Toric Code and Random Bond Ising Model

Question

It was established by Dennis, Kitaev et al. that the 2D Toric Code can be mapped to a 2D Random Bond Ising Model. The original derivation was given in the paper "Topological quantum memory" which can be found in J. Math. Phys. 43, 4452 (2002); doi: 10.1063/1.1499754 or online at http://dx.doi.org/10.1063/1.1499754 . A free arXiv preprint version is available online here.

The derivation that shows that the 2D Toric Code can actually mapped to a 2D Random Bond Ising Model is given in section IV - D "Derivation of the model" (Starting page 4469).

The point where I'm getting lost is when he is trying to set up a function that for a fixed chain E outputs the probability of a "homotopically equivalent" chain $E'$ (which he calls $p(E'|E)$).

To obtain this function he first calculates the probability of a link being occupied that lies on $E'$ but not on $E$. He obtains (Eq. 13) that the probability is equal to

\begin{equation} \left(\frac{p}{1-p}\right)^{n_{C}(\ell)} \end{equation}

up to an overall normalization and with $C$ being the cycle corresponding to $E'$.

I don't really see how he arrives at that results or what the overall normalization factor actually is (after all "normalization" is crucial when talking about probabilities - what is the meaning of probability 35 if I don't know the normalization?) Then he obtains in Eq. (15) an explicit exponential-like form for $p(E'|E)$ which isn't clear to me neither. He also seems to omit a lot of explanations at least at this stage.

Could anyone explain to me in a bit more details how the derivation is carried out in more detail? This would really help me a lot.

Answer 1

I'm not completely sure of the following - if you have any comments/questions about it I'd be very happy to hear them.

Calculating $P(E'|E)$ comes from the standard expression $P(E') = P(E'|E)P(E)$, so this is why we're looking for the ratio of $P(E')/P(E)$.

Start from equation 12:

$P(E) = \prod_{\ell} (1-p) \prod_{\ell} \left(\frac{p}{1-p}\right)^{n_E(\ell)}$,

and

$P(E') = \prod_{\ell} (1-p) \prod_{\ell} \left(\frac{p}{1-p}\right)^{n_{E'}(\ell)}$.

Dividing these through, we see

$\frac{P(E')}{P(E)} = \prod_{\ell} \left(\frac{p}{1-p}\right)^{n_{E'}(\ell)-n_E(\ell)}$.

So depending on whether $n_{E'}(\ell) = 0, n_E(\ell) =1$ or $n_{E'}(\ell) = 1, n_E(\ell) = 0$ we recover equation 13 or 14. I think they say it's proportional because you have to take the product across the whole lattice.

I'm not sure how they get to equation 15 yet but if I figure it out I'll post it.

Answer 2

I'm not sure if this quite solves your problem, but maybe it gives a bit of insight. Suppose you have have a single spin in a magnetic field, such that the energy gap between spin up (low energy) and spin down (high energy) is $1$ (in some units). In thermal equilibrium at some inverse temperature $\beta$, the probability to be spin down is then,

$$ p = \frac{e^{-\beta}}{1+e^{-\beta}} $$

rearranging, we find,

$$ \frac{p}{1-p} = e^{-\beta}. $$

So you could also go the other way. Given an event that occurs with probability $p$, like the outcome 'heads' on a biased coin, you can think of it as a spin in a field at temperature $\beta (p)$. This allows us to map probabilistic events to statistical physics models. This is what is going on when we map errors on the TC to the RBIM. There's other complications, of course, but hopefully this explanation will help you navigate them a little better.