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In Wen's Quantum Field theory of many body systems, on page 254, it discusses Z2 gauge theory, and states that

Count the number of states in the Z2 gauge theory on a finite square lattice. We assume that the lattice has a periodic boundary condition in both directions (i.e the lattice forms a torus). If the lattice has $N_{site}$ sites, then it has $2N_{site}$ nearest neighbor links.

This is very confusing. If I take a square lattice, I see that each point has two nearest neighbors. However, if I extend the lattice into a 3rd dimension, I would also expect nearest neighbors behind and after the lattice "slice", giving me 4N nearest neighbors. I was told that it has to do with combinatorics and the prefactor for feynman diagrams, but I'm wondering if there isn't a simpler explanation. And if there isn't, where can I learn this type of information?

Any help greatly appreciated!

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In three dimensions, there are three nearest-neighbor links per site on a cubic lattice: in the $\hat x$, $\hat y$, and $\hat z$ directions. You don’t count the one in the $-\hat z$ direction, just like you don’t fount the ones in the $-\hat x$ and $-\hat y$ directions. They are the positive-direction links for neighboring sites, and you would be double-counting.

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  • $\begingroup$ Wouldn’t this lead to 3N nearest neighbors instead of 2N? $\endgroup$
    – Некто
    Commented Sep 19, 2019 at 18:49
  • $\begingroup$ $N$ in one dimension; $2N$ in two dimensions; $3N$ in three dimensions. You don’t get $4N$ in three dimensions like you thought. $\endgroup$
    – G. Smith
    Commented Sep 19, 2019 at 18:55
  • $\begingroup$ And this has nothing to do with combinatorics or Feynman diagrams. It’s just geometry. $\endgroup$
    – G. Smith
    Commented Sep 19, 2019 at 18:57
  • $\begingroup$ Hmm okay so when it says the lattice forms a torus it means 2D? But isn’t that increasing the dimension by going from a 2D lattice to a 3D torus? $\endgroup$
    – Некто
    Commented Sep 21, 2019 at 3:12
  • $\begingroup$ I’m trying to reconcile what you said with what the text says $\endgroup$
    – Некто
    Commented Sep 21, 2019 at 3:12

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