Large charge sector of 3d parity-violating CFTs

Abstract

Certain CFTs with a global U(1) symmetry become superfluids when coupled to a chemical potential. When this happens, a Goldstone effective field theory controls the spectrum and correlators of the lightest large charge operators. We show that in 3d, this EFT contains a single parity-violating 1-derivative term with quantized coefficient. This term forces the superfluid ground state to have vortices on the sphere, leading to a spectrum of large charge operators that is remarkably richer than in parity-invariant CFTs. We test our predictions in a weakly coupled Chern-Simons matter theory.

Spontaneously broken boosts in CFTs

“…We end this section with a remark about three-dimensional parity violating superfluids [30]. The ground state contains vortices, but it is homogenous and isotropic, hence our sum rules apply.…”

“…(Here we assume that the underlying CFT preserves parity, hence the parity violating terms of [30] are forbidden.) The saddle point that describes a ground state with finite homogenous charge density ρ is given by…”

“…A natural question is what saturates the sum rules in parity violating superfluids. The answer is that it is still the phonons, because [30] found that the OPE coefficients involving the phonon modes, the ground state Ω and T 00 (or other relevant components of T ) stay unchanged compared to the parity preserving one. Thus the softer vortex modes should not contribute in the macroscopic limit and should not play any role in the sum rule.…”

Boundary conformal field theory at large charge

“…where here and the rest of the paper we use the shorthand √ g ≡ |det(g µν )| and the dots stand for terms which vanish on the classical profile χ = µt. We are also assuming that the bulk theory is parity invariant for simplicity; for discussion of the parity breaking theories see [41]. The Wilson coefficients c i 's are the same as in the CFT without a boundary and they are expected to be O(1) for strongly coupled theories, but may take parametrically…”

“…(3.2). Notice that we are also assuming that the bulk theory is parity invariant, otherwise in d = 3 it would be possible to write terms which are first order in derivatives in terms of the gauge field dual to χ [41].…”