Dirac Representation for Lattice Spin Operators: Spin-$1/2$ and Spin-$1$ cases

A novel quantum representation of lattice spin operators (LSOs) is achieved by mapping quantum spins onto their classical analogues for spin size $S=1/2$ and $S=1$. The "braket" representations of LSOs are attained thanks to a profound inspection into the binary/ternary distribution of classical bits/trits in non-negative integers. We claim the possility of getting the $j$th digit of a positive integer without performing any binary/ternary decomposition. Analytical formulas returning the $j$th bits/trits of an integer are presented. Impacts of our achievements in Physics are highlighted by revisiting the $1D$ spin-$1/2$ {XXZ} Heisenberg model with open boundaries in a magnetic field in both absence (uniform magnetic field) and presence of disorder (random magnetic field). In the absence of disorder (clean system), we demonstrate that the corresponding eigenvalues problem can be reduced to a tight-binding problem on a graph and solved without resorting to any spinless transformation nor the Bethe Anzath. In the presence of disorder, a convergent perturbation theory is elaborated. Our analytical results are compared with data from exact diagonalization for relatively large spin systems ($K\leq 18$ spins with $K$ denoting the total number of spins) obtained by implementing both the global $U(1)$ symmetry to block-diagonalize the Hamiltonian and the spin-inversion symmetry for two-fold block-diagonalization in the sector with total magnetization $\mathcal{J}^z=0$. We observe a good agreement between both results.