Spin hamiltonian matrix representation

Question

To preface, I'm an applied mathematician trying to parse the meaning of physics notation I've come across in a paper. My goal is to understand the setting in terms of matrices and vectors so that I can test an algorithm I'm studying. Since I don't know the terminology or notation, I haven't been able to figure out how to read more about the topic.

I'm reading about spin systems and I see expressions like: $$ \mathbf{H} = -\sum_{i,j} J_{i,j} \mathbf{s}(i) \cdot \mathbf{s}(j) $$ where $\mathbf{s}(i)$ is the spin operator at site $i$.

My understanding is that $\mathbf{H}$ is can be represented as a matrix of size $(2s+1)^N$ where $N$ is the number of spins and and $s$ is the spin number. I have also seem the spin matrices for specific values of $s$, which are of size $2s+1$ (although I'm not sure if these are the same as the $\mathbf{s}(i)$ since they seem to have $x, y, z, +, -)$. So, the missing piece for me is what is the meaning of $\mathbf{s}(i) \cdot \mathbf{s}(j)$ as well as the meaning of the sum.

Answer 1

To start: What is $\mathbf{s}(i)$? This is not an operator (not a matrix) but rather meant to denote a vector of operators. \begin{equation}\mathbf{s}(i)=(s_x(i),s_y(i),s_z(i)),\end{equation} where $s_\sigma(i)$ is an operator, a $(2s+1)^N$-dimensional matrix.

What does the dot product between them mean? It means to imitate the usual dot product, as in \begin{equation} \mathbf{s}(i)\cdot\mathbf{s}(j) = s_x(i)s_x(j)+s_y(i)s_y(j)+s_z(i)s_z(j) \end{equation} where the multiplication $s_\sigma(i)s_\sigma(j)$ is simply matrix-matrix multiplication.

Finally: What are the matrices $s_\sigma(i)$, and how are they $(2s+1)^N$-dimensional? You probably already know the form of the matrix $s_\sigma$ for $\sigma=x,y,z$; these are $(2s+1)$-dimensional matrices. When we describe multiple spins, the total Hilbert space is a tensor product of the Hilbert space of the individual spins, so our total Hilbert space for $N$ particles is $\bigotimes^N\mathbb{C}^{2s+1}$, where the $i$th copy of $\mathbb{C}^{2s+1}$ in the tensor product represents the state of the $i$th particle. This is a $(2s+1)^N$-dimensional space, which is why the overall Hamiltonian is a $(2s+1)^N$-dimensional matrix. When we write $s_\sigma(i)$, we mean the spin operator $s_\sigma$ that acts only on the part of the Hilbert space associated to the $i$th spin. So, strictly speaking, we have \begin{equation} s_\sigma(i) = I\otimes I\otimes\cdots I\otimes \underbrace{s_\sigma}_{i\text{th position}}\otimes I\cdots \otimes I \end{equation} which makes $s_\sigma(i)$ a $(2s+1)^N$ dimensional matrix.