• Kibble-Zurek dynamics
• out-of-equilibrium dynamics
• quantum Ising model
• quantum transitions

We analyze the out-of-equilibrium dynamics of closed many-body systems emerging when one Hamiltonian parameter (say w) is, slowly in time, driven across the critical value of a zero-temperature continuous quantum transition (CQT).
The specific dynamic protocol we adopt is the Kibble-Zurek protocol involving such slow changes.
As a numerical laboratory, we consider the quantum Ising chain with periodic boundary conditions in both a transverse and a longitudinal field. The Hamiltonian H[w(t)] driving the unitary dynamics is the sum of the critical Hamiltonian of the one-dimensional Ising model with the two relevant perturbations coupled
respectively with the deviations of the longitudinal and the transverse field from their critical values. We take in turn one of the couplings as the tunable parameter w while the other one acts as a time-independent perturbation.
The unitary dynamics starts in the ground state (assumed not degenerate) of the system at wi < 0, which is sufficiently far from its critical value wc = 0, enabling an adiabatic evolution at the beginning. Then w is slowly driven across wc, up to a final value wf > 0.
In the thermodynamic limit, a CQT is associated with nonanalicities of the low-energy properties of the system due to the vanishing of the energy gap of the lowest states of the Hamiltonian at the critical point. When approaching the phase boundary the system develops critical modes with a divergent length scale.
Within the critical region, the approach of w to wc gives rise to a departure from the adiabatic evolution. The system is inevitably driven out of equilibrium, no matter how slowly the parameter is changed, because long-range modes cannot equilibrate as the system changes phase. In the limit ts → ∞, with ts the time scale of the tunable parameter w, the dynamics across the critical point develops an universal scaling regime determined by power laws of ts, with critical exponents entirely controlled by the universality class of the equilibrium CQT.
The scaling regime develops in finite-size systems, as well. In this case the emergence of the out-of-equilibrium dynamics is ruled by the interplay between the time scale of the driven parameter, the size of the system and the strength of the time-independent perturbation. In particular, for a sufficiently small strength
of the time-independent perturbation, the ratio between ts^1/ζ (where ζ = yw + z depends on the dynamic exponent z and the renormalization-group dimension yw of the driven parameter) and the finite size L of the system determines the emergence of the out-of-equilibrium dynamics and eventually the crossover to
an adiabatic evolution. This is the core of our work.
At the beginning, we analyze the departure from the adiabatic dynamics as the tunable parameter crosses its critical value by monitoring some relevant observables as the overlap of the state of the system at time t with the ground state of H[w(t)], the surplus energy of the system relative to its instantaneous ground
state and the longitudinal magnetization of the system.
Then, for the KZ protocol driving the quantum Ising chain in a weak longitudinal field from the disordered (paramagnetic) to the ordered (ferromagnetic) phase, we analyze the crossover to the adiabatic dynamics as the nonequilibrium regime is simultaneously suppressed.