  We study the interplay between the Kitaev and Ising interactions on both
ladder and two dimensional lattices. We show that the ground state of the
Kitaev ladder is a symmetry-protected topological (SPT) phase, which is
protected by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. It is confirmed by
the degeneracy of the entanglement spectrum and non-trivial phase factors
(inequivalent projective representations of the symmetries), which are obtained
within infinite matrix-product representation of numerical density matrix
renormalization group. We derive the effective theory to describe the
topological phase transition on both ladder and two-dimensional lattices, which
is given by the transverse field Ising model with/without next-nearest neighbor
coupling. The ladder has three phases, namely, the Kitaev SPT, symmetry broken
ferro/antiferromagnetic order and classical spin-liquid. The non-zero quantum
critical point and its corresponding central charge are provided by the
effective theory, which are in full agreement with the numerical results, i.e.,
the divergence of entanglement entropy at the critical point, change of the
entanglement spectrum degeneracy and a drop in the ground-state fidelity. The
central charge of the critical points are either c=1 or c=2, with the
magnetization and correlation exponents being 1/4 and 1/2, respectively. In the
absence of frustration, the 2D lattice shows a topological phase transition
from the $\mathbb{Z}_2$ spin-liquid state to the long-range ordered Ising phase
at finite ratio of couplings, while in the presence of frustration, an
order-by-disorder transition is induced by the Kitaev term. The 2D classical
spin-liquid phase is unstable against the addition of Kitaev term toward an
ordered phase before the transition to the $\mathbb{Z}_2$ spin-liquid state.

Strongly Correlated Electrons;Quantum Physics