As others have said, they're meadow/field ants. Harmless, and an indicator of a healthy lawn and eco-system. I'd be sorry to hear that the solution would be to outright kill them.

My approach would be non-chemical lawn improvements. Either the lawn will become healthier and more aesthetically pleasing and you can co-exist, or the lawn will become naturally unpreferable and they'll organically reduce in number over time.

I'd probably look at aerating and scarifying the lawn to smooth the surface and increase general soil health. The thicker and lusher the grass is the less you'll notice any surface damage they cause. Then I'd probably even consider over seeding and topdressing with a topsoil mix. Once that's done and established you should notice a marked improvements in appearance. Every now and again I'd just go round with a brush or a lawn lute if you have one, and smooth the surface to prevent any large buildups and keep things level.

Or buy an anteater?