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.