Universal frozen spectra after time-dependent symmetry restoring phase transitions.

For a general O(N) model, we study the time-dependent phase transition from a state with broken symmetry <φ> ≠ 0 to the symmetric phase <φ> = 0. During this non-equilibrium process, the primordial quantum (or thermal) fluctuations of the initial Goldstone modes are frozen and result in a deviation from the final ground (or thermal) state. For very slow transitions, we find that these fluctuations display a universal scaling behaviour. Their spectra are universal functions of a single parameter, which combines the initial frequency of the Goldstone modes and the sweep rate. As a result, the final two-point function [φa(r)φb(r')] is not exponentially suppressed at large distances Δr = r - r' (as it would be in the ground state) but decays polynomially in 1/|Δr|. Finally, we exemplify this universal behaviour for the transition from the super-fluid phase to the Mott state in the Bose-Hubbard model.