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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.

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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.

In higher dimensions, Jordan-Wigner transformation can still be defined, but it will not preserve locality and therefore is not very useful. For any lattice structure in any dimension, you can choose a canonical ordering of sites, such as

enter image description here

You can still define fermion operators as a Jordan-Wigner string followed by a $\sigma^{\pm}$ operator, for example,

enter image description here

However local fermion hopping operators on the 2D lattice will be translated to a nonlocal operator in the spin basis involving a long segment of Jordan-Wigner string wrapping along one direction.

enter image description here

If your goal is to represent fermion operators as Pauli matrices for numerical modeling the many-body Hamiltonian, then this is the correct way to go (this is how we perform exact-diagonalization). But if your goal is to use it to define some spin model, then the nonlocality will make the problem even harder to solve.

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  • $\begingroup$ This is an awesome response thank you. I guess my goal is less to represent fermion operators as Pauli matrices and more to do with simply numerically modeling second quantized Hamiltonians. I thought the JW transformation was the best way to do that in order to explicitly write the annihilation/creation operators, otherwise, I'm not sure what they look like for higher dimensions. Would having it be nonlocal ruin this and so using the JW transformation is not the best way to do this? Sorry if that is more of a programming question than a physics one. $\endgroup$
    – user571688
    Jul 20 at 14:44

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