Learn English – Start a subordinate clause with “thus”

It is appropriate when you have simply flipped the clauses:

I stayed inside because it was raining.

Because it was raining, I stayed inside.

What the teacher was trying to teach was that a a subordinate clause is not a complete sentence:

I stayed inside.

This is a complete sentence.

Because it was raining.

This is not; the conjunction because makes it subordinate and therefore, it requires an associated independent clause.

While beginning a text with the word while is perfectly acceptable, your primary concern, it seems to me, should be its effect on your audience, and whether or not the sentence is sufficiently strong to draw them into what you have to say.

"While electron-electron correlations and interactions . . ." has an informal and slightly chatty feel to it. The sentence sounds as if it would be more appropriate in the middle of a paragraph, and not at the beginning. Here's a hypothetical and illustrative paragraph, which for simplicity's sake contains blanks:

The phenomenon of __ has been well attested in the literature. Scientists and theorists in the field, from John __ in the 1990s to Carol __ early in the new century, have described the phenomenon in elegant detail. While we should be grateful for their efforts to date, we still have a great deal to learn, specifically in the phenomenon's __ . It is my purpose in this book, therefore, to use the discoveries of the past as a springboard for __ , in the hopes of stimulating further explorations into __ . With a little luck and a lot of hard work, perhaps we could be instrumental in ushering in a veritable paradigm shift in this exciting stage of discovery. We are arguably on the cusp of doing just that!

In a perhaps overly simplified explanation of the above, you have the following: a solid and foundational statement, followed by a few prominent examples to buttress your first statement. Then there comes a transitional statement that accomplishes two things. First, it praises the findings of past researchers (assuming their contributions to the field are considered unassailable by your peers), and second, it points to what needs to be done if your community of scientists is to push the envelope of discovery to the next level. That's where your studies in the field, which you plan to explain and develop in your text, become relevant to your readers. Make sense?