other endeavors?
Here are a few suggestions. It provides very concrete systems to address, at both the mathematical and experimental level, the following issues, which turn out to be often inter-related.

A. Qualitative distinctions are defined by discontinuities. (Different phases of matter).

B. Simple models of complex systems. (Landau theory of phase transitions; Effective Hamiltonians).

C. Universality versus particularity. (Universality classes for critical phenomena).

D. Emergence and the hierarchal nature of reality. (Effective interactions. Renormalisation.)

What do you think are the great intellectual achievements of condensed matter that people need to know about?

Strategies for minimal effective Hamiltonians

An important step in understanding any class of complex materials is to find/discover the simplest possible effective Hamiltonian that can be used to describe the main properties of interest (e.g. a phase diagram).

Doing this well is a non-trivial and subjective process. I am thinking about this because I am currently trying to figure out the appropriate Hamiltonian for spin-crossover compounds.

Here are some key elements of the process. 

"Simplest possible" means having the fewest possible degrees of freedom and parameters.

1. What are the key degrees of freedom (molecular orbitals, vibrations, spins, ...)?

2. What are the key interactions and the associated Hamiltonian?

3. What approximation scheme can be used to calculate properties of the many-body Hamiltonian (ground state, thermodynamics, electronic, magnetic, ...)?

4. How do the calculated properties compare to experiment?

5. Can we estimate the values of the Hamiltonian parameters from the comparison of the calculated properties with experiments? 

6. Can we estimate the values of the Hamiltonian parameters from ab initio electronic structure methods, such as those based on density functional theory (DFT)?

Inevitably, things do not work out perfectly, sometimes qualitatively and always somewhat quantitatively. Then one has to face the difficult task of deciding what the problem is and what the next step is. There are several options.

A. There are some missing degrees of freedom in the original Hamiltonian.

B. There are some missing interactions.

C. The approximation scheme used to calculate properties was not reliable enough.

D. There is a problem with the experiments.

E. This is really the best one can hope to do and you should move on to other problems. i.e, know when to quit and face the law of diminishing returns.

This plethora of options is why falsifiability is so hard in the theory of strongly correlated electron materials. But, it does not mean we should give up on it.

The flow diagram below is one way of looking at the process. Some people like the picture. Others do not. As usual, real science is not quite so algorithmic.

Key ideas in solid state physics

I have had some interesting discussions with an editor at Oxford University Press about the Very Short Introductions series. The upshot is that I have been asked to write a VSI Condensed Matter Physics. I find it amazing and concerning that after 500 titles there wasn't one about CMP. There are excellent ones on Magnetism, Superconductivity, Complexity, and Crystallography.
I am very happy about this and will post more about it later. At first, we discussed a VSI on Solid State Physics. Here is my outline for that.

1. Introduction
    Solid state physics
   - is central to technology (diodes, transistors, LEDs, photovoltaic cells, and computer memories)
   - provides important lessons in scientific model building
   - is one of the largest fields of physics
   - is a rich source of ideas and concepts that have cross-fertilised with other fields of science

2. Solids are quantum matter
Solids are made of atoms (nuclei and electrons).
Electrons are waves. Electrons are fermions. Quantum degeneracy
How is a metal like a white dwarf star?

3. Symmetry matters
Crystal structures. Think in reciprocal space, not in real space.
Why is it possible to determine a crystal structure from x-ray diffraction?
Internal symmetries of electrons: spin, gauge symmetries.

4. Electron waves in a crystal
Bragg scattering. Extended states.
Energy gaps: metals, semiconductors, and insulators
Why is copper a metal while diamond is an insulator?
Why can an electron go through a crystal and pass millions of atoms without being scattered?

5. Multitudes of solid phases
Phase diagrams. Allotropes.
When is graphite less stable than diamond?
Magnetic and superconducting phases
Classifications of phases through "broken symmetry"

6. Emergence
Quasi-particles: electrons and holes, phonons, magnons
How does structure (chemical and crystal) determines electronic and structural properties?
Why does magnesium seem to have positively charged electrical currents?

