Why don't polar bears live in the southern hemisphere?

Because they evolved from brown bears, and this species is found only in the northern hemisphere.

Also polar bears are primarily aquatic feeders - feeding on and around ice floes - and the South Pole is a LONG ways from any open water or ice pack. Antarctica is HUGE. In theory polar bears might survive in the seas around Antarctica, but it's not their native habitat and there is much less ice pack than in the Arctic because most of the ice is deposited on the land mass of Antarctica.