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Let's say, that we have some spin system with multiple sites. This system is in a quantum state denoted by $|\psi \rangle$. When we compute e.g. $\langle S_1^z \rangle = \langle \psi | S_1^z | \psi \rangle$, this can be understood as measuring the expected value of spin in the $Z$ axis on the first site.

But we can also calculate the value $\langle S_1^z S_2^z \rangle$. What is its physical interpretation? Can we compare this to measuring expected value of combined spins at sites 1 and 2 in the $Z$ direction?

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    $\begingroup$ The "angular brackets" as you say, corresponds to measuring the expected value. I.e., if you measure the observable many times, collect the results and take the average, that is what you would obtain in the limit of many measurements. However I would say the best intuition you get is that your quantity is a "correlator" $\endgroup$ – lcv Sep 10 at 13:15
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The physical interpretation is that this is an approximation to the nearest-neighbor spin-spin interaction energy.

A charged particle with spin behaves as a magnetic dipole whose dipole moment is proportional to the spin, and the interaction energy of two magnetic dipoles $\mathbf{m}_1$ and $\mathbf{m}_2$ with separation $\mathbf{r}$ is known to be

$$-\frac{\mu_0}{4\pi}\frac{3(\mathbf{m}_1\cdot\hat{\mathbf{r}})(\mathbf{m}_2\cdot\hat{\mathbf{r}})-\mathbf{m}_1\cdot\mathbf{m}_2}{r^3}.$$

If you take the spins to be on a lattice in the $xy$ plane, and oriented at right angles to that plane, then this becomes proportional to $S_1^zS_2^z$.

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Now I believe, that it's hard to assign some physical meaning to this value, but it can be used to calulate the correlation (as pointed by @lcv), that is:

$$ Corr(S_i^z, S_j^z) = \frac{\langle S_i^z \cdot S_j^z \rangle - \langle S_i^z \rangle \cdot \langle S_j^z \rangle}{\sqrt{\langle {(S_i^z)}^2 \rangle - \langle S_i^z \rangle ^2}\sqrt{\langle {(S_j^z)}^2 \rangle - \langle S_j^z \rangle ^2}} $$

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