They stepped out in Hollywood last week, looking casual and relaxed together. As a friend of the actor tells PEOPLE, "He is smitten."
But as the couple gets cozier, some fans of the Twilight star have gone on the attack, sending racist and hateful Tweets. twigs, whose debut album LP1 was released in August, has borne the brunt of the insults.
On Sunday, twigs (Real name: Tahliah Barnett) finally spoke out about the vitriol.
"I am genuinely shocked and disgusted at the amount of racism that has been infecting my account the past week," the 28-year-old singer Tweeted. "Racism is unacceptable in the real world and it's unacceptable online."
It's not the first time that twigs, who is half Jamaican and half Spanish, has dealt with taunts about her race.
As the only mixed-race girl in her school, "people said horrible things about something I had no control of, which was tough," she told Dazed. "Half of my life, I've had people staring at me because they think I'm funny-looking and ugly. The other half of my life, I've had people staring at me because they think I'm fascinating. Everything neutralizes."