Monday, December 10, 2012

A key concept in condensed matter: energy scales

To the experienced this post may seem a bit basic but I think it does concern something really important that students must learn and researchers should not forget.
It is a very simple idea but when continually applied it can be quite fruitful. Understanding and teaching condensed matter became a lot easier when I began to appreciate this.

In considering any phenomena in condensed matter it is important to have good estimates (at least within an order of magnitude) of the different energy scales associated with different interactions and effects.

I give several concrete examples to illustrate.

To understand why Fermi liquid theory works so well for elemental metals (sodium, magnesium, tin, ...) the first step is estimating the Fermi energy, the thermal energy (k_B T), the Zeeman energy in a typical laboratory field, ...

A step towards the BCS theory of superconductivity was appreciation of the profound disparity of energy scales, condensation energy much less than k_B T_c comparable to the energy gap, much less than a phonon energy, which in turn is much less than the Fermi energy.
Similarily in the Kondo effect one has the emergence of a low energy scale that is much less than the Fermi energy and the antiferromagnetic Kondo coupling J.

In my own research this issue was a key step in realising that the metallic phase of organic charge transfer salts was a bad metal and could be described by dynamical mean-field theory of the Hubbard model. Specifically it was a puzzle as to why the thermal energy at which the Drude peak disappeared was so much less than the Fermi energy. I first discussed the issues here.

Furthermore, I often find that this simple approach can often rule out exotic phenomena that theorists propose or simplistic explanations that experimentalists make. For example, this post discusses how phenomena discussed in several theory papers require magnetic fields orders of magnitude larger than laboratory fields.

Some may say this skill and approach is important in any area of physics (e.g. fluid dynamics, nuclear physics, optics, ...). However, I suspect it is even more crucial in condensed matter because of the incredible diversity of interactions and emergent phenomena and the associated diversity of energy scales,

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