the three fundamental laws of logic: (1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity. These are the three fundamental laws of logic and while they seem to be true all the time, I am wondering if there are certain fields where one of the three laws doesn't quite apply, especially in physics and mathematics. Is this the case?
You will find that there is no "law" of logic that is held universally. And to speak "of logic" confuses the issue because there isn't a single logic but rather numerous logics.
Technically all you need to have a logic is a language (syntax or rules for what counts as a well formed formula), definitions of interpretation (or semantics), operators and their interpretations, and rules of inference.
There is incredible flexibility here and you can be arbitrary although the systems are typically judged by their usefulness in solving problems. This is much like mathematics, which has systems where parallel lines intersect, or where division by zero is allowed.
Graham Priest did a great interview with 3am magazine a few years back on non-traditional logics, but 3am does not seem to keep an archive of their publications. Fortunately a copy can be found at a site maintained by Richard Marshall the journalist who did the interview. The interview is an accessible overview of what paraconsistent logics and dialetheism are and why people care about them.
Francesco Berto wrote a journal article "Meaning, Metaphysics and Contradiction" in American Philosophical Quarterly, Vol 43, no. 4 (October 2006) looking at paraconsistent logics and dialetheism which have both challenged the law of non-contradiction Article available on Jstor.
Quantum physics inspired the need for logics that handle more than binary values or a scale from 0 to 1 (such as various fuzzy logics). This lead to eight valued logics. Here is a journal article on work done on 8-value logics The focus in this article is from a computer science perspective.
Studia Logica: An International Journal for Symbolic Logic Vol. 92, No. 2, Jul., 2009 was an entire volume you may find of use on truth, semantics, and some work on non-traditional logic. Here is a link to that volume/issue
And finally Hegel used the dialectic to challenge Aristotalean assumptions on logic. Here is an article on Hegel's logic
Jumping into professional-level logic is not easy. What I found is only an example of scholarship published in journals on these topics, and it may serve as an entry for you. Good luck!
I am just going to add this : these laws might best be considered axioms, an axiom is not true in the acceptance you're using, rather, it's self-refering; you can say it's truth is out of question, hence you cannot speak of its truth (since truth is, most usally, considered a compliance-relation between a "refering object" and a "refered object" (this is a very weak analogy, it's just to convey the point).
In many ways, your are free to drop an axiom, but in doing so : you'll have then construct "logical coherence" without it, something that has been done is mathematics and other fields.