I consider a simple non-equilibrium problem, where a critical one-dimensional system is prepared in a state with two different densities on the left and on the right, and let evolve with a Hamiltonian that conserves the number of particles. A typical example would be a fermionic system prepared with different left/right chemical potentials. For free systems a lot can, and has been understood by making use of a semiclassical picture, in which particles carrying a momentum k propagate ballistically with velocity v(k). Generalization to interacting systems is very much an open problem. I will discuss attempts at understanding such dynamics using field theory. A possible strategy is to study the behavior in imaginary time, the real time dynamics being recovered by performing the Wick rotation. I will show that all degrees of freedom outside a certain region may freeze in imaginary time, contrary to naive expectations. This behavior is analogous to the celebrated “arctic circle” phenomenon found in the study of two-dimensional classical dimer or vertex models. I will also show that the fluctuating region is described by a massless field theory with a position-dependent metric, a field theory in curved space. Such imaginary time pictures can be used to make predictions about the behavior of correlation functions, entanglement entropies, or return probabilities after the quench.

Dresden