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I want to study the Ising Gauge Theory which has the Hamiltonian \begin{equation} \mathcal{H} = -J \sum_{n,\mu,\nu} \sigma_3(n,\mu)\sigma_3(n+\mu,\nu)\sigma_3(n+\mu+\nu, -\mu)\sigma_3(n+\nu, -\nu) \end{equation} where $n$ labels a node and $\mu, \nu$ are unit vectors along the lattice. The spins ($\sigma_3 = \pm 1$) are placed on the bonds between two nodes. I wanted to study this model in 2 and 3 dimensions by calculating it's partition function where I naively want to define the $i$th plaquette as

plaq[i_] = sxb[i] syb[i] sxt[i] syt[i]

where sxt[i] is the spin on the x axis for the 'top' bond of the $i$th square and sxb[i] is the spin on the bottom bond. And similar definition for the y bonds. But this term cannot be summed easily over the squares since Sum does not sum over both the spin variables and their label i. Hard coding the sum is a very inefficient way to do this, so I was looking for better ways. Is there a way Mathematica can do this sum? Efficiency is not an issue since I will not examine very large lattices. Also, help for the same with a 3D lattice would be greatly appreciated. Thanks a lot.

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One option is to specify each node as a pair of integers, {x,y}. You might also consider placing your spins at the nodes rather than on the edges. There will be four spins/plaquette either way, and the spin associated with a link can be though of as the product of the two spins on the nodes.

Start by randomly assigning spins to your lattice. I'll assume it runs from {1,xMax} and {1,yMax}:

lattice = 2 RandomInteger[{0,1},{xMax,yMax}] -1/2;

The generation is handled this way to omit the zero that results if I simply try to use RandomInteger[{-1,1}].

You can define a plaquette by passing the coordinates of the lower left corner, with spins positioned at the four corners. To retrieve the current values, define getPlaquette:

getPlaquette[{x_,y_}] := lattice[[Sequence@@#]]& /@{{x,y},{x+1,y},{x+1,y+1},{x,y+1}};

To set an individual node's spin to a new value, use setNodeSpin:

setNodeSpin[{x_,y_}, value_Integer] := lattice[[x,y]] = value;

You can also write a function to set all four spins at the corner of a plaquette using by calling setNodeSpin multiple times with the appropriate coordinates passed in.

Your Hamiltonian sum then becomes a sum over {x,1,xMax}, {y,1,yMax}.

If you need to specify a single identifier for each node, you can take the {x,y} pair and convert it in a similar manner to that used to index a multi-dimensional array as a 1D array, for example:

nodeId[x_,y_] := (x-1)*xMax + y;
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  • $\begingroup$ Getting {1, 1, 1, -1} on the bonds cannot be done by multiplying spins on the nodes. $\endgroup$
    – QFTheorist
    Nov 18, 2023 at 12:35

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