Identification and optimization of accurate spin models for Fermi-Hubbard ladders using matrix product states

Andoni Agirre; Geza Giedke; Thomas Frederiksen; Tobias Grass

REVIEW 3 major objections 4 minor 40 references

Identification and optimization of accurate spin models for Fermi-Hubbard ladders using matrix product states

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Oligo(indenoindene) ladders have an effective description as a frustrated J1–J2 spin chain: optimized delocalized modes act as emergent spins with fidelity above 0.98.

desk verdict Solid DMRG validation of a J1–J2 spin picture for OInIn ladders, with a fitted-spectrum caveat that needs an independent convergence check before the quantitative claims fully land. read the letter →

arxiv 2512.18695 v2 pith:376GO5RD submitted 2025-12-21 cond-mat.mes-hall cond-mat.str-elquant-ph

Andoni Agirre , Thomas Frederiksen , Geza Giedke , Tobias Grass This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.str-elquant-ph

keywords Fermi-HubbardmodeldensitymatrixrenormalizationgroupproductstatesemergentspinsJ1–J2Heisenbergchainoligo(indenoindene)frustratedmagnetismnanographenes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the low-energy physics of oligo(indenoindene) ladders — non-bipartite carbon structures whose tight-binding spectra contain a flat band of quasi-zero modes — is effectively that of a set of interacting spin-1/2 degrees of freedom. Using density matrix renormalization group simulations of the full Fermi-Hubbard model, the authors construct optimized delocalized fermionic modes that stand in for the spins, with single-occupation probabilities above 0.98. They show these emergent spins interact according to a frustrated J1–J2 Heisenberg chain, reproducing the Hubbard model's energy gaps, spin multiplicities, and correlations across system sizes. The value is a compact and systematically tunable spin picture of strongly correlated carbon magnetism, with a trade-off between mode simplicity and accuracy.

What carries the argument

The central object is the optimized delocalized mode c_{M(p),σ} = Σ_i α_i^{(p)} c_{i,σ}, a normalized fermionic mode supported on a subset of lattice sites around pentagon p. The coefficients α are optimized to maximize the spin fidelity ⟨4(S^z_{M(p)})^2⟩, the probability that the mode is singly occupied; this quantity limits every spin observable of the effective description. Intermediate 3-site and 8-site symmetric modes provide controlled complexity/accuracy trade-offs, and a Gram-Schmidt-style correction cancels overlaps between neighboring modes. The effective Hamiltonian is the frustrated J1–J2 chain, H = J1 Σ S_p·S_{p+1} + J2 Σ S_p·S_{p+2}, with ferromagnetic J1 and antiferromagnetic

What would settle it

Compute the low-lying singlet–triplet gap of a small OInIn Hubbard ladder (e.g., P=4 or P=6) by exact diagonalization, and compare with the DMRG value used for fitting; a discrepancy larger than the fitted J values would invalidate the spectral matching and the fitted couplings. On the experimental side, STM inelastic tunneling spectroscopy of a single OInIn molecule should reveal a spin excitation at the chain-predicted energy.

Watch

Extended reading notes

Core claim

The paper establishes a quantitative correspondence between the Fermi-Hubbard model on oligo(indenoindene) ladders and a J1–J2 Heisenberg spin chain. The carriers of the correspondence are delocalized fermionic modes: each emergent spin p is a normalized linear combination of lattice sites, optimized to maximize the expectation value of single occupation. With modes spread over about a dozen sites around each pentagon, the spin fidelity reaches 0.98, and the optimized modes transfer across ladders of different lengths. Fitting the spin-chain couplings J1 and J2 to the DMRG spectra yields ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor exchange with J2/|J1| > 1/4, p

Load-bearing premise

The whole correspondence rests on the DMRG-computed low-energy spectrum of the Hubbard ladder being converged, in particular the tiny singlet–triplet splittings that determine the fitted values of J1 and J2; the paper reports a Schmidt truncation of 10^-9 and energy convergence of 10^-6, but provides no independent check of these small gaps against another method.

