I am a fourth year PhD candidate in theoretical physics at a U.S. institution. Two and a half years ago I began working with my advisor on a topic I found interesting, despite my limited prior knowledge. A few months into research my advisor experienced a health issue that lead to them taking a semester off from work. Upon their return, I noticed a subtle shift in their research capabilities. For instance, they would assign me problems that seemed peculiar and unreasonable, but I attributed my concerns to my own lack of expertise in the field.
As I became more familiar with the field, I began to realize that the work we were doing was flawed and that my instincts were correct. When I raised these concerns with my advisor, they did not take what I was saying seriously. They insisted on the correctness of our approach, dismissing my scientific objections and refusing to allow me to work on other problems.
I shared my situation with my advisor's only other student, who is working on an independent problem, and they shared similar doubts about our advisor's guidance and their own lack of progress. I also shared my situation with a professor who supervised my master's thesis, and they were surprised by the kind of advisement I have been receiving and agreed that our current work is undoubtedly incorrect. I have also overheard whispers from other faculty members expressing similar concerns, but no one has directly said anything to me. It appears that my advisor is no longer able to do research or advise students like before. I am not sure if this is related to the health issue they faced two years ago but it seems likely.
Given that what we have been doing is incorrect, I have no published papers and have made virtually no research progress despite completing my fourth year. While it took me longer than it should have to recognize the problem, I hope it is not too late for me to graduate with some good results. My field of research is not pursued by anyone at my institution other than my advisor, so my options are either contacting professors at other institutions or changing fields/advisor with someone else in my department.
I would like some advice on how I should handle my situation and complete my degree. How should I handle this with my advisor without ruining my relationship with them? Is it reasonable to approach professors at other universities, and if so, how can I explain my situation to them without tarnishing my advisor's reputation?