What symbol do I use to go from common time to half time when notating that particular change?
To change the meter, use whatever symbol denotes the meter you want to change to. To change the tempo, you can either use numerical metronome marks or give a relative indication.
So if you start in 4/4 or 𝄴 ("common time" symbol), and you want to switch to 2/2 or 𝄵 ("cut time" symbol), and you want the half note in 2/2 to be as fast as the quarter note in 4/4, you can start with 𝅘𝅥=80 (quarter=80) and then use 𝅗𝅥=80 (half note=80 -- or whatever actual tempo you want). A problem with the traditional approach of equating the half note with the quarter note is that practice has changed over the years as to whether one should write 𝅗𝅥=𝅘𝅥 (half=quarter) or 𝅘𝅥=𝅗𝅥 (quarter=half). In this case it's unlikely to be ambiguous, but in some cases it can be.