Editorial extensions

If this is right

The fitted J1–J2 chain reproduces the low-energy spectrum, spin multiplicities, and singlet–triplet gap oscillations of the Hubbard ladder for even P, indicating the ground state alternates between singlet and triplet with the parity of P/2.
The spin-chain description provides a compact surrogate for the full electron model, enabling larger-system simulations and further low-energy modeling such as effective t–J extensions with charge fluctuations and doping.
The optimized delocalized modes are transferable across system sizes: modes optimized on a P=4 ladder retain fidelities close to 1 when used to construct all effective spins of a P=24 ladder.
The mode picture improves dynamic observables: spin-flip fidelity (probability that flipping one effective spin connects two molecular eigenstates) rises from 0.19 for localized pentagon-tip spins to 0.94 for fully optimized modes, making STM-based spin-probing schemes more realistic.
The fitted parameters lie in the regime where the spin chain hosts a Haldane-dimer phase, implying the OInIn ladder may exhibit the associated valence-bond and topological features.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the optimized modes are essentially system-size independent, the emergent-spin construction could be lifted to other non-bipartite nanographenes with nearly flat mid-gap bands, giving a generic protocol for distilling spin models from Hubbard models without active-space choices.
The reported decrease of spin fidelity with decreasing total spin (0.98 for S=3 down to 0.81 for S=0 at P=6) suggests the effective spin description is state-dependent; singlet ground states are the hardest case, so corrections beyond the J1–J2 chain may be needed for precise quantitative predictions of the ground-state singlet.
The small asymmetric coupling δ2 needed to reproduce the magnetization profiles of the two low-lying triplets (two orders of magnitude smaller than J2) indicates the perfect uniform chain is an idealization; observables that are sensitive to the splitting of near-degenerate states may require symmetry-broken extensions.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does a genuinely useful thing: it takes the J1–J2 mapping for oligo(indenoindene) ladders that Ortiz et al. proposed from mean-field/CAS work and puts it under full-Hubbard DMRG, with a careful look at how the effective spins are actually distributed. The central new assets are the optimized delocalized modes — spin fidelities above 0.98, stable across system sizes and transferable to P=24 — and the demonstration that these modes capture not just energy gaps but local magnetizations, spin correlations, and a spin-flip fidelity that matters for STM drives. That is a real step beyond the original mapping.

The paper is also honest about its own soft spots: it explicitly notes the coupling asymmetry and the degradation of fidelity in low-spin sectors. Good.

Where I would push back: the spectral matching is partly constructed. J1 and J2 are obtained by least-squares fitting to the DMRG energy gaps (SM S6), so the side-by-side spectra in Fig. 1(c)-(d) are not a beyond-the-fit prediction. The stress-test note frames this as a circularity burden, and that's fair — the correlations and fidelities in Fig. 3 are not entirely independent of the fitted couplings either, though they do test the form of the spin Hamiltonian and the mode structure.

The more concrete worry is convergence of the small singlet–triplet gaps. At the fitted parameters J2/|J1| ~ 4 the chain is close to decoupled AFM chains, and the gap that changes sign with P is tiny. The DMRG is converged to 10^-6 in energy, but with no independent cross-check — no ED on a small system, no second tensor-network implementation, no data release — that single number has to carry a lot of weight. The paper itself mentions needing extra Lanczos iterations to converge the first two triplet states, which tells you these states are delicate. If those gaps are off by a fraction of a meV, the fitted J1 values and the claim that the spectral matching is quantitative would weaken, though the qualitative physics — gap sign alternation, spin multiplicities, Haldane-dimer regime — would likely survive.

Bottom line: this is a serious contribution to the nanographene spin-model literature, with a method (optimized delocalized modes) that is likely to be reused. It deserves peer review, but a referee should ask for an independent convergence check and ideally for the code/parameters to be released. If the small-gap convergence holds up, the quantitative mapping becomes quite persuasive.

