Yamibo: Darkness, The Hat and the Travelers of the Books

ヤミと帽子と本の旅人

Hatsuki is a highschool student living with her sister, Hatsumi, who she has a huge crush on. On Hatsumi's 16th birthday, she is suddenly surrounded by a green light and disappears in front of Hatsuki. She manages to follow Hatsumi with the help of a being resembling a fat baby chick (literally), ending up in a place called "The Great Library", which is full of different worlds stored in books. Hatsumi wasn't there, though, so the search for Hatsuki's great love begins and involves traveling from book to book. (Source: AniDB)