YU-NO: A Girl Who Chants Love at the Bound of This World

Takuya Arima lost his mother when he was very young and his archeologist father in a much more recent cave-in, so he now lives with his stepmother Ayumi, who manages an unpopular construction project near the archeological site his father had been studying. Strange events, including accidents and unexplained lightning strikes, have been happening, but things get personally weird for Takuya when first a bizarre relic and then a treatise on parallel worlds gets shipped to him, apparently from his deceased father, all at the same time that a new transfer student arrives. Stranger still, he briefly meets a naked, elfin-eared girl at a time and place specified by his father, while one of father's research associates demands the relic he has at gunpoint and insists that his parents are alive but somewhere else. After more lightning strikes lead to an apparent time-and-space shift, no one else present at the incident seems to remember it the next day.

Synopsis from Anime News Network