Referee Report

3 major / 4 minor

Summary. The paper studies oligo(indenoindene) (OInIn) Fermi-Hubbard ladders at half filling using DMRG. It identifies a flat manifold of P quasi-zero modes and proposes that their low-energy physics is captured by an effective spin-1/2 system. The authors optimize delocalized fermionic modes with maximal single occupation ('spin fidelities'), report values up to 0.98, and fit a uniform J1-J2 Heisenberg chain to the low-lying DMRG energy gaps. The chains are claimed to reproduce the spin multiplicities, the alternating singlet/triplet ground state, energy gaps, and spin correlations of the full Hubbard model. The paper also introduces simplified 3-site and 8-site modes and shows that the optimized modes can be transferred across system sizes.

Significance. If the central claim holds, this is a useful methodological advance: it connects full-DMRG electronic-structure calculations to effective spin models in non-bipartite nanographenes, and the idea of optimized, transferable delocalized spin modes is likely to be of broader interest. The paper is commendably explicit about DMRG truncation thresholds and about the distinction between qualitative and quantitative agreement. The transfer of optimized modes from P=4 to P=24 is a strong, nontrivial check. However, the quantitative spectral matching rests on fits to the very DMRG gaps that are then compared, and on convergence of extremely small singlet-triplet gaps that are not independently validated. The central claim is defensible but needs additional verification before it can be accepted as quantitatively accurate.

major comments (3)

[SM S6, Fig. 1(e)] The fitted J1*,J2* are obtained by minimizing the squared distance to the DMRG energy gaps, and the same gaps are then used to claim spectral matching. The optimal parameters place the system at J2/|J1| ~ 4, close to two decoupled antiferromagnetic chains, where the singlet-triplet gap is very small. The stated DMRG convergence is 10^-6 (no units) but no independent check is provided (e.g., exact diagonalization for a small P, or a second MPS implementation). The paper itself notes that the first two excited triplet states required many Lanczos iterations to converge, underscoring the delicacy of these excitations. An energy error of 10^-6 could be a large fraction of such a tiny gap. Please report the convergence of the individual gaps versus bond dimension and provide an independent cross-check for at least the P=4 ladder.
[Fig. 1(c)-(e), SM S6] The P=4 spectral match is evaluated on the same data used to optimize the spin-chain parameters; the transfer of P=4 parameters to P=8 and P=12 is a good cross-validation step, but it still uses DMRG gaps from the same implementation and same convergence criteria. To substantiate the quantitative claim, the paper should either perform a true hold-out test (fit on a subset of levels or system sizes and test on the rest) or compare the fitted J1,J2 with values obtained from a perturbative/effective-exchange derivation. As it stands, the agreement in Fig. 1(c)-(e) is partly constructed by the fitting procedure.
[Fig. 3, Table I] The spin correlations, magnetization ratios, and spin-flip fidelities appear to be computed with modes that were optimized on the same eigenstates and system sizes used in the benchmark (e.g., P=4 in Fig. 3 and Table I). The transfer across sizes in Fig. 2(b) is a good control, but the in-sample benchmark leaves open how the reported 0.98 and 0.94 figures degrade when the modes are fixed before the target state is known. State explicitly whether the modes used in Fig. 3/Table I are optimized for the very state being analyzed or transferred from another state; if in-sample, add a leave-one-state-out test.

minor comments (4)

[Abstract, SM S5] The abstract's 'spin fidelities above 0.98' should be qualified: it holds only for the S=P/2 sector. For P=6 the fidelity drops to 0.81 in the S=0 sector, as stated in SM S5.
[Main text near Fig. 3] The magnetization discussion refers to 'Fig. 3(b)', but panel (b) shows correlations; the relevant panel appears to be Fig. 3(d).
[Conclusions] The sentence 'extensions to effective t-J descriptions that that incorporate charge fluctuations' contains a duplicated 'that'.
[SM S2, Eq. (2)] The asymmetry term introduced in SM S2 (δ(S1·S3 - S2·S4)) is absent from Eq. (2). The main text should state explicitly that Eq. (2) is the symmetric effective model and that a small symmetry-breaking correction is needed for certain Sz≠0 states.

