COMPLETE Derivation of Heisenberg and Hubbard Models?

[maverick_starstrider]

Does anyone know of a COMPLETE derivation of the Hubbard Model and then the Heisenberg model from it. What I mean by complete is pedagogical including all (or at least most) steps. Books like Assa Auerbach's and Altland and Simons are worthless for these kind of things (in fact IMHO those books are worthless to learn ANYTHING from), they generally pull out some somewhat arcane perturbation technique and transformation and matrix relation out of the ether and then skip all steps and write down the "answer". This seems to be par for the course in all expositions I've found. Furthermore every "derivation" I've seen is only for a two state system and ends with the claim that generalizing to N-states is "trivial", anyone know of a place where someone degrades themselves so much to actually attempt this generalization in a textbook or lecture note? I suppose it's a pipe dream but the key questions I'd like to find a source to answer is stuff like:

-What is this esoteric perturbation theory involving projectors, is it equivalent to standard perturbation theory? (If so why bother with it?)
-What is the complete expansion? What are the higher order terms (isn't this where ring exchange comes from)? Why are they discarded?
-How does the derivation change if it's not spin 1/2?

I know it's generally the attitude of most condensed matter books for authors not to stoop so low as to provide such trifling details but is there maybe some lecture notes somewhere or the likes that derives these things?

 

[azntoon]

Unfortunately, I'm not aware of any complete derivations of the Hubbard Model and Heisenberg model from it. However, there are some good resources that can help you better understand the models. For the Hubbard Model, I would recommend the book "Introduction to the Hubbard Model" by D. J. Scalapino. This book provides a comprehensive overview of the model, from its historical development to its various applications in condensed matter physics. It also includes detailed mathematical derivations of the model's equations and their corresponding solutions. The book "Quantum Many-Body Systems: From Strongly Correlated Systems to Quantum Phase Transitions" by Jean-Louis Barrat is also a great resource for understanding the Heisenberg model. The book covers the fundamentals of the model, including its mathematical derivation, as well as its applications in condensed matter physics. It also discusses the various types of spin exchange interactions and their effects on the system's behavior. If you want to go even deeper into the details of the models, I would suggest looking at the lecture notes from some of the recent courses taught by leading researchers in the field. For example, the lecture notes from the 2020 course "Strongly Correlated Electrons: From Localized Spin Systems to High Temperature Superconductivity" by Prof. Piers Coleman are available online and provide an in-depth look into the Hubbard Model and Heisenberg Model. Finally, I would also suggest reading some of the original papers that introduced the models. These papers often contain detailed mathematical derivations and provide valuable insight into the underlying principles of the models.