emphasize that here we considered the periodic boundary condition in order to better explain the topological order in the Toric code. However, in the following, we will return to the open boundary condition which is more convenient for our calculations.

![(Color online) A schematic of phase diagram of random bond Ising model where ferromagnetic and paramagnetic phases are separated by a critical curve. There is also a Nishimori line where $p=\frac{e^{-2\beta J}}{1+e^{-2\beta J}}$. []{data-label="schem"}](schem){width="6cm" height="4cm"}

Quantum formalism for the order parameter {#sec2}
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2D Ising model is a well-known statistical mechanical model which can be used for describing the ferromagnetic-paramagnetic phase transition when all couplings in Eq.(\[ran\]) are fixed to $J$. However, one can consider a quenched disorder for couplings to generate a random bond Ising model where each coupling might be $J$ with a probability of $1-p$ and might be $-J$ with a probability of $p$. The set of all edges $E$ then includes two subsets of $E_1$ and $E_2$ where $E_1 \cup E_2 =E$; all edge couplings in subset $E_1$ are fixed to $+J$ and the couplings in subset $E_2 $ are fixed to $-J$. It has been shown that such a model can still show a ferromagnetic-paramagnetic 