Circularity Check

1 steps flagged · score 6.0 of 10

The J1–J2 agreement with the Hubbard gaps is partly the objective of the fit; the delocalized-mode construction and cross-size transfer provide remaining independent content.

fitted input called prediction [Spectral matching (main text, Fig. 1(c)–(e)); SM Sec. S6]
"and then tune the parameters of the spin chain in Eq. (2) to find the best match for the energy gaps of the ladder for each size. ... Notably, the energy gaps of the much simpler spin chain are in good agreement with the gaps of the OInIn, and the spin multiplicities fully match. ... [SM S6] we optimize the effective spin chain (SC) parameters by finding the ones that give the energy spectrum that minimizes the square distance between the respective gaps of the system."

J1 and J2 are obtained by minimizing the squared distance to the DMRG energy gaps of the same ladder; therefore the subsequent statement that the spin-chain gaps are 'in good agreement' with those Hubbard gaps is a restatement of the fit objective, not an independent confirmation of the model. The paper does not derive J1,J2 from t and U, but optimizes them to the target spectrum. This is partial circularity: the fit has only two parameters and the paper also validates spin multiplicities, the P-dependent gap alternation, transfer of parameters across sizes, and correlations that were not fit targets.

full rationale

The central quantitative demonstration—that the OInIn low-energy spectrum is accurately reproduced by the J1–J2 Heisenberg chain—rests on J1,J2 optimized against the very DMRG gaps used for comparison (SM S6), so that particular spectral agreement is partly constructed. I do not score this higher because (i) the paper openly states that the parameters are tuned, (ii) the match includes spin multiplicities and gap-sign alternations not directly enforced by a two-parameter fit, (iii) parameters obtained at one P are tested at other P values (Fig. 1(e), SM S7), and (iv) the spin-correlation benchmarks in Fig. 3 are not fit objectives. The self-citation to Ref. [15] (Ortiz, Giedke, Frederiksen) supplies the H_SC ansatz, but the present DMRG fitting and optimized-mode construction provide independent evidence, so it is not a load-bearing circularity. The DMRG convergence of small singlet-triplet gaps (SM S6) is a correctness risk rather than a circularity. Overall: one central fit-as-validation reduces the spectral claim by construction, giving score 6.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central quantitative claims rest on: the standard Hubbard model with literature parameter values; the quasi-zero-mode/spin truncation picture; a fitted uniform J1–J2 Hamiltonian; and a specific DMRG convergence assumption. The optimized mode amplitudes are additional variational objects. This is a typical effective-model identification, so the paper contributes a framework and numerical validation rather than a first-principles derivation of the couplings.

free parameters (4)

Hubbard parameters t and U = t = 2.7 eV, U = 1.5t
Chosen from the nanographene literature (Ref. [30]) as representative values, not fitted to OInIn targets. Central results are shown for this single U value, with U-dependence in SM S3.
Spin-chain couplings J1*, J2* = P=4: J1*=-0.06, J2*=0.26 (eV); SM S6 reports J2* ~ 0.23 with J1* decreasing slowly with P
Fitted for each system size by minimizing the squared distance between spin-chain and DMRG energy gaps (SM S6). The spectral agreement is thus partly obtained by construction.
Coupling asymmetry δ = small, two orders of magnitude below J2* (SM S2)
Introduced ad hoc as H' = δ(S1·S3 - S2·S4) to reproduce magnetization profiles of low-lying triplet states. It is not part of the main J1–J2 model.
Optimized mode amplitudes α_i^(p) = not tabulated; spatial pattern shown in inset of Fig. 2(a)
Optimized with NLopt to maximize spin fidelity of each effective mode. These amplitudes are the core of the delocalized-spin construction and are chosen per system and per eigenstate.

