  It is well-known that the partition function of a classical spin model can be
mapped to a quantum entangled state where some properties on one side can be
used to find new properties on the other side. However, the consequences of the
existence of a classical (critical) phase transition on the corresponding
quantum state has been mostly ignored. This is particularly interesting since
the classical partition function exhibits non-analytic behavior at the critical
point and such behavior could have important consequences on the quantum side.
In this paper, we consider this problem for an important example of Kitaev
toric code model which has been shown to correspond to the two-dimensional (2D)
Ising model though a duality transformation. Through such duality
transformation, it is shown that the temperature on the classical side is
mapped to bit-flip noise on the quantum side. It is then shown that a
transition from a coherent superposition of a given quantum state to a
non-coherent mixture corresponds exactly to paramagnetic-ferromagnetic phase
transition in the Ising model. To identify such a transition further, we define
an order parameter to characterize the decoherency of such a mixture and show
that it behaves similar to the order parameter (magnetization) of 2D Ising
model, a behavior that is interpreted as a robust coherency in the toric code
model. Furthermore, we consider other properties of the noisy toric code model
exactly at the critical point. We show that there is a relative stability to
noise for the toric code state at the critical noise which is revealed by a
relative reduction in susceptibility to noise. We close the paper with a
discussion on connection between the robust coherency as well as the critical
stability with topological order of the toric code model.
