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Here the parameter µ is chosen, so that the dispersion of the linearized XY model is recovered: 

which we also write as ω$_{k}$ = v | k | , where the velocity v is given by v = √ µλT $^{2}$. The nonlinear term cos θ in Eq. (34) describes the vortex field. If this term is im- portant (large g ) then the field θ localizes corresponding to a highly disordered phase of the dual field φ , i.e. to the normal state. Conversely small g corresponds to the superfluid algebraic regime. The starting point of our RG analysis will be supercritical superfluid state which emerges after short time light cone dynamics. Because the Kosterlitz-Thouless transition is classical in nature occuring at high temperatures the quantum fluctuations are no longer expected to be important and instead of Wigner function as the new initial condition we can use its classical Boltzmann’s limit. The initial state for the vortex dynamics, described by the effective temperature T , is thus fully characterized by the quadratures of the spectrum: 

The equations of motion corresponding to the Hamilto- nian (34) are given by 

We now apply the following renormalization procedure to these equations. We rescale the spatial and temporal variables as r → r (1 + d Λ / Λ) and t → t (1 + d Λ / Λ), and the p -field as p p (1 − d Λ / Λ). This implies that the mo- mentum cut-off Λ is rescaled as Λ → Λ ′ ≡ Λ(1 − d Λ / Λ), so the momentum degrees of freedom between Λ ′ and Λ are removed. Without the non-linear term in Eq. 38 these rescalings leave the equations of motion invariant. The linear dynamical evolution can therefore be considered to be the non-interacting fixed point of the RG. We now ask the question, how this dynamical evolution is affected by the non-linear term. Specifically we want to determine how the equations of motion behave at long times and distances. For this, we go beyond the bare rescaling and correct for the integrated-out degrees of freedom up to second order in g . The resulting flow equations are of the well-known BKT form: 

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where l = ln Λ, and α is a non-universal prefactor. The RG step generated the equations of motion at time t ′ and distance r ′ from the equations at time t and distance r , with renormalized coefficients, according to Eqs. 40 and 41. Therefore the time dependence of the coefficients can be read off the solution of the RG flow, by realizing that: dt/dl = t or t = t$_{0}$e $^{l}$. In Fig. 10 we show a schematic representation of our RG process. In the Appendix we discuss the derivation of the flow equations and give their more complete form. 

One conclusion from Eqs. 40 and 41 is that the crit- ical exponent of the dynamical process is equal to the one of the equilibrium system. We see from Eq. 40 that the critical value of λ is λ$_{c}$ = 1 / 8 π , which corresponds to T = T$_{KT}$ as can be seen from Eq. 32. Another important observation is that the RG equations (40) and (41) pre- dict a non-equilibrium analogue of the BKT transition, where depending on the initial fluctuations in the sys- tem, the vortex-antivortex pairs can either unbind in the long time limit or remain bounded. This transition, as in the equilibrium case, is characterized by exponentially divergent time and length scales. Physically these diver- gencies correspond to a very slow process of equilibration of vortices near the nonequilibrium phase transition. 

We can also use the RG flow to determine the time scale of vortex unbinding by using 

When T is well above T$_{KT}$ , the time scale can be deter- mined from Eq. 40, 

where E$_{c}$ = S$_{c}$T . Away from the transition, the time scale of vortex unbinding is therefore exponentially in- creased, because of the energy cost given by the vortex core energy. Very close to the transition t ∗ scales as: 

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