assumptions (5)

domain assumption The Fermi-Hubbard model (Eq. 1) with the chosen t and U is an adequate description of the π-electron system of OInIn ladders.
Invoked in the Model section; relies on standard nanographene modeling practice and Ref. [30], with no ab initio benchmark for this specific molecule family.
domain assumption There are P gapped, weakly dispersing quasi-zero modes at half filling that become singly occupied, forming a Mott insulator.
Model section and Fig. 1(b). This is the physical basis for one spin per pentagon; it is not proven for all U and is itself part of what the paper demonstrates numerically.
domain assumption The low-energy Hilbert space can be truncated to the singly occupied sector of P effective modes; charge fluctuations are negligible.
Used in the delocalized-spins construction. Spin fidelities below 1 (e.g., 0.81 for the S=0 sector at P=6, SM S5) show this truncation is approximate.
domain assumption DMRG results are converged with Schmidt truncation 10^-9 and energy convergence 10^-6 for all states used.
SM S6. No independent exact-diagonalization benchmark for small systems is provided in the main text.
ad hoc to paper The effective Heisenberg Hamiltonian has only uniform nearest-neighbor and next-nearest-neighbor couplings (Eq. 2).
Motivated by Ref. [15]. SM S2 shows an additional asymmetry term is needed for some observables, so the uniform ansatz is not exact.

invented entities (1)

Delocalized effective spin modes (sets M(p) with amplitudes α^(p))
purpose: Define spin operators S_M(p) that behave as emergent spin-1/2 degrees of freedom with high single occupancy.
These are constructed objects optimized on DMRG eigenstates. They are supported by internal spin-fidelity metrics and correlation matching, but no external experimental measurement is reported.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Identification and optimization of accurate spin models for Fermi-Hubbard ladders using matrix product states." pith.science (2026). https://pith.science/paper/376GO5RD

@misc{pith2026251218695,
  author       = {Pith},
  title        = {Pith review of: Identification and optimization of accurate spin models for Fermi-Hubbard ladders using matrix product states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/376GO5RD}},
  note         = {Machine review of arXiv:2512.18695}
}

abstract

Open-shell nanographenes offer a controlled setting to study correlated magnetism emerging from $\pi$-electron systems. Here, we study non-bipartite Fermi-Hubbard ladders describing oligo(indenoindene) molecules. These feature a gapped, weakly dispersing manifold of quasizero modes in their single-particle spectra, and we 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 demonstrate 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

Figures reproduced from arXiv: 2512.18695 by the authors.

**Figure 2.** (a) Maximum spin fidelity of the leftmost ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**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… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 2 linked inside Pith

[1]

Auerbach,Interacting Electrons and Quantum Mag- netism(Springer-Verlag, Berlin, Heidelberg, 1994)

A. Auerbach,Interacting Electrons and Quantum Mag- netism(Springer-Verlag, Berlin, Heidelberg, 1994)

1994
[2]

P. A. McClarty, Topological magnons: A review, Annu. Rev. Condens. Matter Phys.13, 171 (2022)

2022
[3]

O. V. Yazyev, Emergence of magnetism in graphene ma- terials and nanostructures, Rep. Prog. Phys.73, 056501 (2010)

2010
[4]

D. G. de Oteyza and T. Frederiksen, Carbon-based nanostructures as a versatile platform for tunableπ- magnetism, J. Phys.: Condens. Matter34, 443001 (2022)

2022
[5]

J.Li, S.Sanz, M.Corso, D.J.Choi, D.Peña, T.Frederik- sen, and J. I. Pascual, Single spin localization and ma- nipulation in graphene open-shell nanostructures, Nat. Commun.10, 200 (2019)

2019
[6]

T. Wang, S. Sanz, J. Castro-Esteban, J. Lawrence, A. Berdonces-Layunta, M. S. G. Mohammed, M. Vilas- Varela, M. Corso, D. Peña, T. Frederiksen, and D. G. de Oteyza, Magnetic interactions between radical pairs in chiral graphene nanoribbons, Nano Lett.22, 164 (2022)

