Kibble-Zurek problem: Universality and the scaling limit

Abstract

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.

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

“…where ξ v is the KZ correlation length corresponding to finite velocity in the thermodynamic limit [17][18][19];…”

“…(5), a physical quantity A evaluated at the critical point can be written in the following finitesize scaling form [19,20,22]:…”

“…Given the difficulties in studying the critical behavior through equilibrium simulations, the recently developed non-equilibrium approach based on generalized KibbleZurek (KZ) scaling [15][16][17][18][19][20][21][22] provides a powerful alternative method for studies of spin-glass models. KZ scaling of simulated annealing (SA) results has been successfully applied to 3D and 2D spin glass systems in order to extract the dynamic exponent z and other critical exponents [23,24].…”

Universal postquench coarsening and aging at a quantum critical point

“…For the specific ansatz in Eq. (70), this amounts to an expansion in the dimensionless light-cone amplitude a:…”

“…In the Kibble-Zurek description of such a parameter sweep through a critical point, the correlation length is assumed to remain constant at this freeze-out length scale for the remainder of the sweep 65,66 . It then follows that the number of (topological) excitations depends on the rate via a universal scaling law that solely contains equilibrium critical exponents [67][68][69][70][71] . Similarly, it follows that the long time approach to equilibrium of a system that is suddenly quenched close to a (quantum) critical point is governed by equilibrium exponents 72 .…”