9 ergy, high momentum degrees of freedom. In equilibrium, this is done by the means of usual perturbation theory (or Gaussian integration), which is justified because of the large energy gap separating high energy states from the low-energy degrees of freedom we are interested in. In real time the idea of renormalization is quite similar. High energy (momentum) phonons are not very sensi- tive to slow processes leading to vortex formations. Thus these phonons can be well treated within the linearized approach. However, due to nonlinearities such phonons slightly renormalize the parameters governing dynamics of low energy degrees of freedom. This renormalization is precisely what we are interested in. Note that techni- cally in the RG procedure we perform averaging of the equations of motion over short times. Then odd powers of highly oscillating fields average to zero while averag- ing of the even powers gives some constant contribution. This contribution is precisely what renormalizes coupling constants governing the low temperature dynamics. FIG. 9: Time dependence of the exponent τ extracted from fitting the long-time correlation function G ( x, t ), for different initial couplings. In all four examples we use T = 1 and κ = 8. The initial couplings V are chosen as V = 80 , 70 , 50, and 20, corresponding to curves I to IV. The curve I corresponds to the example shown in Fig. 8. In this case the correlation function can be well fitted with an algebraic function for up to t ≈ 60, after that G ( x, t ) is better fitted by an exponen- tial function with a decay length of the order of the lattice constant. For the other cases, G ( x, t ) is well fitted with an algebraic function throughout the whole time interval. 20 40 60 0 80 100 1 2 5 0 1 2 5 t τ I II III IV We point out that typical RG flow diagrams contain mostly non-equilibrium points, in fact, all except for the fixed points. As we have seen in the previous sec- tions, one can associate an effective temperature to the metastable state that emerges after the dephasing of the phonon modes. In turn with this effective parameter we can associate a location of the transient state in the RG flow of the equilibrium system. This effective tempera- ture can then either gradually increase, until the system starts to show exponential scaling, or the system can al- ways remain superfluid, if the algebraic scaling is subcrit- ical and the effective temperature always remains below T$_{KT}$ . This behavior resembles the equilibrium RG flow of a Kosterlitz-Thouless transition (which now occurs in real, not imaginary, time), on which we elaborate in this section. ponent. After that, the exponent gradually increases on much longer time scales. During this process, the decay of the correlation function is still fitted well with the alge- braic function. Eventually the algebraic scalings breaks down and G ( x, τ ) develops exponential scaling, indicat- ing vortex unbinding. This regime of exponential scaling is reached for V = 80 (I) within the time interval shown in Fig. 9. For V = 70 (II) and V = 50 (III) the time scale of the vortex unbinding is longer then the time in- terval shown. For V = 20 (IV) the system equilibrates to the superfluid state. Because in this case the exponent τ is less than one, vortices never unbind and the algebraic scaling persists at all times. We conclude from these ex- amples that there can be a sizeable range of initial values of V which generates the scenario of a supercritical super- fluid, and of dynamically suppressed vortex unbinding. Furthermore, the algebraic scaling exponents that can occur in the metastable state are well above criticality, and should be easily distinguishable from subcritical val- ues. These supercritical exponents can be detected using interference experiments along the lines of Refs. 5,19. Instead of directly analyzing the rotor model to de- scribe the Kosterlitz-Thouless physics and vortex unbind- ing, we will work with the dual Z$_{1}$ clock model (or equiv- alently 2D sine-Gordon model), described by the action For the details of the duality transformation see Ref. [17]. The parameters of this model can be related to those of the XY model by S = ∫ d $^{2}$r ( λ 2 ( ∂$_{x}$θ ) 2 − g a 2 cos θ ) . (31) λ = 1 8 π T T$_{KT}$ = 1 4 π 2 T J$_{KT}$ (32) g 2 = exp( − S$_{c}$ ) , (33) B. Renormalization group approach H/T = ∫ d $^{2}$r ( µ 2 p 2 − λ 2 ( ∂$_{x}$θ ) 2 + g a 2 cos θ ) . (34) where E$_{c}$ = S$_{c}$T is the vortex core energy, and λ is a measure of the relative temperature. We note that the action in Eq. 31 has a high-momentum cut-off Λ, which is the inverse of the short-range cut-off a , i.e. we set Λ a = 1. To describe the dynamics of this model we use the effective 2D sine Gordon Hamiltonian: In this section we develop the renormalization group (RG) approach to dynamical vortex unbinding. We find that the dynamical evolution of the system can be re- lated to the RG flow of the equilibrium system. The idea of RG in real time is quite similar in spirit to the RG in imaginary time. Namely our goal is to eliminate high en-