2022
[7]

J. Li, S. Sanz, J. Castro-Esteban, M. Vilas-Varela, N. Friedrich, T. Frederiksen, D. Peña, and J. I. Pas- cual, Uncovering the triplet ground state of triangular graphene nanoflakes engineered with atomic precision on a metal surface, Phys. Rev. Lett.124, 177201 (2020)

2020
[8]

Mishra, D

S. Mishra, D. Beyer, K. Eimre, R. Ortiz, J. Fernández- Rossier, R. Berger, O. Gröning, C. A. Pignedoli, R. Fasel, X. Feng, and P. Ruffieux, Collective all-carbon mag- netism in triangulene dimers, Angew. Chem. Int. Ed.59, 12041 (2020)

2020

Show all 40 references

[9]

Mishra, G

S. Mishra, G. Catarina, F. Wu, R. Ortiz, D. Jacob, K. Eimre, J. Ma, C. A. Pignedoli, X. Feng, P. Ruffieux, J. Fernández-Rossier, and R. Fasel, Observation of frac- tional edge excitations in nanographene spin chains, Na- 6 ture598, 287 (2021)

2021
[10]

Hieulle, S

J. Hieulle, S. Castro, N. Friedrich, A. Vegliante, F. R. Lara, S. Sanz, D. Rey, M. Corso, T. Frederiksen, J. I. Pascual, and D. Peña, On-surface synthesis and collective spin excitations of a triangulene-based nanostar, Angew. Chem. Int. Ed.60, 25224 (2021)

2021
[11]

Majzik, N

Z. Majzik, N. Pavliček, M. Vilas-Varela, D. Pérez, N. Moll, E. Guitián, G. Meyer, D. Peña, and L. Gross, Studying an antiaromatic polycyclic hydrocarbon ad- sorbed on different surfaces, Nat. Commun.9, 1198 (2018)

2018
[12]

Di Giovannantonio, Q

M. Di Giovannantonio, Q. Chen, J. I. Urgel, P. Ruffieux, C. A. Pignedoli, K. Müllen, A. Narita, and R. Fasel, On- surface synthesis of oligo(indenoindene), J. Am. Chem. Soc.142, 12925 (2020)

2020
[13]

Mishra, M

S. Mishra, M. Vilas-Varela, L.-A. Lieske, R. Ortiz, S. Fa- tayer, I. Rončević, F. Albrecht, T. Frederiksen, D. Peña, and L. Gross, Bistability betweenπ-diradical open- shell and closed-shell states in indeno[1,2-a]fluorene, Nat. Chem.16, 755 (2024)

2024
[14]

Khandarkhaeva, T

S. Khandarkhaeva, T. Fedotenko, A. Aslandukova, F. I. Akbar, M. Bykov, D. Laniel, A. Aslandukov, U. Rusche- witz, C.Tobeck, B.Winkler, S.Chariton, V.Prakapenka, K. Glazyrin, C. Giacobbe, E. L. Bright, M. Belov, N. Dubrovinskaia, and L. Dubrovinsky, Extending car- bon chemistry...

2024
[15]

Ortiz, G

R. Ortiz, G. Giedke, and T. Frederiksen, Magnetic frustration and fractionalization in oligo(indenoindenes), Phys. Rev. B107, L100416 (2023)

2023
[16]

Ortiz, Topological edge states in a double isomeric class-II oligo(indenoindene), Theor

R. Ortiz, Topological edge states in a double isomeric class-II oligo(indenoindene), Theor. Chem. Acc.144, 91 (2025)

2025
[17]

E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett.62, 1201 (1989)

1989
[18]

S. R. White, Density matrix formulation for quantum renormalizationgroups,Phys.Rev.Lett.69,2863(1992)

1992
[19]

Phys.326, 96 (2011)

