Hamiltonian for a 1D-spin chain

[vbrasic]

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.

 

[KataKoniK]

First of all, let's clarify the notation. In the Hamiltonian $$H=\sum_{i=0}^{L-1}S_i\cdot S_{i+1},$$ the index $i$ represents the position of the particle on the spin chain, not the particle itself. So, for example, if we have a spin chain with 5 particles, the Hamiltonian would look like this: $$H=S_0\cdot S_1+S_1\cdot S_2+S_2\cdot S_3+S_3\cdot S_4.$$

Now, to compute the action of the Hamiltonian on a random vector $v$, we can use the following steps:

1. Write the vector $v$ as a linear combination of basis vectors. For example, if we have a spin chain with 5 particles, we can write $v$ as $$v=c_1|0\rangle+c_2|1\rangle+c_3|2\rangle+c_4|3\rangle+c_5|4\rangle,$$ where $c_i$ are coefficients and $|i\rangle$ represents a basis vector with all particles in the spin up state except for the $i$th particle which is in the spin down state.

2. Use the definition of the spin matrices to write out the action of each $S_i$ on the basis vectors. For example, we have $$S_0|0\rangle=\frac{1}{2}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\begin{pmatrix}1 \\ 0\end{pmatrix}=\frac{1}{2}\begin{pmatrix}0 \\ 1\end{pmatrix}=|1\rangle,$$ since $S_0|0\rangle=|1\rangle$ and similarly for the other spin matrices.

3. Use the definition of the dot product to write out the action of $S_i\cdot S_{i+1}$ on the basis vectors. For example, we have $$S_0\cdot S_1|0\rangle=S_0(S_1|0\rangle)=S_0|1\rangle=S_0(|0\rangle)=\frac{1}{2}\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}\