# What does the notation $\sigma_j^z$ mean for Pauli matrices?

In multiples papers or online article on the QAOA algorithm (such as this one), I found notation for the Hamiltonian similar to this one :

$$\sum_{ij} \frac{1}{2} (I-\sigma_i^z \sigma_j^z)$$

I don't know how to interpret the notation $$\sigma_j^z$$. I guess it has something to do with the Pauli matrix $$\sigma_z$$ but that's all I can understand.

What $$\sigma^z_i$$ means is that you've got a Pauli-$$Z$$ applied to qubit $$i$$, and nothing else on the other qubits (i.e. the identity). So, you could expand it as $$I^{\otimes(i-1)}\otimes\sigma^z\otimes I^{n-i}$$ if your system has $$n$$ qubits. A term such as $$\sigma^z_i\sigma^z_j$$ is then a product of two of these, which is equivalent to the tensor product of $$\sigma^z$$ on qubits $$i$$ and $$j$$ and identity everywhere else. The summation is usually taken over pairs $$i,j$$ which are nearest-neighbours on some underlying geometry of the qubits.