U.Schollwöck,Thedensity-matrixrenormalizationgroup in the age of matrix product states, Ann. Phys.326, 96 (2011)

2011
[20]

C. Zhao, G. Catarina, J.-J. Zhang, J. C. G. Henriques, L. Yang, J. Ma, X. Feng, O. Gröning, P. Ruffieux, J. Fernández-Rossier, and R. Fasel, Tunable topologi- cal phases in nanographene-based spin-1/2 alternating- exchange Heisenberg chains, Nat. Nanotechnol.19, 1789 (2024)

2024
[21]

C. Zhao, L. Yang, J. C. G. Henriques, M. Ferri-Cortés, G.Catarina, C.A.Pignedoli, J.Ma, X.Feng, P.Ruffieux, J. Fernández-Rossier, and R. Fasel, Spin excitations in nanographene-based antiferromagnetic spin-1/2 Heisen- berg chains, Nat. Mater.24, 722 (2025)

2025
[22]

J. C. G. Henriques, Y. del Castillo, R. Segundo, J. Phillips, and J. Fernández-Rossier, Prediction of a topological phase transition in exchange alternating spin- 1 nanographene chains (2025), arXiv:2510.23555

2025 arXiv
[23]

del Castillo, A

Y. del Castillo, A. Ferrón, and J. Fernández-Rossier, Remote spin control in Haldane spin chains (2025), arXiv:2508.21544

2025 arXiv
[24]

Catarina and J

G. Catarina and J. Fernández-Rossier, Hubbard model for spin-1 Haldane chains, Phys. Rev. B105, L081116 (2022)

2022
[25]

Catarina, E

G. Catarina, E. Turco, N. Krane, M. Bommert, A. Ortega-Guerrero, O. Gröning, P. Ruffieux, R. Fasel, and C. A. Pignedoli, Conformational tuning of magnetic interactions in coupled nanographenes, Nano Lett.24, 12536 (2024)

2024
[26]

Bursill, G

R. Bursill, G. A. Gehring, D. J. J. Farnell, J. B. Parkin- son, T. Xiang, and C. Zeng, Numerical and approximate analytical results for the frustrated spin- 1/2 quantum spin chain, J. Phys.: Condens. Matter7, 8605 (1995)

1995
[27]

Furukawa, M

S. Furukawa, M. Sato, S. Onoda, and A. Furusaki, Ground-state phase diagram of a spin-1 2 frustrated fer- romagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases, Phys. Rev. B86, 094417 (2012)

2012
[28]

C. E. Agrapidis, S.-L. Drechsler, J. van den Brink, and S. Nishimoto, Coexistence of valence-bond formation and topological order in the Frustrated FerromagneticJ1-J2 Chain, SciPost Phys.6, 019 (2019)

2019
[29]

See Supplemental Material at [URL or DOI] for a de- scription of effective mode design and singlet formation, coupling asymmetries, on-site interaction effects, entan- glement entropy, and DMRG/spectral matching details
[30]

Schüler, M

M. Schüler, M. Rösner, T. O. Wehling, A. I. Lichtenstein, and M.I. Katsnelson, OptimalHubbard modelsfor mate- rials with nonlocal Coulomb interactions: Graphene, sil- icene, and benzene, Phys. Rev. Lett.111, 036601 (2013)

2013
[31]

Fishman, S

M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor Software Library for Tensor Network Calcula- tions, SciPost Phys. Codebases4(2022)

2022
[32]

Due to the weak FM coupling, the two doublets split into a lower triplet and a slightly higher-energy singlet

This behavior can be understood from the limiting case J2/|J1| → ∞: ForPa multiple of four, the chain splits into two spin chains with even numbers of spins (both with singlet ground states), whereas for even non- multiples of four, it splits into two chains with odd num- ber ...
[33]

Clar,The Aromatic Sextet(Wiley, 1972)

E. Clar,The Aromatic Sextet(Wiley, 1972)

1972
[34]

