Carbuncle
Carbuncle is a bunny/fox-esque companion summoned by Arcanist and Summoner. Arcanist and Summoner can spawn their pets with the actions 
Summon Carbuncle, 
Summon Ruby, 
Summon Topaz, and 
Summon Emerald. Each type has a different spell style and associated element.
Alphinaud has a special Moonstone Carbuncle, that heals with the same spells as Eos
Additional Information
Carbuncle is also a server in the Japanese Elemental Data Center.
Carbuncle is a term for a red gemstone, such as Garnet. In this case, it refers to the gem in the head of the creature.
Lore
“Using geometric symbols to lend shape to living aether, certain mages can unlock the latent power within certain gemstones-a power which can be directed to the "summoning" of aetherial beings known as Carbuncles. The inherent traits displayed by these familiars differ depending on the stone used in their creation- emerald and topaz being the most common, with ruby and diamond also proving popular.
— Encyclopædia Eorzea vol 1, page 301