I'm going to make a very brief answer. We have to understand why Rome was able to reach such a substantial population in the first place. The estimate of 1 million was reached only in the early scond century CE, partially because it was the capital of a vast empire of 50-60 million inhabitants and partially because of the relative peace and stability it experienced in the period. The main reason, in my view, however, is the Cura Annonae or the grain dole. The government simply gave out free food that had been imported form the extremely fertile provinces in North Africa and Egypt. The dole was enough to feed hundreds of thousands of citizens of course more people to come to the city.

The population of Rome probably dipped below 1 million already during the third century crisis, a period of widespread civil unrest throughout the empire. In Late Antiquity, Rome was no longer capital with Constantinople being seat of the imperial government, and with Milan as regional capital in the Italy. The city of Rome still retained its symbolic value, with the senate and important temples and monuments. During the fifth and sixth centuries moreover, Rome was sacked numerous times, by Visigoths, Vandals, and Ostrogoths and was even recaptured by the Eastern Emperor Justinian.

Most importantly, however, the grain dole was severly diminished and eventually dried up entirely after the Vandal conquest of the North African provinces in the fifth century and the muslim capture of Egypt and the rest of North Africa in the seventh century.