Entanglement and fidelity across quantum phase transitions in locally perturbed topological codes with open boundaries

Abstract

We investigate the topological-to-nontopological quantum phase transitions (QPTs) occurring in the Kitaev code under local perturbations in the form of a local magnetic field and spin-spin interactions of the Ising-type using fidelity susceptibility (FS) and entanglement as the probes. We assume the code to be embedded on the surface of a wide cylinder of height M and circumference D with M≪D. We demonstrate a power-law divergence of FS across the QPT and determine the quantum critical points via a finite-size-scaling analysis. We verify these results by mapping the perturbed Kitaev code to the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions and computing the single-site magnetization as the order parameter using the quantum Monte Carlo technique. We also demonstrate a finite-size odd-even dichotomy in the occurrence of the QPT in the Kitaev ladder with respect to the odd and even values of D, when the system is perturbed with only Ising interaction. Our results also indicate a higher robustness of the topological phase of the Kitaev code against local perturbations if the boundary is made open along one direction. We further consider a local entanglement witness operator designed specifically to capture a lower bound to the localizable entanglement on the vertical nontrivial loop of the code. We show that the first derivative of the expectation value of the witness operator exhibits a logarithmic divergence across the QPT and perform the finite-size-scaling analysis. We demonstrate similar behavior of the expectation value of the appropriately constructed witness operator also in the case of a locally perturbed color code with open boundaries.