From Exact Diagonalization to DMRG: A Complete Numerical Study of the Transverse-Field Ising Model

We present a self-contained numerical study of the one-dimensional transverse-field Ising model (TFIM), tracing its ground-state entanglement structure from exact diagonalization at small system size (L=8,14,20) through density-matrix renormalization group (DMRG) calculations up to L=100. A single, consistent methodology - identical field grid, identical convergence diagnostic (independent runs at bond dimensions chi=100 and chi=200 at every point) - is used across all seven system sizes, and we show explicitly where exact diagonalization and DMRG must agree exactly rather than merely approximately. Along the way we document and resolve a bond-dimension convergence artifact that produced a spurious discontinuity in an earlier, less systematic dataset, as a worked cautionary example for practitioners. Using the resolved dataset we extract the central charge of the transition via two complementary routes - the leading logarithmic finite-size scaling of the mid-chain entropy, and the full Calabrese-Cardy formula applied to every bond of every system size simultaneously - obtaining c_eff to 0.51-0.52 as short-distance lattice corrections are systematically excluded, consistent with the exact two-dimensional Ising value c=1/2. We review the quantum-classical (Suzuki-Trotter) correspondence that underlies this agreement and discuss how the Widom-Kadanoff scaling hypothesis transplants from the classical to the quantum problem. The manuscript is intended as both a physics result and a worked methodological example of careful finite-size numerics.