- #1
vbrasic
- 73
- 3
Homework Statement
[/B]
A 1D spin chain corresponds to the following figure:
Suppose there are ##L## particles on the spin chain and that the ##i##th particle has spin corresponding to ##S=\frac{1}{2}(\sigma_i^x,\sigma_i^y,\sigma_i^z)##, where the ##\sigma##'s correspond to the Pauli spin matrices in the $z$-basis, so that ##\sigma_i^z## is diagonal. The goal of my code is to implement the Lanczos algorithm to tri-diagonalize the Hamiltonian for a 1D spin chain. However, to do so, I need to know the action of the Hamiltonian on a random vector ##v##. However, I'm having a lot of trouble computing the Hamiltonian/it's action to begin with.
Homework Equations
The Attempt at a Solution
[/B]
Here is where I am getting confused. My professor defines $$S_{i}\cdot S_j=\frac{1}{4}\begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & -1 & 2 & 0 \\ 0 & 2 & -1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$$ which is just the standard tensor product between two spin matrices. He then goes onto say that $$P_{ij}=2S_i\cdot S_j-\frac{1}{2}I=\begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix},$$ where ##I## is the identity matrix. Clearly it is the case that ##P_{ij}## is just a permutation matrix which permutes ##i## with ##j##. He then goes onto say that the Hamiltonian for the spin chain is $$H=\sum_{i=0}^{L-1}S_i\cdot S_{i+1}=\frac{1}{2}\bigg(\sum_{i=0}^{L-1}P_{ij}-\frac{L}{2}I\bigg).$$ My question is, how do I compute this Hamiltonian? What is ##j## in the summation? Alternatively, how can I figure out the action of this Hamiltonian on a random vector ##v##, as is necessary for the Lanczos algorithm?
You can see my prof's full notes here.