Spectral functions of the Higgs mode near two-dimensional quantum critical points

We study the Higgs excitation in the Goldstone phase of the relativistic O(N) model in two spatial dimensions at zero temperature. The response functions of the order parameter, and its magnitude squared, become universal functions of frequency in the vicinity of the quantum critical point described by the Wilson-Fisher fixed point, and we compute them to next-to-leading order in 1/N. The Higgs particle has an infrared singular decay to gapless Goldstone excitations, and its response functions are characterized by a pole in the lower half of the complex frequency plane. The pole acquires a nonzero real part only at next-to-leading order in 1/N, demonstrating that the Higgs excitation has an oscillatory component even in the scaling limit. Both the real and imaginary parts of the pole position vanish with the correlation length exponent ν upon approaching the critical point. We present evidence that the spectral density of the O(N)-invariant amplitude-squared of the order parameter has a peak at a nonzero frequency in the scaling limit. We connect our results to recent experimental studies of the superfluid-insulator quantum phase transition of ultracold bosonic atoms in optical lattices.