On the universality of Landau theories

There is a lot of buzz going on about topological materials and the quantum Hall effect these days that mark the 40 years since its discovery. If you know a few things about topological materials, you will definitely know that the theory behind them is about a macroscopic mechanism that originates from microscopic (quantum) effects.

There is a similar class of problems, maybe less famous at the present time, namely that of polarization in materials, which has its own counterpart “The modern theory of polarization” that was developed in the ’90s. What these two have in common is exactly the emergence of macroscopic phenomena from microscopic ones.

What I find amazing, is how both phenomena have their own Landau theory associated with them. For the latter it is the Landau-Devonshire and Landau-Ginzbug variations, which are both related to the symmetry of materials. Symmetries are also at the center of research especially in computational science with both high-throughput calculations and Machine Learning algorithms.

If an external force is applied to an electron, we find (omitting the proof) that this force is equal to \hbar\frac{d\mathbf{k}}{dt}, where t is the time that the force is applied. This relation leads to the definition of its effective mass \frac{\hbar^2}{\frac{d^2E}{d\mathbf{k}^2}}.

In simple words, there is an inherent relationship between the crystal potential energy as a force that acts on the electron that is incorporated in the models with the use of k, the crystal quantum number (crystal momentum) and through which, we can express the velocity of the electron using \frac{1}{\hbar}\frac{dE}{dk}.

On the bigger scale, that is, when it comes to how the electron behaves in relation to its environment (i.e. the nucleus), we know that the wavefunction acquires from elementary adiabatic perturbation theory, a first-order correction

\left| \delta \psi_{n\textbf{k}}\right> = -i\hbar\dot{\lambda}\sum_{m\ne n}\frac{\left<\psi_{m\textbf{k}}|\partial_{\lambda}\psi_{n\textbf{k}}\right>}{E_{n\textbf{k}}-E_{m\textbf{k}}}\left|\psi_{m\textbf{k}}\right>

where \dot{\lambda} = \frac{d\lambda}{dt} is the change of the parameter that changes slowly with time.

As a result of this, there is a first order current arising from the entire band, which is also equal to the change of polarization in the case where the material that we examine presents spontaneous polarization.

And here we start talking about the Landau family of models that are related to the symmetries of the material and appear to be so important for the description of phase transition of materials, as from our point of view, and in the way that the theories are built, all the electrons seem to do is respond to the forces that thread them.

You can find more information about spontaneous polarization and ferroelectrics in this nice book:

K. M. Rabe, C. H. Ahn, and J.-M. Triscone, Physics of Ferroelectrics, vol. 105. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007.

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