This page contains materials for teaching condensed matter theory concepts to a broad audience, particularly experimental graduate students. These can either be used for self-study, or as part of a course. This page will be updated with more material shortly.
Landau theory
Landau and Landau-Ginzburg theories provide an excellent phenomelogical approach to understanding phase transitions and classical critical behavior based on symmetry-breaking. The basic concept is that a phase transition involves the development of an order parameter, M that lowers the symmetry of the system, and that the free energy can be written as a functional of this order parameter, F[M]. For temperatures close to the phase transition, F[M] can be expanded in powers of M, and then minimized with respect to M to find the value of the order parameter as a function of temperature, field etc. Then a wide variety of thermodynamic quantities can be calculated, and used to compare to experiment. In Landau theories, the order parameter is assumed to be uniform, while Landau-Ginzburg theory includes spatial fluctuations of the order parameter.
There are many sets of lecture notes on Landau theory available in textbooks [1-3] or on the web. Here, I take a different approach, which is to provide jupyter notebooks that implement increasingly complex Landau theory problems, where you can play around with the effects of different parameters and see how to calculate a wide variety of responses that can be connected to experimentally measurable quantities.
What you will need: In order to run these notebooks, you will need python3, numpy, matplotlib and jupyter notebook installed on your computer. If you are not already familiar with python, I would suggest installing Anaconda, an open source package that can be installed on Windows, Mac or Linux, which contains all of the necessary packages.
Notebooks:
Introduction to Landau theory - single component order parameters.ipynb
This notebook introduces the single-component order parameter, using an Ising magnetization as a simple example. You will calculate the free energy, magnetization, entropy, specific heat and magnetic susceptibility and see their critical behavior.
Landau theory for two-component order parameters.ipynb
This notebook introduces Landau theories with two coupled order parameters, which may be two components of the same order parameter, or two distinct order parameters.
More complicated Landau examples.ipynb
Finally, we cover the more complicated case of three order parameters, where we use the example of a two-component antiferromagnetic order parameter, (Nx,Ny) in square symmetry and the induced C4 lattice distortion as a third order parameter. We discuss the effect on the magnetic susceptibility tensor, anisotropic thermal expansion and magnetostriction.
References
[1] Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Chapters 5,6
[2] Coleman, Introduction to Many Body Physics Chapter 11
[3] Dove, Structure and Dynamics Chapter 12
This work was supported by NSF CAREER grant CMMT-DMR-1555163.