We study the out-of-equilibrium dynamics of one-dimensional quantum Ising models in a transverse field
g, driven by a time-dependent longitudinal field
h across their {\em magnetic} first-order quantum transition at
h=0, for sufficiently small values of
∣g∣. We consider nearest-neighbor Ising chains of size
L with periodic boundary conditions. We focus on the out-of-equilibrium behavior arising from Kibble-Zurek protocols, in which
h is varied linearly in time with time scale
ts, i.e.,
h(t)=t/ts. The system starts from the ground state at
hi≡h(ti)<0, where the longitudinal magnetization
M is negative. Then it evolves unitarily up to positive values of
h(t), where
M(t) becomes eventually positive. We identify several scaling regimes characterized by a nontrivial interplay between the size
L and the time scale
ts, which can be observed when the system is close to one of the many avoided level crossings that occur for
h≥0. In the
L→∞ limit, all these crossings approach
h=0+, making the study of the thermodynamic limit, defined as the limit
L→∞ keeping
t and
ts constant, problematic. We study such limit numerically, by first determining the large-
L quantum evolution at fixed
ts, and then analyzing its behavior with increasing
ts. Our analysis shows that the system switches from the initial state with
M<0 to a positively magnetized state at
h=h∗(ts)>0, where
h∗(ts) decreases with increasing
ts, apparently as
h∗∼1/lnts. This suggests the existence of a scaling behavior in terms of the rescaled time
Ω=tlnts/ts. The numerical results also show that the system converges to a nontrivial stationary state in the large-
t limit, characterized by an energy significantly larger than that of the corresponding homogeneously magnetized ground state.
We study nonequilibrium dynamics of the quantum Ising chain at zero temperature when the transverse field is varied stochastically. In the equivalent fermion representation, the equation of motion of Majorana operators is derived in the form of a one-dimensional, continuous-time quantum random walk with stochastic, time-dependent transition amplitudes. This type of external noise gives rise to decoherence in the associated quantum walk and the semiclassical wave packet generally has a diffusive behavior. As a consequence, in the quantum Ising chain, the average entanglement entropy grows in time as
t1/2 and the logarithmic average magnetization decays in the same form. In the case of a dichotomous noise, when the transverse field is changed in discrete time steps, $\ensuremath{\tau}$, there can be excitation modes, for which coherence is maintained, provided their energy satisfies ${\ensuremath{\epsilon}}_{k}\ensuremath{\tau}\ensuremath{\approx}n\ensuremath{\pi}$ with a positive integer
n. If the dispersion of ${\ensuremath{\epsilon}}_{k}$ is quadratic, the long-time behavior of the entanglement entropy and the logarithmic magnetization is dominated by these ballistically traveling coherent modes and both will have a
t3/4 time dependence.
We study an infinite-range ferromagnetic Ising model in the presence of a transverse magnetic field, which exhibits a quantum paramagnetic-ferromagnetic phase transition at a critical value of the transverse field. In the thermodynamic limit, the low-temperature properties of this model are dominated by the behavior of a single large classical spin governed by an anisotropic Hamiltonian. Using this property, we study the quench and ac dynamics of the model both numerically and analytically, and develop a correspondence between the classical phase-space dynamics of a single spin and the quantum dynamics of the infinite-range ferromagnetic Ising model. In particular, we compare the behavior of the equal-time order-parameter correlation function both near to and away from the quantum critical point in the presence of a quench or ac transverse field. We explicitly demonstrate that a clear signature of the quantum critical point can be obtained by studying the ac dynamics of the system even in the classical limit. We discuss possible realizations of our model in experimental systems.
Based on the obtained exact results we systematically study the quench dynamics of a one-dimensional spin-1/2 transverse field Ising model with zero- and finite-temperature initial states. We focus on the magnetization of the system after a sudden change of the external field and a coherent time-evolution process. With a zero-temperature initial state, the quench magnetic susceptibility as a function of the initial field strength exhibits strongly similar scaling behaviors to those of the static magnetic susceptibility, and the quench magnetic susceptibility as a function of the final field strength shows a discontinuity at the quantum critical point. This discontinuity remains robust and always occurs at the quantum critical point even for the case of finite-temperature initial systems, which indicates a great advantage of employing quench dynamics to study quantum phase transitions.
Out-of-equilibrium behavior is explored in the one-dimensional anisotropic XY model. Initially preparing the system in the isotropic XX model with a linearly varying magnetic field to create a domain-wall magnetization profile, dynamics is generated by rapidly changing the exchange interaction anisotropy and external magnetic field. Relaxation to a nonequilibrium steady state is studied analytically at the critical transverse Ising point, where correlation functions may be computed in closed form. For arbitrary values of anisotropy and external field, an effective generalized Gibbs' ensemble is shown to accurately describe observables in the long-time limit. Additionally, we find spatial oscillations in the exponentially decaying, transverse spin-spin correlation functions with wavelength set by the magnetization jump across the initial domain wall. This wavelength depends only weakly on anisotropy and magnetic field in contrast to the current, which is highly dependent on these parameters.
We consider the
d-dimensional transverse-field Ising model with power-law interactions $J/{r}^{d+\ensuremath{\sigma}}$ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength $\ensuremath{\kappa}$ and distances longer than $a{(J/\ensuremath{\kappa})}^{2/\ensuremath{\sigma}}$ with
a being the lattice spacing, we find a critical scaling regime of correlation and response functions consistent with the model A dynamical universality class with an initial-slip exponent $\ensuremath{\theta}=1$ and dynamical critical exponent $z=\ensuremath{\sigma}/2$. We obtain our results analytically by deriving an effective action for the magnetization field including the noise in a nonperturbative way. The above scaling regime is governed by a nonequilibrium fixed point dominated by the noise fluctuations.