7. Beyond perfect infinite crystals
a. Impurities, disorder, localisation, glasses: the value of imperfection
b. Flatland. Surfaces and dimensionality

8. Topology matters
Quantum Hall effects, Topological insulators, Quantum magnetism

9. Solid state technology
 Diodes, transistors, LEDs, photovoltaic cells, and computer memories

10. Solid concepts
What have we learned about scientific model building?

This is too much. But what would you add or subtract?

Solid State or Condensed Matter Physics?

The two terms are often used interchangeably, but that is not appropriate. Condensed matter physics does not just involve solids but also phenomena in liquids, liquid crystals, superfluids, and polymer melts.  Solid state physics is a subset of condensed matter physics. The latter term was arguably coined by Phil Anderson, when he and Mott renamed their research group at Cambridge in the 1970s. One can view research fields or course titles as a list of topics or as a way of thinking about certain parts of reality. Solids exhibit rich phenomena including magnetism and superconductivity. However, it is best to actually view the solids as (an almost irrelevant) substrate for the phenomena.

Like many things, this perspective arguably started with Landau. His theory of phase transitions in the 1930s did not consider atomic structure or chemical composition. Even structural phase transitions were viewed in terms of symmetry change, not in terms of explicit microscopic details. In 1950 this led to the Ginzburg-Landau theory of superconductivity. This all suggested a unified approach to phase transitions.
Furthermore, Landau's Fermi liquid theory papers were originally concerned with understanding liquid 3He, not electrons in metallic crystals.

This idea was further highlighted in the 1970s with the study of critical phenomena and the associated idea of universality. Specifically, the critical behaviour of an XY magnet, a superconductor, and a superfluid, are the same (i.e. they have the same critical exponents). The critical behaviour of the liquid-gas transition, an Ising magnet, and the order-disorder transition in a binary alloy are the same. The view that the solid state might actually not be the key feature for understanding and describing superconductivity was highlighted in the 1950s by Fritz London in his two-volume book, Superfluids, which suggested the two phenomena were intimately connected. Beginning in 1968, De Gennes took a condensed matter perspective in applying order parameters and scaling ideas to “soft matter”: liquid crystals, polymers, wetting, …

The important element to this conceptual view of condensed matter is that it provides a unifying perspective on phenomena in a diverse range of materials. It also brings to the fore how a wide suite of powerful theoretical and experimental tools (esp. neutron and x-ray scattering) can be used to study diverse materials. One of the key theoretical strategies is that of effective Hamiltonians, which is not unique to condensed matter, because it just reflects the hierarchy of energy, length, and times scales that result from emergence. This then leads to an intellectually rich interchange of ideas and techniques from other fields of physics, particularly quantum field theory.

More recently, this unity is illustrated by ultracold atomic gases which can be used to study some phenomena that had previously only been studied in solids.

Computational density functional theory (DFT) in a nutshell

My recent post, Computational Quantum Chemistry in a nutshell, was quite popular. There are two distinct approaches to computational approaches: those based on calculating the wavefunction, which I described in that post, and those based on calculating the local charge density [one particle density matrix of the many-body system]. Here I describe the latter which is based on density functional theory (DFT). Here are the steps and choices one makes.

First, as for wave-function based methods, one assumes the Born-Oppenheimer approximation, where the atomic nuclei are treated classically and the electrons quantum mechanically.

Next, one makes use of the famous (and profound) Hohenberg-Kohn theorem which says that the total energy of the ground state of a many-body system is a unique functional of the local electronic charge density, E[n(r)]. This means that if one can calculate the local density n(r) one can calculate the total energy of the ground state of the system. Although this is an exact result, the problem is that one needs to know the exchange-correlational functional, and one does not. One has to approximate it.

The next step is to choose a particular exchange-correlation functional. The simplest one is the local density approximation [LDA] where one writes E_xc[n(r)] = f(n(r)), where f(x) is the corresponding energy for a uniform electron gas with constant density x. Kohn and Sham showed that if one minimises the total energy as a function of n(r) then one ends up with a set of eigenvalue equations for some functions phi_i(r) which have the identical mathematical structure to the Schrodinger equation for the molecular orbitals that one calculates in a wave-function based approach with the Hartree-Fock approximation. However, it should be stressed that the phi_i(r) are just a mathematical convenience and are not wave functions. The similarity to the Hartree-Fock equations means the problem is not just computationally tractable but also relatively cheap.

