Dynamics of longitudinal magnetization in transverse-field quantum Ising model: from symmetry-breaking gap to Kibble-Zurek mechanism

We show that the symmetry-breaking gap of the quantum Ising model in the transverse field can be extracted from free evolution of the longitudinal magnetization taking place after a gradual quench of the magnetic field. We perform for this purpose numerical simulations of the Ising chains with either periodic or open boundaries. We also study the condition for adiabaticity of evolution of the longitudinal magnetization finding excellent agreement between our simulations and the prediction based on the Kibble-Zurek theory of non-equilibrium phase transitions. Our results should be relevant for ongoing cold atom and ion experiments targeting either equilibrium or dynamical aspects of quantum phase transitions. They could be also useful for benchmarking D-Wave machines.