A haul rope could be as small as being in the 6-8 mm range (depending upon how much shit/weight you're going to be hauling), but your jug rope would need to be of a large enough diameter to be clamped safely by your Jumar/Clog/whatever brand ascender you use. Also remember, if you're going to be using your Jumar as part of the hauling/ratchet/locking system, then you need to consider a larger diameter rope for hauling (otherwise, use a locking knot/prussic).
If the rope diameter is too small, you're stepping on a slippery slope, so to speak. My rec would be to not go smaller than 10 mm, but then I did aid climbs back in the 80s with Jumars, and I'm just a washed up, old skewl, fat boy now. Just ask anybody who was at RopeUp.