The main themes usually revolve around adaptation. Percy has to adapt to the rules and lifestyle of the vampire world, which is very different from his own. Carlisle, on the other hand, has to understand Percy's connection to the Greek gods and the whole demigod culture. Another theme is the forging of a new family. Since they are mates, they create a unique bond that might not be fully understood by either of their original families or groups. And love in the face of differences is also a big theme, as they come from such disparate fictional backgrounds yet find a connection with each other.
Yes, there are some out there. You can try searching on popular fanfiction websites like Archive of Our Own or Fanfiction.net. Just be aware that the quality can vary greatly. Some might focus more on the relationship development between Percy and Carlisle, while others could be part of a larger crossover plot with other characters from both universes.
It could start with an unexpected encounter. Perhaps Percy somehow ends up in the Twilight world or vice versa. Their relationship might develop slowly as they try to understand each other's worlds. Carlisle, being an old and wise vampire, could be intrigued by Percy's youth and his connection to the gods. Percy, on the other hand, would have to deal with the fact that Carlisle is a vampire, which is different from the monsters he's used to in his own world. And as their relationship progresses, they would face challenges not only from their own worlds but also from the differences in their natures.
You could try searching on popular fanfiction websites like Archive of Our Own (AO3). There's a large collection of fanfics there, and you can use the search function to look for this specific pairing. Just type in 'Percy Jackson Carlisle Cullen mate' and it should show relevant results if any exist.
One possible plotline could be how Harry and Carlisle meet in a world where the two fictional universes collide. Maybe they meet during a magical event that draws vampires and wizards together.
The theme of protection also often comes up. Carlisle, being the ethical and caring vampire he is, may feel a strong sense of protecting Bella from various threats in the vampire world. Additionally, there can be a theme of self - discovery for Bella. Through her interactions with Carlisle, she might learn more about herself, her values, and her place in this strange new world she's been thrust into.
One common theme is loyalty. Both Percy and the werewolf characters often show loyalty to their friends and mates. Another theme is the struggle between different worlds. The werewolves come from their own world, and Percy from his demigod world, and there's often a struggle to co - exist.
One common theme is loyalty. The werewolf mates often show extreme loyalty to each other and to their friends in the Percy Jackson world. Another theme is the struggle between different natures. The werewolves have their wild side, and the demigods have their own powers and responsibilities, and the fanfictions often explore how they deal with these differences.
Love and devotion are very common themes. These fanfictions often explore how Carlisle and his mate are deeply in love and devoted to each other, through various situations.
One common theme is the exploration of a forbidden relationship. Since Carlisle is much older and in a different position in the vampire hierarchy compared to Bella, it creates an interesting dynamic. Another theme is the idea of mentorship. Carlisle often acts as a mentor figure to Bella, guiding her through the vampire world.
The theme of sacrifice often appears. Carlisle may have to sacrifice his principles or his position in the vampire community for Bella. And Bella might sacrifice her normal human life for Carlisle. Also, the theme of self - discovery is common. As Bella gets involved with Carlisle, she discovers new aspects of herself, like her ability to adapt to the vampire world. And Carlisle discovers new emotions and a different way of loving through his relationship with Bella.