Identification and Optimization of Accurate Spin Models for Open-Shell Carbon Ladders with Matrix Product States

Open-shell nanographenes offer a controlled setting to study correlated magnetism emerging from $\pi$-electron systems. We analyze oligo(indenoindene) molecules, non-bipartite carbon ladders whose tight-binding spectra feature a gapped, weakly dispersing manifold of quasi-zero modes, and show that their low-energy properties can be effectively mapped onto an interacting set of spin-1/2 degrees of freedom. Using Density Matrix Renormalization Group simulations of the full Fermi-Hubbard model, we obtain their excitation spectra, entanglement profiles, and spin-spin correlations. We then construct optimized delocalized fermionic modes that act as emergent spins and show that their interactions are well described by a frustrated $J_1$-$J_2$ Heisenberg chain. This effective description clarifies how spin degrees of freedom arise and interact in non-bipartite nanographene ladders, providing a compact and accurate representation of their correlated behavior.