To harvest root vegetable crops, you can use the Root Digger attachment (also known as the Potato Digger).
The round tines that extend from the back of the Root Digger sift through previously-worked soil, pushing your potatoes or other root crop toward the soil surface, so that you can harvest them. Typically, after using the Root Digger, your potatoes will be at least partially exposed.
The Root Digger can burrow as deeply as you have previously worked the soil, but it cannot burrow deeper than that. In other words, it is used for harvesting, not for plowing.