13 in powers of , i.e. θ $^{<}$( x , t ) ≈ θ $^{<}$( r , t ) + ξ ∇ θ $^{<}$( r , t ) + 1 $_{2}$ξ$_{α}$ ξ$_{β}$ ∂θ $^{<}$( r ,t ) ∂r$_{α}$∂r$_{β}$ . Then g a 2 cos( θ $^{<}$( r , t )) θ > 1 ( r , t ) ≈ C$_{2}$ 8 π g 2 λ δ Λ Λ △ θ − C$_{1}$ 4 π g 2 λ $^{2}$µ δ Λ Λ ¨ θ, (A17) where C$_{1}$ Λ 2 = ∫ d $^{2}$ξ J$_{0}$ (Λ ξ ) , C$_{2}$ Λ 4 = ∫ d $^{2}$ξ ξ$^{2}$J$_{0}$ (Λ ξ ) . (A18) where l = ln Λ. We can read off from Eq. A23, that if the system contains a fixed velocity, for example in relativistic systems, we need to have C$_{2}$ = 2 C$_{1}$ , to enforce that the velocity is invariant under the flow. Note that if the initial system is already close to the critical point then the RG equations above simplify to When we need to substitute these expressions back into Eq. (A9), we find that the term containing △ θ renormal- izes the coupling λ as In addition there is an extra term proportional to ¨ θ gen- erated in Eq. (A9), which renormalizes µ : dg dl ≈ g ( 2 − 1 4 πλ ) , (A24) dλ dl ≈ C$_{2}$ 8 π g 2 λ , (A25) λ → λ + C$_{2}$ 8 π g 2 λ δ Λ Λ . (A19) 1 µ → 1 µ + C$_{1}$ 4 π g 2 λ $^{2}$µ δ Λ Λ . (A20) which are equivalent to Eqs. (40) and (41). Note that a more complete set of RG equations (A21) - (A23) has the same universal predictions of the dynamical phase tran- sitions and exponential divergence of the time scales as the simplified equations above. Also note that the real RG equations bear close analogy to the flow equations in imaginary time characterizing the equilibrium Kosterlitz- Thouless transition $^{35}$. Thus the non-equilibrium KT transition discussed here is characterized by exponen- tially divergent length and time scales. Physically these long scales characterize very slow process of vortex un- binding and equilibration at long distances. Note that the RG equations (A21) - (A23) also implicitly take into account renormalization of the temperature in the sys- tem. This comes from the fact that creating vortex- antivortex pairs removes the energy from the phonon de- grees of freedom. We are going to investigate this issue in more detail in a separate publication. Finally we restore the cutoff by rescaling k → k (1 − δ Λ / Λ), r → r (1 + δ Λ / Λ), t → t (1 + δ Λ / Λ), and p → p (1 − δ Λ / Λ). This rescaling additionally renormalizes the coupling g : g → g (1 + 2 δ Λ / Λ). Combining this re- sult with Eqs. (A11), (A19), (A20) we find the following renormalization group equations: dg dl = g ( 2 − 1 4 πv$_{0}$ 1 K ) (A21) dK dl = 1 16 πK g 2 v $_{2}$( C$_{2}$ + 2 C$_{1}$ ) , (A22) dv dl = g 2 16 πvK $_{2}$( C$_{2}$ − 2 C$_{1}$ ) . (A23) 1 K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Phys. Rev. Lett. 75 3969 (1995); M.H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, Science 269 198 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, Phys. Rev. Lett. 75 1687 (1995). 2 M. Greiner, et al. , Nature (London) 415 , 39 (2002). 3 C. A. Regal, M. Greiner and D. S. Jin, Phys. Rev. Lett. 92 040403 (2004); M.W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman and W. Ketterle, Phys. Rev. Lett. 92 120403 (2004); M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. Hecker Denschlag and R. Grimm, Phys. Rev. Lett. 92 120401 (2004); T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Te- ichmann, L. Tarruell, S. J. J. M. F. Kokkelmans and C. Salomon Phys. Rev. Lett. 93 050401 (2004). 4 J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6 , 1181 (1973); V. S. Berezinskii, Sov. Phys. JETP 34 , 610 (1972). 5 Z. Hadzibabic, P. Kr¨ uger, M. Cheneau, B. Battelier, and J. B. Dalibard, Nature 441 , 1118 (2006). 6 P. Clade, C. Ryu, A. Ramanathan, K. Helmerson, and W.D. Phillips, Phys. Rev. Lett. 102 , 170401 (2009). 7 C.D. Fertig, K.M. O’Hara, J.H. Huckans, S.L. Rolston, W.D. Phillips, and J.V. Porto, Phys. Rev. Lett. 94 , 120403 (2005); T. St¨ oferle, H. Moritz, C. Schori, M .K¨ ohl, and T. Esslinger, Phys. Rev. Lett. 92 , 130403 (2004). 8 M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E. Wieman and E.A. Cornell, Phys. Rev. Lett. 83 , 2498 (1999); K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84 , 806 (2000); Z. Dut- ton, M. Budde, C. Slowe, and L.V. Hau, Science 293 , 663 (2001). 9 S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Seng- stock, A. Sanpera, G.V. Shlyapnikov, and M. Lewenstein, Phys. Rev. Lett. 83 , 5198 (1999); B.P. Anderson, P.C. Hal- jan, C.A. Regal, D.L. Feder, L.A. Collins, C.W. Clark, and E.A. Cornell, Phys. Rev. Lett. 86 , 2926 (2001); J. Denschlag, J.E. Simsarian, D.L. Feder, C.W. Clark,