When one solves the Kohn-Sham equations on the computer one has to choose a finite basis set. Often they are similar to the atomic-centred basis sets used in wave-function based calculations. For crystals, one sometimes uses plane waves. Generally, the bigger and the more sophisticated and chemical appropriate the basis set, the better the results.

With the above uncontrolled approximations, one might not necessarily expect to get anything that proximates reality (i.e. experiment). Nevertheless, I would say the results are often surprisingly good. If you pick a random molecule LDA can give a reasonable answer (say within 20 per cent) of the geometry, bond lengths, heats of formation, and vibrational frequencies... However, it does have spectacular failures, both qualitative and quantitative, for many systems, particularly those involving strong electron correlations.

Over the past two decades, there have been two significant improvements to LDA.
First, the generalised gradient approximation (GGA) which has an exchange-correlation functional that allows for the spatial variations in the density that are neglected in LDA.
Second, hybrid functionals (such as B3LYP) which contain a linear combination of the Hartree- Fock exchange functional and other functionals that have been parametrised to increase agreement with experimental properties.
It should be stressed that this means that the calculation is no longer ab initio, i.e. one where you start from just Schrodinger's equation and Coulomb's law and attempts to calculate properties.

It should be stressed that for interesting systems the results can depend significantly on the choice of exchange-correlational functional. Thus, it is important to calculate results for a range of functionals and basis sets and not just report results that are close to experiment.

DFT-based calculations have the significant advantage over wave-function based approaches that they are computationally cheaper (and so are widely used). However, they cannot be systematically improved [the dream of Jacob's ladder is more like a nightmare], and become problematic for charge transfer and the description of excited states.

Computational quantum chemistry in a nutshell

To the uninitiated (and particularly physicists) computational quantum chemistry can just seem to be a bewildering zoo of multiple letter acronyms (CCSD(T), MP4, aug-CC-pVZ, ...).

However, the basic ingredients and key assumptions can be simply explained.

First, one makes the Born-Oppenheimer approximation, i.e. one assumes that the positions of the N_n nuclei in a particular molecule are a classical variable [R is a 3N_n dimensional vector] and the electrons are quantum. One wants to find the eigenenergy of the N electrons. The corresponding Hamiltonian and Schrodinger equation is

The electronic energy eigenvalues E_n(R) define the potential energy surfaces associated with the ground and excited states. From the ground state surface one can understand most of chemistry! (e.g., molecular geometries, reaction mechanisms, transition states, heats of reaction, activation energies, ....)
As Laughlin and Pines say, the equation above is the Theory of Everything!
The problem is that one can't solve it exactly.

Second, one chooses whether one wants to calculate the complete wave function for the electrons or just the local charge density (one-particle density matrix). The latter is what one does in density functional theory (DFT). I will just discuss the former.

Now we want to solve this eigenvalue problem on a computer and the Hilbert space is huge, even for a simple molecule such as water. We want to reduce the problem to a discrete matrix problem. The Hilbert space for a single electron involves a wavefunction in real space and so we want a finite basis set of L spatial wave functions, "orbitals". Then there is the many-particle Hilbert space for N-electrons, which has dimensions of order L^N. We need a judicious way to truncate this and find the best possible orbitals.

The single particle orbitals can be introduced

where the a's are annihilation operators to give the Hamiltonian

These are known as Coulomb and exchange integrals. Sometimes they are denoted (ij|kl).
Computing them efficiently is a big deal.
In semi-empirical theories one neglects many of these integrals and treats the others as parameters that are determined from experiment.
For example, if one only keeps a single term (ii|ii) one is left with the Hubbard model!

Equivalently, the many-particle wave function can be written in this form.

Now one makes two important choices of approximations.

1. atomic basis set
One picks a small set of orbitals centered on each of the atoms in the molecule. Often these have the traditional s-p-d-f rotational symmetry and a Gaussian dependence on distance.

2. "level of theory"
This concerns how one solves the many-body problem or equivalently how one truncates the Hilbert space (electronic configurations) or equivalently uses an approximate variational wavefunction. Examples include Hartree-Fock (HF), second-order perturbation theory (MP2),  a Gutzwiller-type wavefunction (CC = Coupled Cluster), or Complete Active Space (CAS(K,L)) (one uses HF for higher and low energies and exact diagonalisation for a small subset of K electrons in L orbitals.
Full-CI (configuration interaction) is exact diagonalisation. This only possible for very small systems.

