The nonthermal quantum relaxation of the magnetization under nonequilibrium initial conditions is studied for the transverse Ising chain and for the one-dimensional isotropic XY model. In the absence of a heat bath the transverse Ising system exhibits an oscillating relaxation with a decaying amplitude. At the critical transverse field some stationary properties become independent of the initial state. The isotropic XY chain with a spatially inhomogeneous initial state shows a slow, modulated algebraic decay of the magnetization. Analysis of the two-time autocorrelation function reveals the phenomenon of aging.
We introduce a model with conserved dynamics, where nearest neighbor pairs of spins ↑ ↓ (↓↑) can exchange to assume the configuration ↓ ↑ (↑↓), with rate β (α), through energy decreasing moves only. We report exact solution for the case when one of the rates, α or β, is zero. The irreversibility of such dynamics results in strong dependence on the initial conditions. Domain wall arguments suggest that for more general models with steady states the dynamical critical exponent for the anisotropic spin exchange is different from the isotropic value. – 1
Exact analytic calculations in spin-1/2
XY chains show the presence of long-time tails in the asymptotic dynamics of spatially inhomogeneous excitations. The decay of inhomogeneities for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty},$ is given in the form of a power law $(t/{\ensuremath{\tau}}_{Q}{)}^{\ensuremath{-}{\ensuremath{\nu}}_{Q}},$ where the relaxation time ${\ensuremath{\tau}}_{Q}$ and the exponent ${\ensuremath{\nu}}_{Q}$ depend on the wave vector Q, characterizing the spatial modulation of the initial excitation. We consider several variants of the
XY model (dimerized, with staggered magnetic field, with bond alternation, and with isotropic and uniform interactions), that are grouped into two families, whether the energy spectrum has a gap or not. Once the initial condition is given, the nonequilibrium problem for the magnetization is solved in closed form, without any other assumption. The long-time behavior for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ can be obtained systematically in a form of an asymptotic series through the stationary phase method. We found that gapped models show critical behavior with respect to Q, in the sense that there exist critical values
Qc where the relaxation time ${\ensuremath{\tau}}_{Q}$ diverges and the exponent ${\ensuremath{\nu}}_{Q}$ changes discontinuously. At those points, a slowing down of the relaxation process is induced, similarly to phenomena occurring near phase transitions. Long-lived excitations are identified as incommensurate spin density waves that emerge in gapped systems, as a consequence of both approximate nesting of the spectrum and the degeneracy of some stationary points. In contrast, gapless models do not present the above anomalies as a function of the wave vector Q.
We introduce a model with conserved dynamics, where nearest neighbor pairs of spins ↑↓ (↓↑) can exchange to assume the configuration ↓↑ (↑↓), with rate β(α), through energy decreasing moves only. We report exact solution for the case when one of the rates, α or β, is zero. The irreversibility of such zero-temperature dynamics results in strong dependence on the initial conditions. Domain wall arguments suggest that for more general, finite-temperature models with steady states the dynamical critical exponent for the anisotropic spin exchange is different from the isotropic value.
We reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap
We derive the exact contribution of two spinons to the dynamical correlation function of the spin-\textonehalf{} Heisenberg model. For this, we use the isotropic limits of the exact form factors that have been recently computed through the quantum affine symmetry of the anisotropic Heisenberg model
XXZ.