I am taking a course in mathematical physics, and we've just begun a section in the lecture notes where we want to describe free electrons in the lattice $\mathbb{Z}^d$ for some $d$, which, to have physical meaning, is either $0,1,2$ or $3$. After a 'failed', naive definition, we introduce a large but finite volume in the following way.
"[$\ldots$] we consider a finite lattice of linear dimension $L$ with periodic boundary conditions, \begin{align} \Lambda_L :=\mathbb{Z}^d/(L\mathbb{Z}^d)\,,\qquad (1) \end{align} i.e. a discrete torus of with $L^d$ sides".
After this, the author then introduces the single particle Hilbert space $$h_L=l^2(\Lambda_L;\mathbb{C}^N)\,,\qquad (2)$$ which is supposedly isomorphic to $\mathbb{C}^{L^D\times N}$, where $N$ are the available orbitals of nuclei. Now, my questions are the following;
What is meant by "linear dimension $L$" ? What is a discrete torus, and how is the space in $(1)$ a such torus with $L^d$ sides? And lastly, how do I interpret the space in $(2)$? Usually I know $l^2$ as the space of sequences, but do we find the elements of said sequences in $\Lambda_L;\mathbb{C}^N$, and if so, what do these elements look like?
I apologize for all the questions, maybe there is some notation that I am not used to, but most of this is completely new to me and is not explained with any further comments in this section.
Thanks a lot.
I am not sure I understand the phrase "$L^d$ sides". Your (1) defines a discrete torus: a discrete hypercube of side-length $L$ in $d$ dimensions whose boundaries are identified in a specific way.
I suppose the phrase "linear dimension $L$" just means that each side of the hypercube is of length $L$ (as opposed to the size being specified by the volume of the hypercube). A discrete torus is a subset of $\mathbb{Z}^d$ where the boundaries are identified in a similar way to the way it is done for the flat torus $\mathbb{R}^2/(\mathbb{Z}^2)$: the two horizontal boundaries are identified with each other, and the two vertical boundaries are identified with each other according to the following picutre:
For any set $S$ (including your torus), the definition of the $\ell^2$ space is $$ \ell^2(S;\mathbb{C}^N)\equiv\ell^2(S\to\mathbb{C}^N)\cong\ell^2(S\to\mathbb{C})\otimes\mathbb{C}^N \cong\left\{\psi:S\to\mathbb{C}^N\,|\,\sum_{x\in S}\|\psi(x)\|_{\mathbb{C}^N}^2 <\infty\right\}\,. $$
Now if $S$ is finite then the requirement is trivial, but the $\ell^2$ Hilbert space can in principle also accommodate infinite $S$.