Defects and their Time Scales in Quantum and Classical Annealing of the Two-Dimensional Ising Model

Abstract

We investigate defects in the two-dimensional transverse-field Ising ferromagnet on periodic $L\times L$ lattices after quantum annealing from high to vanishing field. With exact numerical solutions for $L \le 6$, we observe the expected critical Kibble-Zurek (KZ) time scale $\propto L^{z+1/\nu}$ (with $z=1$ and $1/\nu \approx 1.59$) at the quantum phase transition. We also observe KZ scaling of the ground-state fidelity at the end of the process. The excitations evolve by coarsening dynamics of confined defects, with a time scale $\propto L^2$, and interface fluctuations of system-spanning defects, with life time $\propto L^3$. We build on analogies with classical simulated annealing, where we characterize system-spanning defects in detail and find differences in the dynamic scales of domain walls with winding numbers $W=(1,0)/(0,1)$ (horizontal/vertical) and $W=(1,1)$ (diagonal). They decay on time scales $\propto L^3$ (which applies also to system-spanning domains in systems with open boundaries) and $\propto L^{3.4}$, respectively, when imposed in the ordered phase. As a consequence of $L^{3.4}$ exceeding the classical KZ scale $L^{z+1/\nu}=L^{3.17}$ the probability of $W=(1,1)$ domains in SA scales with the KZ exponent even in the final $T=0$ state. In QA, also the $W=(1,0)/(0,1)$ domains are controlled by the KZ time scale $L^{2.59}$. The $L^3$ scale can nevertheless be detected in the excited states, using a method that we develop that should also be applicable in QA experiments.