Confusion about notation for block transformation in Ising model

Question

I'm going through Cardy's "Scaling and Renormalization in Statistical Physics", and I've run across a notational confusion. Consider a 2D Ising system with the following Hamiltonian

$$\mathcal{H}(s)=-\frac{1}{2}\sum_{r,\bar{r}}J(r,\bar{r})s(r)s(\bar{r})-\mu H\sum_{r}s(r).$$

We now would like to perform a block transformation, where we take 9x9 blocks and project them down to a single lattice site $s'$. The spin of this new site is defined by a majority rule, which can be implemented by the following projection operator.

$$ T(s';s_1,s_2,...,s_9)=\begin{cases} 1, & s'\sum_i s_i>0; \\ 0 & \mathrm{otherwise.} \\ \end{cases} $$

The following is where I get confused. He then defines the new block Hamiltonian by

$$e^{-\mathcal{H}'(s')}=\mathrm{Tr}_s\prod_{\mathrm{blocks}}T(s';s_i)e^{-\mathcal{H}(s)}.$$

Is $s_i$ supposed to stand for individual lattice sites? Or does it stand for a collection of lattice sites? The latter makes sense, since we are talking about a product of projection operators for each block.

Answer 1

A more explicit way of writing this would be

$$e^{-H'(s')}=\mathrm{Tr}_s \prod_{q=0}^{N'} T(s'_q;s^{B_q}_1,s^{B_q}_2,\dots s_9^{B_q})e^{-H(s)} $$

where $B_q$ denotes the blocks, $s_k^{B_q}$ is the $k$-th spin in the $q$-th block and $N'=N/9$ is the number of blocks (or of new spins)

Answer 2

It looks like $T(s'; s_i)$ is shorthand for $T(s';s_1,s_2,s_3,s_4,s_5,s_6,s_7,s_8,s_9)$. This sort of abbreviation is pretty common.