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

TL;DR

The paper analyzes open-shell, non-bipartite oligo(indenoindene) carbon ladders by mapping their low-energy π-electron magnetism—computed via DMRG on the full Fermi-Hubbard model—onto an emergent set of spin-1/2 degrees of freedom. By constructing delocalized, optimized spin modes and fitting a frustrated J1–J2 Heisenberg chain, the authors achieve a compact, accurate description of the system's spectra, entanglement, and spin correlations across sizes. The work demonstrates robust, transferable effective-spin descriptions even in nontrivial topologies, clarifying how spin physics arises from correlated electrons and providing a foundation for future t–J extensions and STM-accessible dynamical probes of carbon-based spin chains. Overall, the method offers a scalable route to distill complex π-electron magnetism into a tractable spin model with predictive power for nanographene-based spin systems.

Abstract

Open-shell nanographenes offer a controlled setting to study correlated magnetism emerging from $π$-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.

Figures (10)

• Figure 1: (a) Schematic of a regular oligo(indenoindene) (OInIn), made up of $P$ pentagons and $P+1$ hexagons, with an unpaired electron at each pentagon in the representation with the maximum Clar sextets (one at each hexagon). (b) The single-particle spectra of OInIn for different $P$, with the $P$ quasi-zero modes (offset in the $x$ axis for better visualization) highlighted in blue. (c) Parts of the energy spectra (positive $S^z$ only) of the $P=8$ OInIn chain (62 sites), and (d) the 8-spin spin chain (SC). The highlighted states are those corresponding to panel (c). (e) Difference between the lowest energy singlet and lowest energy triplet for various system sizes, for the OInIn ladder (blue circles) and the SC with parameters optimized for each $P$ (black $+$ markers) and with fixed optimal $P=4$ parameters $J_1^{*}=-0.06$, $J_2^{*}=0.26$ (gray $\times$ markers).
• Figure 2: (a) Maximum spin fidelity of the leftmost ($p=1$) effective mode versus the number of lattice sites used to span it for $P=2$ (blue), $P=4$ (orange) and $P=6$ (green) OInIn ladders. The size of each circle on the $P=4$ chain in the inset is proportional to the magnitude of the corresponding optimized component of $\bm{\alpha}^*$. The spin fidelity systematically increases by considering more sites in the set $\mathcal{M}^{(p=1)}$, starting from the pentagon tip, where the number over each point denotes the new site index that is added in MPS order (see inset). The sites are grouped into three different groups, depending on their relevance in describing the effective mode. (b) Spin fidelities obtained for four different effective mode descriptions (see SM Sec. \ref{['app:othermodes']}supplementary) by reusing and transferring the optimized effective modes for $P=4$ for a eigenstate of a much larger system of 24 pentagons.
• Figure 3: (a)-(c) Spin correlations in the $P=4$ system for different effective mode descriptions (pentagon tip, 3-site, 8-site, and all-site modes), compared with the target spin chain correlations (dashed black crosses). Results are shown for three eigenstates with $S^z = 0$: (a) the singlet ground state, dominated by singlets between second neighbors; (b) the first excited triplet, mainly composed of configurations where one of the singlets of the ground state is broken $\ket{\uparrow\uparrow\downarrow\downarrow}-\ket{\downarrow\downarrow\uparrow\uparrow}$; (c) the $S^z=0$ state of the $S=2$ multiplet, corresponding to a fully symmetric Dicke state with two spins up and two spins down. The structures on top of each plot show the site-wise correlation values of the corresponding eigenstate, where the reference spin has been placed on the first pentagon tip. The area of the circles on each site is proportional to the magnitude of the correlation and the color indicates its sign. (d) The measured and maximum possible magnetizations for different effective mode types for all effective spins within the state lowest energy $\ket{3,3}$ state for $P=6$. A perfect fidelity spin that perfectly mimics the target spin would lie on the top right corner of the plot. We also show the averaged magnetization approximation ratios $\bar{r}=\sum_pr_p/P$ for each mode type.
• Figure S1: Schematics of 3-site and 8-site modes used throughout the main paper. (a) Left: 3-site symmetric modes with one parameter ($\alpha$) to optimize over. Right: change in spin-fidelity with respect to the pentagon tip's for all values of $\alpha$. (b) 8-site symmetric modes with 3 parameters to optimize. The small weights on the next pentagon tips over exactly cancel out all overlaps.
• Figure S2: Spectra comparison for $P=4$. Note the slightly smaller gap between the first two triplet states for OInIn.