Most commonly refer to it as an emotional affair. The emotional affair best corresponds to my affair type #4 (out of 7 unique types of affairs): “I Fell out of Love…and just love being in love.”
Here is an interview of a psychiatrist who talks about the emotional affair.
It is a good overview of the characteristics of an emotional affair, and might be helpful for someone who feels an attraction to someone and questions whether s/he is engaged in this type of affair.
Click here for other blogs on infidelity and emotional affairs.