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I've simulated a few 2d Ising models at critical temperature on triangular lattice and I'm now trying to check that the correlation functions are right. I alraedy did it for the spin operator ($\sigma$) and I'd like to do the same for the local energy operator that should satisfy the following space dependence:

$$\langle\epsilon (x) \epsilon (y) \rangle \propto \frac{1}{(x-y)^2}$$

(scaling weight of 1)

but I'm not sure of the definition of this operator, I would guess something as:

$$\epsilon(x)=-J\,\sigma(x)\sum_{\text{neighbours}}\sigma$$

where J is the critical coupling ($=\frac{1}{T_{\text{crit}}}$)

Is that right?

Edit: new research led me to reject my first proposition: This energy operator should be odd under the $\text{Z}_2$ symmetry of the system $\sigma \leftrightarrow - \sigma$ which is clearly not the case of my previous guess.

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  • $\begingroup$ You have to recenter your random variable, i.e., subtract $\langle\epsilon(x)\rangle$ from $\epsilon(x)$. That's analogous to Wick ordering for the square of the elementary spin field. $\endgroup$ – Abdelmalek Abdesselam Apr 11 '17 at 15:36
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The energy operator is even under the $\mathbb{Z}_2$ that changes the sign of the spin operator. Look at the Hamiltonian in the absence of an external magnetic field -- does it change sign under the symmetry? It doesn't.

I think your working definition of the energy operator is fine -- you might want to divide your expression by $2$ since an edge is shared by two sites. Either way, that will not affect the scaling exponent which is the main objective of the computation. For simplicity, you may even consider dropping $J$ from the definition.

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