On the dynamics of continuous phase transitions

Guy Battle

A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one-to-one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.

G. Toulouse

Simon A. Weidinger, Markus Heyl, Alessandro Silva +1 more

Interacting many-body systems that are driven far away from equilibrium can exhibit phase transitions between dynamically emerging quantum phases, which manifest as singularities in the Loschmidt echo. Whether and under which conditions such dynamical transitions occur in higher-dimensional systems with spontaneously broken continuous symmetries is largely elusive thus far. Here, we study the dynamics of the Loschmidt echo in the three-dimensional O($N$) model following a quantum quench from a symmetry-breaking initial state. The O($N$) model exhibits a dynamical transition in the asymptotic steady state, separating two phases with a finite and vanishing order parameter, that is associated with the broken symmetry. We analytically calculate the rate function of the Loschmidt echo and find that it exhibits periodic kink singularities when this dynamical steady-state transition is crossed. The singularities arise exactly at the zero crossings of the oscillating order parameter. As a consequence, the appearance of the kink singularities in the transient dynamics is directly linked to a dynamical transition in the order parameter. Furthermore, we argue, that our results for dynamical quantum phase transitions in the O($N$) model are general and apply to generic systems with continuous symmetry breaking.

Srikanth Sastry

Aron J. Beekman, Louk Rademaker, Jasper van Wezel

In this chapter, we focus on the time dynamics encountered during a symmetry-breaking phase transition. After introducing the central concept of a relaxation time, the Kibble–Zurek mechanism arising in continuous phase transitions and the nucleation dynamics of discontinuous transitions are discussed.