Abstract
Ergodicity sits at the heart of the connection between statistical mechanics and dynamics of a physical system. By fixing the initial state of the system into the ground state of the Hamiltonian at zero temperature and tuning a control parameter, we consider the occurrence of the ergodicity with quench dynamics in the one-dimensional (1D) spin-1/2 XY model in a transverse magnetic field. The ground-state phase diagram consists of two ferromagnetic and paramagnetic phases. It is known the magnetization in this spin system is non-ergodic. We set up two different experiments as we call them single and double quenches and test the dynamics of the magnetization along the Z-axis and the spin-spin correlation function along the X-axis which are the order parameters of the zero-temperature phases . Our exact results reveal that for single quenches at zero-temperature, the ergodicity depends on the initial state and the order parameter. In single quenches for a given order parameter, ergodicity will be observed with an ergodic-region for quenches from another phase, non-correspond to the phase of the order parameter, into itself. In addition, a quench from a ground-state phase point corresponding to the order parameter into or very close to the quantum critical point, hc = 1.0, discloses an ergodic behavior. Otherwise, for all other single quenches, the system behaves non-ergodic. Interestingly on the other setup, a double quench on a cyclic path, ergodicity is completely broken for starting from the phase corresponding to the order parameter. Otherwise, it depends on the first quenched point, and the quench time T when the model spent before a second quench in the way back which gives an ability to controlling the ergodicity in the system. Therefore, and contrary to expectations, in the mentioned model the ergodicity can be observed with probing quench dynamics at zero-temperature. Our results provide further insight into the zero-temperature dynamical behavior of quantum systems and their connections to the ergodicity phenomenon.
Highlights
One of the most controversial topics is how the statistical mechanics behavior could emerge in quantum-mechanical systems evolving under unitary dynamics[1,2,3,4,5,6,7,8,9,10,11,12]
In addition to the 1D XY model, the non-ergodicity has been studied in quantum chaos15, 1D XXZ model to show ergodicity breaking that can create a many-body localization[16] and its extended17, 1D system of spinless and interacting fermions with a disordered potential[18], the anisotropic Dicke model[19], and in a small quantum system consisting of three superconducting qubits by measuring the evolution of the entanglement entropy[20]
We are going to study quantum ergodicity at zero-temperature with the use of quench dynamics of the two quantities, the magnetization along the Z-axis, and the spin-spin correlation function along the X-axis which reveals the magnetization along the X-axis
Summary
Ergodicity sits at the heart of the connection between statistical mechanics and dynamics of a physical system. On the other setup, a double quench on a cyclic path, ergodicity is completely broken for starting from the phase corresponding to the order parameter Otherwise, it depends on the first quenched point, and the quench time T when the model spent before a second quench in the way back which gives an ability to controlling the ergodicity in the system. In 1970, for the first time, Barouch and coworkers[13] studied the dynamics of the magnetization of the anisotropic spin-1/2 XY chain They used a single quench at finite temperature where their initial and final states were thermal states. By a quench from the paramagnetic phase into itself they showed that the equilibrium is not reached at the final evolutionary time and the magnetization is a non-ergodic observable This non-ergodic behavior was later confirmed for the entanglement between the nearest neighbor pair spins of the evolved states[14].
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