# What is the proper use of Jordan-Wigner transformation for second quantized Hamiltonian simulations?

I am trying to learn how to perform simulations on fermionic second quantized Hamiltonians of the form
$$H =\sum_{i,j}^{NN} Jc_{i}^{\dagger}c_j$$ Where $$c_{i}^{\dagger}(c_j)$$ are the creation/annihilation operators. However, a few questions came up when I was trying to figure this out. The best way I know how to do this is to essentially write the Hamiltonian as one big matrix in MATLAB, however, I am struggling to completely understand how to do that when going to dimensions greater than 1. For the 1D case, it is my understanding that the 1D Jordan-Wigner (JW) transformation needs to be used. A lot of papers I have seen list the JW transformation being of the following form
$$c_{i}^{\dagger} = \exp(-i\pi\sum_{k=1}^{i-1}\sigma_{i}^{\dagger}\sigma_i)\times\sigma_{i}^{+}$$ $$c_{i} = \exp(i\pi\sum_{k=1}^{i-1}\sigma_{i}^{\dagger}\sigma_i)\times\sigma_{i}^{-}$$ Where $$\sigma_{i}^{\pm}$$ are the Pauli raising/lowering operators. My first question is this: I see many authors stating the above, then I see others saying that the transformation is given by the following: $$c_{i}^{\dagger} = \overbrace{\sigma_z \otimes \sigma_z \otimes ...\otimes \sigma_z}^{i-1}\otimes \sigma_{i}^{+}\otimes I ... \otimes I$$ $$c_{i} = \overbrace{\sigma_z \otimes \sigma_z \otimes ...\otimes \sigma_z}^{i-1}\otimes \sigma_{i}^{-}\otimes I ... \otimes I$$ Where $$\sigma_z$$ is the Pauli matrix in the z-direction. Where does the tensor product come from? I see some authors placing a minus sign in front of the Pauli matrix in the z-direction, when does this occur? The above relation is easy to implement in MATLAB but I wanted to understand better how the exponential relation gets to the tensor product relation.

Lastly, what would the JW transformation look like in higher dimensions? Being a visual learner and liking examples, I think I am being bogged down by some of the notation in journal articles that I have read on higher dimensional JW transformations. From what I have seen, a similar exponential relation occurs, but how does this look in tensor product notation, if applicable? If you are trying to find the correct annihilation/creation operators in the (mn)th site, do you just have (mn-1) Pauli z matrices before the raising/lowering operator in the tensor product when considering a square lattice? Lastly, for more general lattices (such as a triangular lattice), how is it represented? I'd imagine the exponential relation would have some sort of phase relation between the unit vectors of the lattice that would move it away from being related to the Pauli z matrix, but some other matrix that you could maybe then use the tensor product notation with?

Any help would be appreciated.

The two different ways of expressing the fermion operator in terms of spin (qubit) operators are equivalent, because of the simple fact that $$\exp(a+b)=\exp(a)\exp(b)$$, which applies if $$a$$ and $$b$$ are commuting operators (such that they can be treated as ordinary numbers). Applying this general idea to the expression, one can check that $$\begin{split} \exp\left(\pm\mathrm{i}\pi\sum_{k=1}^{i-1}\sigma_k^+\sigma_k^-\right)&=\prod_{k=1}^{i-1}\exp\left(\pm i\pi\sigma_k^+\sigma_k^-\right)\\ &=\bigotimes_{k=1}^{i-1}\exp\left(\pm i\pi\left[\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\right]\left[\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right]\right)\\ &=\bigotimes_{k=1}^{i-1}\exp\left(\pm i\pi\left[\begin{smallmatrix}0&0\\0&1\end{smallmatrix}\right]\right)\\ &=\bigotimes_{k=1}^{i-1}\left[\begin{smallmatrix}e^{0}&0\\0&e^{\pm i\pi}\end{smallmatrix}\right]\\ &=\bigotimes_{k=1}^{i-1}\left[\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\right]\\ &=\bigotimes_{k=1}^{i-1}\sigma^z \end{split}$$ Note that the $$\pm$$ sign in front of $$\mathrm{i}\pi$$ does not make any difference in the final result, so you can choose the sign as you wish. This string of product of $$\sigma^z$$ operators is also called the Jordan-Wigner string.
You can still define fermion operators as a Jordan-Wigner string followed by a $$\sigma^{\pm}$$ operator, for example,