Sabbatical in mathematics in the US
The main thing to keep in mind is that from the host department’s point of view they aren’t inviting you to come “do a sabbatical” - that’s not an official status at any US department I know. They are inviting you to come and do a certain kind of work (usually with a title such as Visiting Professor/Researcher/Scholar, which is usually but not always unpaid), whether it’s teaching or research or some combination of the two. The fact that you’re on sabbatical from your home institution (as opposed to some other status like being on leave) is immaterial.

is it necessary to have people interested in a collaboration with them in the department they wants to visit?

I think it’s fair to say that it’s necessary to give the department a reason to invite you. They need to have some concrete expectation that your presence will be beneficial to them. The most common such reason (in my very anecdotal experience) is that you will collaborate with some of the department’s faculty, but if you’re offering to teach, run a seminar, mentor graduate students etc, those could all be useful things that might make the department receptive to inviting you. And a collaboration might not necessarily be very concrete - it may be that you are simply an expert in some field that’s of interest to someone and they feel that having you around would be nice for them and their grad students, postdocs etc, regardless of whether there is an official intent to collaborate.
It could also be that you are a famous researcher at the top of your field. In that case I doubt you will need to offer anything concrete. Just say “I was wondering if it would interest your department to have me visit next year”... Your reputation will do the convincing for you.

Is it possible to find a department for sabbatical in order to start working in a new field?

I don’t know, but I don’t see why not. Again, the key is that you have to give people a reason to want to have you around. If you want to expand from field X to field Y and the department has people who are experts on Y but are interested in learning more about X (or about techniques from X that might help them attack the problems of Y) and think you could help them, I think that could be a win-win for everyone and a perfectly good reason to invite you for a visit.