By what method do ants dissect and hold soil
Short answer  

Ants use specialized facial structures called mandibles to cut, shape and carry soil and other matter.

Background

you say "I know ants break off soil in small quantities, so that a single ant carries very little soil...". On the contrary I would say - Ants can carry relatively a lot of soil, as they can lift up to about 100 times their own body weight. In absolute terms, however, the weight carried is about 500 mg (Spectrum). 


Source: The World's Best Photos

The structure that ants use to break off and carry soil are their mandibles:


Source: Tree of Life Web Project

Mandibles are specialized bodily structures typical for insects, and I quote from the University of Sydney website:


  The mandibles are the first pair of jaws in insects, moving laterally (i.e. from side to side) when in use. Mandible shape is strongly influenced by function and is therefore highly variable. For example a chewing insect will have strong tooth-shaped mandibles while in a sucking insect they will be thin and needle-shaped. Mandibles are used to cut, chew and tear food as well as to carry objects, fight and mould materials such as wax or soil.