  We investigate the Ising and Heisenberg models using the Block
Renormalization Group Method (BRGM), focusing on its behavior across different
system sizes. The BRGM reduces the number of spins by a factor of 1/2 (1/3) for
the Ising (Heisenberg) model, effectively preserving essential physical
features of the model while using only a fraction of the spins. Through a
comparative analysis, we demonstrate that as the system size increases, there
is an exponential convergence between results obtained from the original and
renormalized Ising Hamiltonians, provided the coupling constants are redefined
accordingly. Remarkably, for a spin chain with 24 spins, all physical features,
including magnetization, correlation function, and entanglement entropy,
exhibit an exact correspondence with the results from the original Hamiltonian.
The study of the Heisenberg model also shows this tendency, although complete
convergence may appear for a size much larger than 24 spins, and is therefore
beyond our computational capabilities. The success of BRGM in accurately
characterizing the Ising model, even with a relatively small number of spins,
underscores its robustness and utility in studying complex physical systems,
and facilitates its simulation on current NISQ computers, where the available
number of qubits is largely constrained.