AlthoughinthisfulloptimizationcasesthesetsM (p) will be the same for different spins corresponding to different pentagonsp, we still label its operators withM (p) for short,M (p) representing the mode with support around thepth pentagon tip
[35]

S. G. Johnson, The NLopt nonlinear-optimization pack- age,https://github.com/stevengj/nlopt(2007)

2007
[36]

M. J. D. Powell, A direct search optimization method that models the objective and constraint functions by lin- ear interpolation, inAdv. Optim. Numer. Anal., Mathe- matics and Its Applications, Vol. 275, edited by S. Gomez and J.-P. Hennart (Springer, 1994) pp. 51–67

1994
[37]

With an appropriate gauge transformation of the MPS, the number of bonds to contract over remains con- stant withP

Only optimizing over the green and purple sites means that the single occupation operator only spans a few ten- sors. With an appropriate gauge transformation of the MPS, the number of bonds to contract over remains con- stant withP
[38]

Ortiz, R

R. Ortiz, R. A. Boto, N. García-Martínez, J. C. Sancho- García, M. Melle-Franco, and J. Fernández-Rossier, Ex- change rules for diradicalπ-conjugated hydrocarbons, Nano Lett.19, 5991–5997 (2019)

2019
[39]

D.JacobandJ.Fernández-Rossier,Theoryofintermolec- ular exchange in coupled spin-1/2 nanographenes, Phys. Rev. B106, 205405 (2022)

2022
[40]

background

Note that the placement of the spin chain markers is not precise due to the overlap of different effective spins. 7 Supplemental Material: Identification and Optimization of Accurate Spin Models for Open-Shell Carbon Ladders with Matrix Product States S1. DESIGNING OTHER EFFEC...

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Auerbach,Interacting Electrons and Quantum Mag- netism(Springer-Verlag, Berlin, Heidelberg, 1994)

A. Auerbach,Interacting Electrons and Quantum Mag- netism(Springer-Verlag, Berlin, Heidelberg, 1994)

1994

[2] [2]

P. A. McClarty, Topological magnons: A review, Annu. Rev. Condens. Matter Phys.13, 171 (2022)

2022

[3] [3]

O. V. Yazyev, Emergence of magnetism in graphene ma- terials and nanostructures, Rep. Prog. Phys.73, 056501 (2010)

2010

[4] [4]

D. G. de Oteyza and T. Frederiksen, Carbon-based nanostructures as a versatile platform for tunableπ- magnetism, J. Phys.: Condens. Matter34, 443001 (2022)

2022

[5] [5]

J.Li, S.Sanz, M.Corso, D.J.Choi, D.Peña, T.Frederik- sen, and J. I. Pascual, Single spin localization and ma- nipulation in graphene open-shell nanostructures, Nat. Commun.10, 200 (2019)

2019

[6] [6]

T. Wang, S. Sanz, J. Castro-Esteban, J. Lawrence, A. Berdonces-Layunta, M. S. G. Mohammed, M. Vilas- Varela, M. Corso, D. Peña, T. Frederiksen, and D. G. de Oteyza, Magnetic interactions between radical pairs in chiral graphene nanoribbons, Nano Lett.22, 164 (2022)

2022

[7] [7]

J. Li, S. Sanz, J. Castro-Esteban, M. Vilas-Varela, N. Friedrich, T. Frederiksen, D. Peña, and J. I. Pas- cual, Uncovering the triplet ground state of triangular graphene nanoflakes engineered with atomic precision on a metal surface, Phys. Rev. Lett.124, 177201 (2020)

2020

[8] [8]

Mishra, D

S. Mishra, D. Beyer, K. Eimre, R. Ortiz, J. Fernández- Rossier, R. Berger, O. Gröning, C. A. Pignedoli, R. Fasel, X. Feng, and P. Ruffieux, Collective all-carbon mag- netism in triangulene dimers, Angew. Chem. Int. Ed.59, 12041 (2020)

2020