I'm applying to PhD programs in physics. However, there are no good research opportunities at my school. I have produced no original results that I can talk about so far, and I'm worried that I will be competing with candidates who have several publications, original results, and so forth. However, I independently studied some subjects in depth. For instance, condensed matter theory topics, such as topological order, the quantum hall effect, the renormalization group technique which, I think, are normally studied at graduate not undergraduate level. Is it appropriate to elaborate in my statement of purpose at length about the topics that I have studied and the calculations that I have done even if these are not original results of mine? Otherwise, I will not have much to say about my academic experience.
What you want to do above all is to make your SoP about the future, not the past. Find a way to include your self study in your CV, say in a special section. Especially if you can say something brief but credible about it: online courses, work of authors x, y, z, correspondence with ....
Then, if you have some backup from the CV, and I don't think it needs to be a lot, then you can say in the SoP how that prior study supports your future plans.
Your admission to the program won't be entirely about your past. A large part of the considerations will be about people's predictions about your future and how successful you are likely to be. So try, as much as possible in the SoP to make that case, not your past accomplishments.
Try to balance it a bit between CV and SoP so that each covers what it is intended to cover and makes your strongest case overall.