# Hamiltonian for a 1D spin chain [closed]

I am trying to implement the Lanczos algorithm to tridiagonalize the Hamiltonian for a 1D spin chain of length $$L$$, but I am unable to decipher from my professor's notes (here's a link), what the action the Hamiltonian has on a random vector (or for that matter what the Hamiltonian is). My touble arises at Eqn. 20 in these notes. They say that the Hamiltonian is $$\frac{1}{2}\bigg(\sum_{i=0}^{L-1}P_{ij}-\frac{L}{2}I\bigg).$$ However, this is really confusing to me since if $$P_{ij}$$ is what he defined in Eqn. 18, then the resulting matrix is just a 4 by 4 matrix and not $$2^L\times 2^L$$ as he claims it should be. If it's not the case that $$P_{ij}$$ is the same as in Eqn. 18, then what is it, and how do I compute this Hamiltonian (or at the very least) the Hamiltonian's action on a vector, $$v$$?

## closed as unclear what you're asking by Norbert Schuch, Kyle Kanos, ZeroTheHero, Jon Custer, BuzzDec 29 '18 at 2:53

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Implicitly each of those summands is $$I^{\otimes (i-1)} \otimes P_{ij} \otimes I^{\otimes k}$$ so that each summand acts as the identity on all but two of the spins so maybe $$P_{ij}$$ is only 4 by 4 but this extension with the identity operators is actually $$2^L$$ by $$2^L$$. I put $$k$$ here just to say the rest it is something like $$L-i$$ but I may be off by 1 or 2 and didn't check which.