How is the Ising model an example of a lattice model as per Kontsevich?

Question

In section 3.2 of Kontsevich's very interesting paper "Notes on motives in finite characteristic,", he gives an axiomatic definition of a "lattice model" attached to a Boltzmann datum $(V_1,V_2,R)$, where $V_1$ and $V_2$ are vector spaces and $R$ is a linear endomorphism of $V_1$ tensor $V_2$. He remarks that the 2-dimensional Ising model is an example. Can someone explain to me what $V_1$, $V_2$, and $R$ are for the 2-dimensional Ising model?

Answer 1

In this section, Kontsevich is describing a pattern of index contraction of a certain tensor which reproduces a statistical mechanics sum over configurations. For his model, you have at each vertex, which I will label by the integer $k$, a copy of the same four index tensor

$$R_{a\mu}^{b\nu}$$

where $a$ and $b$ are indices for one vector space of dimension $d_1$, $\mu$ and $\nu$ indices are for another vector space of dimension $d_2$. The dimensions of the vector space are going to be the number of discrete states at each site.

You have two different directions, he calls them 1 and 2, but I will call them "up" and "left". The image in your head should be a finite square grid, and at each point there are two incoming arrows, one going up and one going left, and two outgoing arrows going up and left.

He multiplies together all the $R$'s at all the vertices, and contracts the $a$-index of each site with the $b$-index of the site which is "left" of it, and the $\mu$ index of every site with the $\nu$ index of the site which is "up" of it. This contracts all the indices in closed cycles, giving a complex number.

In the spirit of maximum generalization, he defines the construction on the most general graph that allows it: the graph needs up arrows and down arrows, and the up arrows make closed oriented cycles, and the left arrows make closed oriented cycles. Maximally generalizing in another direction, he notes that the same definition works for any objects with a tensor product and a notion of contraction, i.e. for tensor categories.

Sliding the vertices along the two kinds of arrows, you get two independent "flows" which are number conserving on a finite graph, so the trajectories of the flows are cycles, and on each cycle, the flow is a cyclic permutations of the vertices. For the Ising model on an square $N$-by-$M$ lattice with periodic boundaries, one of the two flows moves all the vertices left by one unit, and the other moves all the sites up one unit, and these operations commute, and he likes this condition, so he emphasizes it.

The partition function with this pattern of contraction is then

$$Z = \sum_{a_k,\mu_k} \prod_k R_{a_k\mu_k}^{a_{l(k)}\mu_{u(k)}}$$

where $l(k)$ is the left-neighbor of $k$, and $u(k)$ is the up-neighbor of $k$. That is, $Z$ is a sum over all possible configurations of values of the indices $a_k, \mu_k$ on the vertices of the graph, of the product of a certain quantity which depends on the value of the indices at the site and the right and up neighbor. In terms of the logarithm of $R$,

$$R_{a\mu}^{b\nu} = \exp(-E(a,\mu;b,\nu)),$$

where $E$ is the energy function, then

$$Z = \sum_{a_k,\mu_k} \exp(-\sum_k E(a_k,\mu_k;a_{l(k)},\mu_{u(k)}))$$

where the big sum outside is over all the possible assignments of indices $a_k$ and $\mu_k$ to each of the vertices, and $l(k)$ is the left-map taking vertex number $k$ to the number of the left neighbor, while $u(k)$ is the up-map, taking $k$ to the up neighbor.

To reproduce the Ising model, let $a_k$ and $\mu_k$ take the two "spin" values 0 or 1 (i.e. the two vector spaces $V_1$ and $V_2$ are both 2 dimensional). Then you want to make sure you get a zero contribution unless the index value $a_k$ is equal to $\mu_k$. For this purpose, make the energy function $E(a,\mu;b,\nu)$ infinite unless $a=\mu$ (i.e. make the corresponding $R$ element zero). Then the big sum on the outside collapses to a sum over $a_k$. The following values of $E$ are the finite ones:

• $E(00;00) = E(11;11) = 0$ (the up neighbor and the left neighbor are the same)
• $E(00;01) = E(11;10) = J$ (the up neighbor is different)
• $E(00;10) = E(11;01) = J$ (the left neighbor is different)
• $E(00;11) = E(11;00) = 2J$ (both neighbors are different)

The nonzero $R$ tensor elements are the exponentials of these. The coupling $J$ is the standard extra energy cost for mismatched neighboring spins.

Reconstructing what he is doing can be a little difficult because he says "correspond" in a way that is vague on the top of page 22, and "an identification" on the bottom of page 21 without specifying the correspondence or the identification. This is why I went into great detail.

Answer 2

For the 2d-Ising model $V_1=V_2$ is a $2$-dimensional vector space. The formula for $R$ may be find in Baxter's book "Exactly Solved Models in Statistical mechanics" (chapter 7 if I remember well).

The point is that the nearest-neighbor lattice Ising model is not a vertex model... but for a square lattice in 2d it is: one has to consider the dual lattice.

EDIT: this paper of Baxter might help you.