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arXiv:1812.11887

Quench dynamics and defects formation in the Ising chain in a transverse magnetic field

Alexander I. Nesterov, Mónica F. Ramírez

Published Dec 31, 2018• 0 citations
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Abstract

We study analytically and numerically quench dynamics and defects formation in the quantum Ising model in the presence of a time-dependent transverse magnetic field. We generalize the Landau-Ziner formula to the case of non-adiabatic evolution of the quantum system. For a quasi-static magnetic field, with a slow dependence on time, our outcomes are similar to the results predicted by the Landau-Zener formula. However, a quench dynamics under a shock-wave load is more complicated. The final state of the system depends on the amplitude and pulse velocity, resulting in the mixture of ground and excited states and significant density of defects.

DOI
10.48550/arxiv.1812.11887

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We study the quantum dynamics of a one-dimensional spin-12\frac{1}{2}21​ anisotropic XYXYXY model in a transverse field when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. The two quenching schemes are called transverse and anisotropic quenching, respectively. Our emphasis in this paper is on the anisotropic quenching scheme, and we compare the results with those of the other scheme. In the process of anisotropic quenching, the system crosses all the quantum critical lines of the phase diagram where the relaxation time diverges. The evolution is nonadiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to nonadiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau-Zener problem and is generally found to vary as $1∕\sqrt{\ensuremath{\tau}}$, where $\ensuremath{\tau}$ is the characteristic time scale of quenching, a scenario that supports the Kibble-Zurek mechanism. Interestingly, in the case of anisotropic quenching, there exists an additional nonadiabatic transition, in comparison to the transverse quenching case, with the corresponding probability peaking at an incommensurate value of the wave vector. In the special case in which the system passes through a multicritical point, the defect density is found to vary as $1∕{\ensuremath{\tau}}^{1∕6}$. The von Neumann entropy of the final state is shown to maximize at a quenching rate around which the ordering of the final state changes from antiferromagnetic to ferromagnetic.
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We consider the finite-time quench dynamics in the quantum transverse field Ising model which exhibits a second order phase transition from a paramagnetic to a ferromagnetic phase, as the transverse magnetic field is decreased. These dynamics have been thoroughly investigated in previous studies when the critical point is crossed during the quench; here, we quench the system from deep in the paramagnetic phase to just above the critical field so that the system remains in the gapped phase throughout the quench duration. On linearly quenching the infinitely large system, we find that the behavior of mean longitudinal defect density and mean transverse magnetization at the end of the quench falls into three distinct scaling regimes as the quench time is increased. For sufficiently small quench times, these observables remain roughly constant, but for larger quench times, a crossover occurs from the Kibble-Zurek scaling law to the quadratic quench rate law when the Kibble-Zurek time is of the order of relaxation time at the final quench field. These results are shown analytically using power series and uniform asymptotic expansions of the exact solution of the model, and also compared with an adiabatic perturbation theory in the third regime. We find that the above mentioned scaling regimes hold for quenches within the ferromagnetic phase also, and provide a general scaling argument for crossover from the Kibble-Zurek regime to an adiabatic regime for slow quenches within a gapped phase.
We consider the non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that stationary values of local correlation functions can be described by the generalised Gibbs ensemble (GGE). Then we study the full time evo-lution of the transverse magnetisation by means of stationary phase methods. The equal time two-point longitudinal correlation function is analytically derived for a par-ticular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches. ar
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.