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Their first album, Give You the Ghost, was released on Valentine's Day, 2012 and received praise from critics, including Grammy winner and Bon Iver founder Justin Vernon who told Rolling Stone, "They're the best band I've ever heard."[3] Two tracks on the album feature vocals by Bon Iver's Mike Noyce. Rolling Stone's review of the album said that Leaneagh's "focused, tender voice cascades over dense fields of minor key electronics, backed by funked-up bass and a swarm of percussion. It's the sound of heartbreak and celebration happening simultaneously."[4] Consequence of Sound's review said that "the post-prog-rock percussion and mellifluous bass, along with Channy Leanagh's Auto-Tuned swan songs, re-shape the intersection of rhythm and blues."[5]

In March 2012, the band played at SXSW in Austin, Texas, where NPR wrote that

Polica appears on the brink of something more worldwide, thanks in part to polished and propulsive, word-of-mouth-inducing live shows like this one. Where Give You the Ghost gives the suggestion of a subdued live presence, Polica's stage show heads to somewhere approaching the opposite extreme — a welcome jolt of beauty and power.[6]

The Huffington Post review of Poliça's SXSW performance said that "Poliça makes moody music that will stick to your bones. This band is going to be huge."[7]

On April 25, 2012 Poliça appeared on Late Night with Jimmy Fallon. On August 22 the band appeared on Jimmy Kimmel Live!. On October 14 Poliça played at the Austin City Limits Music Festival in Austin, Texas. The band appeared on the UK television show Later... with Jools Holland on October 30th.

Their songs have appeared on the television shows Person of Interest, 666 Park Avenue, Arrow, and the soundtrack to Need for speed: Most Wanted (2012).