algebra on $\Lambda$ is: $$\label{fermifieldalgebra}
{\mathfrak F}_{\Lambda} :=  \CAR \ell^2(\Lambda^0,\Cn). =C^*\big(\mathop{\bigcup}_{x\in\Lambda^0}{\mathfrak F}_x\big)$$ where ${\mathfrak F}_x:=\CAR V_x.$ and $V_x:=\{f\in\ell^2(\Lambda^0,\Cn)\,\mid\, f(y)=0\;\;\hbox{if}\;\; y\not=x\}\cong \Cn.$ We interpret ${\mathfrak F}_x\cong\CAR \Cn.$ as the field algebra for a fermion at $x.$ We denote the generating elements of $\CAR \ell^2(\Lambda^0,\Cn).$ by $a(f),$ $f\in\ell^2(\Lambda^0,\Cn),$ and these satisfy the usual CAR–relations: $$\label{eq:car}
\{a(f), a(g)^*\}  = {\langle}f, g {\rangle}{\mathbf{1}}\quad \hbox{ and } \quad  \{a(f),a(g)\} = 0
\quad \mbox{ for } \quad f, g \in \ell^2(\Lambda^0,\Cn)$$ where $\{A,B\} := AB + BA$.*]{}\
Note that the odd parts of ${\mathfrak F}_x$ and ${\mathfrak F}_y$ w.r.t. the fields $a(f)$ anticommute if $x\not=y.$ Moreover, as $\Lambda^0$ is infinite, ${\mathfrak F}_{\Lambda}$ has inequivalent irreducible representations.

We have the following inductive limit structure. Let $\cl S.$ be a directed set of finite connected subgraphs $S\subset(\Lambda^0, \Lambda^1)$, such that $\bigcup\limits_{S\in\cl S.}S=(\Lambda^0, \Lambda^1)$, where the partial ordering is set inclusion. Note that $S_1\subseteq S_2$ implies $\Lambda_{S_1}^i\subseteq\Lambda_{S_2}^i$ and $\bigcup\limits_{S\in\cl S.}\Lambda_S^i=\Lambda^i.$ Define ${\mathfrak F}_{S}:=C^*\big(\mathop{\cup}\limits_{x\in\Lambda_S^0}{\mathfrak F}_x\big)\subset{\mathfrak F}_{\Lambda}$ and then ${\mathfrak F}_{\Lambda}=\ilim{\mathfrak F}_{S}$ is an inductive limit w.r.t. the partial ordering in $\cl S..$

This defines the quantum matter fields on the lattice sites, and to obtain correspondence with the physics notation, we choose an appropriate orthonormal basis in $\Cn$ and proceed 