The Kibble-Zurek Problem: Universality and the Scaling Limit

Near a critical point, the equilibrium relaxation time of a system diverges and any change of control/thermodynamic parameters leads to non-equilibrium behavior. The Kibble-Zurek problem is to determine the dynamical evolution of the system parametrically close to its critical point when the change is parametrically slow. The non-equilibrium behavior in this limit is controlled entirely by the critical point and the details of the trajectory of the system in parameter space (the protocol) close to the critical point. Together, they define a universality class consisting of critical exponents-discussed in the seminal work by Kibble and Zurek-and scaling functions for physical quantities, which have not been discussed hitherto. In this article, we give an extended and pedagogical discussion of the universal content in the Kibble-Zurek problem. We formally define a scaling limit for physical quantities near classical and quantum transitions for different sets of protocols. We report computations of a few scaling functions in model Gaussian and large-N problems and prove their universality with respect to protocol choice. We also introduce a new protocol in which the critical point is approached asymptotically at late times with the system marginally out of equilibrium, wherein logarithmic violations to scaling and anomalous dimensions occur even in the simple Gaussian problem.