Open XXZ spin-half generators and Hamiltonian convention

Fix $L\ge 2$, $\gamma \in \mathbb{R}$, $q=e^{i\gamma }$, and loop parameter $\delta =q+q^{-1}=2\cos \gamma$. Let ${\mathcal{H}}_L=({\mathbb{C}}^2{)}^{\otimes L}$ with ordered one-site basis $\mathrm{\mid }\uparrow ⟩,\mathrm{\mid }\downarrow ⟩$ and two-site basis $\mathrm{\mid }\uparrow \uparrow ⟩,\mathrm{\mid }\uparrow \downarrow ⟩,\mathrm{\mid }\downarrow \uparrow ⟩,\mathrm{\mid }\downarrow \downarrow ⟩$. With the standard Pauli matrices $X,Y,Z$, define $e_i=I^{\otimes (i-1)}\otimes U(q)\otimes I^{\otimes (L-i-1)},1\le i<L,$ where $U(q)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& q^{-1}& -1& 0\\ 0& -1& q& 0\\ 0& 0& 0& 0\end{array}\right)=-\frac{1}{2}(X\otimes X+Y\otimes Y+\cos \gamma Z\otimes Z-\cos \gamma I_4+i\sin \gamma (Z\otimes I_2-I_2\otimes Z))\mathrm{.}$ The open-chain Temperley–Lieb Hamiltonian convention is $H_L=-{\sum }_{i=1}^{L-1}{e}_i$. No wrap-around term couples site $L$ to site $1$. This defines local operators; the following result proves their Temperley–Lieb relations.

The loop convention is that $e_i^2=\delta e_i$. The basis order and the sign in the off-diagonal matrix entries are fixed as displayed; changing either changes the parameter comparison.