Mechanism of defect formation in the quantum annealing of the random transverse-field Ising chain

Abstract

Based on the strong-disorder renormalization group method, a microscopic mechanism of defect formation in the quantum annealing of the random transverse-field Ising chain is proposed, which represents the annealing process as a gradual aggregation of strongly coupled spin clusters. The ferromagnetic ground state of clusters is either preserved or get excited in pairwise fusions of clusters, depending on the effective annealing rate of the fusion, the latter events being responsible for the appearance of defects in the final state. A consequence of the theory is that, although the Griffiths-McCoy phases surrounding the critical point are gapless, they are still effectively gapped from the point of view of quantum annealing. Thereby we provide an explanation of the finiteness of gap outside of the critical point, which was implicit in an early approach to the problem by Kibble-Zurek scaling [Dziarmaga, Phys. Rev. B {\bf 74}, 064416 (2006)]. Furthermore, by identifying the accessible excitations, we refine the functional form of the vanishing of the gap at the critical point. The defect density in the final state is found to decrease with the annealing time $\tau$, as $n(\tau)\sim \ln^{-2}\left(\frac{\tau}{\ln^2\tau}\right)$ for large $\tau$. In addition to this, our approach gives access also to the density of defects at intermediate times of the annealing process.