Kibble-Zurek mechanism and infinitely slow annealing through critical points

Abstract

We revisit the Kibble-Zurek mechanism by analyzing the dynamics of phase ordering systems during an infinitely slow annealing across a second order phase transition. We elucidate the time and cooling rate dependence of the typical growing length and we use it to predict the number of topological defects left over in the symmetry broken phase as a function of time, both close and far from the critical region. Our results extend the Kibble-Zurek mechanism and reveal its limitations. PACS numbers:The out of equilibrium dynamics induced by a quench are the focus of intense research [1,2]. Interesting realizations are quenches through a second order phase transition, which take the system from the symmetric phase into the symmetry broken one. Below the transition, times scaling with the system size are needed to reach equilibrium and to realize the spontaneous symmetry breaking process. Before this-typically unreachableasymptotic limit the symmetry is broken only locally: the system is formed by ordered regions of size growing with time [3]. Only when this size reaches the order of the volume of the sample the symmetry is broken globally and the spatial average of the order parameter deviates from zero. The majority of theoretical studies focused on the dynamics after infinitely rapid quenches although experimentally quenches are performed at finite speed. Indeed, since the typical time-scale on which the system evolves is its age, i.e. the time elapsed since crossing the critical point, finite quench time-scales (τ Q ) eventually become short compared to the relaxation time. Thus, they alter the out of equilibrium dynamics at short times only. The opposite limit of an extremely slow annealing, corresponding to very long τ Q , needs a separate treatment. Surprisingly, this has not been studied in detail in the statistical physics literature with, however, some exceptions for disordered systems [4][5][6]. It has, instead, attracted a lot of attention within the cosmology and, more recently, the condensed matter communities. An explanation of the slow dynamics induced by this protocol was given by the so-called Kibble-Zurek (KZ) mechanism [7][8][9][10]. This is an equilibrium scaling argument that yields an estimate for the density of topological defects left over in the ordered phase as a function of the quenching rate close to the critical point. The argument has been recently generalized to study very slow 'quantum annealing' across a quantum phase transitions in isolated systems [11][12][13].The aim of this work is to obtain a more complete picture of the slow dynamics induced by an extremely slow annealing. With numerical and analytical arguments we unveil the limitations of the KZ approach and we obtain a full scaling description of the slow dynamics. Our main result is that the dynamic evolution is characterized by a first adiabatic regime in agreement with KZ, followed by critical coarsening and, finally, standard coarsening at very long times. We find a new universal scaling function that characterizes the gro...

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