Entanglement and fidelity across quantum phase transitions in locally perturbed topological codes with open boundaries
K J Harikrishnan, Amit Kumar Pal · Physical Review A · 2025
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 $M\ensuremath{\ll}D$. 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.