I just got a Pi 3, model B. It's running raspbian, and I have it on power and Ethernet only. I have internet connection when on the Pi itself, and can SSH into it for up to ~30 minutes after I last touch it. However, if I leave it running with the monitor disconnected, then it no longer accepts connections. The IP remains correct, and if I return to my room and stir the Pi, then it will respond to SSH again.
I'm new to the Pi. Does it have some kind of sleep mode? Or am I losing my connection from the router.