The many-body wavefunction contains many variational parameters, both the coefficients in from of the atomic orbitals that define the molecular orbitals and the coefficients in front of the Slater determinants that define the electronic configurations.

Obviously, one expects that the larger the atomic basis set and the "higher" the level of theory  (i.e. treatment of electron correlation) one hopes to move closer to reality (experiment). I think Pople first drew a diagram such as the one below (taken from this paper).

However, I stress some basic points.

1. Given how severe the truncation of Hilbert space from the original problem one would not necessarily to expect to get anywhere near reality. The pleasant surprise for the founders of the field was that even with 1950s computers one could get interesting results. Although the electrons are strongly correlated (in some sense), Hartree-Fock can sometimes be useful. It is far from obvious that one would expect such success.

2. The convergence to reality is not necessarily uniform.
This gives rise to Pauling points: "improving" the approximation may give worse answers.

3. The relative trade-off between the horizontal and vertical axes is not clear and may be context dependent.

4. Any computational study should have some "convergence" tests. i.e. use a range of approximations and compare the results to see how robust any conclusions are.

The role of simple models and concepts in computational materials science

Today I am giving the first talk in a session on Computational materials science at the 4th International Conference on Advances in Materials and Materials Processing.

Here are the slides for my talk "The role of simple models and concepts in computational materials science".

I will be referring the audience to the article such as those mentioned here, here and here that give a critical assessment of computer simulations and stress the importance of concepts.

I welcome comments, particularly as I think the talk could be stronger and clearer.

A basic quantum concept: energy level repulsion (avoided crossings)

When I learnt and later taught basic quantum mechanics I don't think the notion of energy level repulsion (or equivalently avoided crossings) was emphasised (or even discussed?).

Much later I encountered the idea in advanced topics in theoretical physics such as random matrix theory and in theoretical chemistry  (non-adiabatic transitions and conical intersections).

Yet level repulsion is a very simple phenomena that can be illustrated with just a two by two matrix describing two coupled quantum states, as nicely discussed on the Wikipedia page.

Last semester when I was teaching Solid State Physics I realised just how central and basic the phenomena is and that the students did not appreciate this.

Level repulsion is the origin of several key phenomena in chemistry and physics.

In solid state physics, it is the origin of the appearance of band gaps at the zone boundary and thus the all important distinction between metals and insulators.

Previously, I posted how Chemistry is quantum science because chemical bonding (the lowering of energy due to interacting atoms) arises due to the superposition principle. This could also be viewed as level repulsion.

Another key idea in chemistry is that of transition states and activation energies for chemical reactions. When one uses a diabatic state picture, particularly as emphasised by Shaik and Warshel, the transition state emerges naturally in terms of level repulsion.

The figure is taken from here.

Can you think of any other nice examples?

Teaching students to think like a condensed matter physicist

Yesterday I heard Carl Wieman give a talk at UQ, Taking a scientific approach to science education. I hope I will say more about it later. Here I just want to highlight one helpful point he made concerning relating teaching to the psychology and practise of "becoming an expert".
We need to teach students to "think like a physicist". This is quite different to imparting (memorising) information in textbooks.

Later in the day I taught my class PHYS4030 Condensed Matter Physics which is really the basics of solid state physics, a la Ashcroft and Mermin. I led a discussion with the students about
"What is the conceptual strategy that we are following in this course?"
We came up with something like the following.

1. Define the simplest possible model.
2. Calculate some properties of materials that are predicted by the model.
3. Compare the predicted properties with experimental results. What are the successes and failures of the model?
4. Refine the model in the hope of better agreement with experiment.
5. Repeat the process.

This is what we are doing as we go from Drude to Sommerfeld to Bloch models, and then consider the role of electron-electron interactions.

I stressed that this is not just what we do in this course but this is the general research strategy in condensed matter physics.

Note this strategy is quite different to how one teaches most physics courses; e.g., quantum mechanics and electromagnetism.
The latter is largely an exercise in the applied mathematics of Maxwell's equations. One never really considers whether they are right or not, or need to be modified.