All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, it should be noted that Islam urged the Muslims to be brothers to each other and love each other, especially those who are relatives; Allaah Says (what means): {The believers are but brothers.}[Quran 49:10] Abu Hurayrah narrated that the Prophet said: “You will not enter Paradise unless you believe, and you will not believe unless you love each other. Shall I tell you about something if you do it you will love each other, spread the Salaam amongst yourselves.” [Muslim, At-Tirmithi, Abu Daawood and Ibn Maajah] For more benefit on the excellence of keeping ties with kinship, please refer to Fataawa 82547 and 83096.
As regards the issue which you mentioned in the question, then the answer will be as follows:
1- The rumors and lies that you mentioned that your sister spread against you: It is your right to defend yourself and refute what she attributed to you falsely and unjustly, and you should clarify that you are innocent of it.
2- With regard to your sister, in principle, she is innocent of what is attributed to her that she spread these lies against you. In principle a Muslim is innocent until the contrary is proven. In any case, it is better if you forgive and forget the past. Investigating this matter may make your dispute with her more complicated and this may lead to unpleasant consequences. For more benefit on the excellence of forgiving, please refer to Fatwa 84351.
Finally, it is not permissible for the sister to cut ties with her brother or desert him, or prevent him from entering her house without a sound reason. Therefore, we advise you to reconcile with her, and if she insists on preventing you from visiting her, then there are many means of keeping ties with her, like telephoning her and the like.
Allaah Knows best.