I've grown potatoes as perennials in the same place for 5 or 6 years in USDA zone 6, but with one winter with a few nights down in the -20F range. When I harvested my crop, I'd leave any that were too small to bother with and remulch. I mulched about a foot deep, usually with old hay. I moved off of my
land for a while and the volunteers quit producing after the mulch got too thin. I think it was a combination of freezing too deep and too many weeds. When I get back to living on my land, I'll start another patch.
One other idea I've used is laying the spuds right on the ground and burying them in a foot or so of mulch (
hay or straw) and then using that bed for other crops the next year. After sitting under the mulch for a year, there was no need to till the soil.
Who rotates plants in nature? If you mulch deep and don't plant large monocultures there
should never be much need to rotate, especially if 'rotate' means move the patch a few feet away.