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arXiv:2609.11591

Robust Spin-1 Haldane Topology under Itinerant Doping

Satoshi Nishimoto

Published Sep 10, 2026• 0 citations
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Abstract

We study itinerant hole doping of the spin-$1$ Haldane symmetry-protected topological (SPT) phase in two $t$--$J$ chains coupled by ferromagnetic (FM) Hund exchange. Density-matrix renormalization group calculations show that string order, a finite spin gap, and the Haldane entanglement structure coexist over a broad doping range. Combined with entanglement-entropy scaling consistent with a single gapless charge mode, these results support a metallic doped Haldane SPT with a $C1S0$ low-energy description. At low doping, short-range charge correlations reduce the density of spin-$1/2$ defect rungs, whereas at larger doping their positions become nearly uncorrelated while their spins are collectively incorporated into the correlated spin sector. The doped Haldane SPT terminates in a fully spin-polarized FM phase, which forms a reentrant pocket at high doping in the global phase diagram. Dilute-limit analyses trace the two FM boundaries to spin--charge factorization at weak Hund coupling and to rung-triplet binding with reduced pair mobility at strong Hund coupling. Our results show how mobile carriers reorganize the Haldane spin structure while preserving its characteristic signatures.

DOI
10.48550/arxiv.2609.11591

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We study itinerant hole doping of the spin-111 Haldane symmetry-protected topological (SPT) phase in two ttt--JJJ chains coupled by ferromagnetic (FM) Hund exchange. Density-matrix renormalization group calculations show that string order, a finite spin gap, and the Haldane entanglement structure coexist over a broad doping range. Combined with entanglement-entropy scaling consistent with a single gapless charge mode, these results support a metallic doped Haldane SPT with a C1S0C1S0C1S0 low-energy description. At low doping, short-range charge correlations reduce the density of spin-1/21/21/2 defect rungs, whereas at larger doping their positions become nearly uncorrelated while their spins are collectively incorporated into the correlated spin sector. The doped Haldane SPT terminates in a fully spin-polarized FM phase, which forms a reentrant pocket at high doping in the global phase diagram. Dilute-limit analyses trace the two FM boundaries to spin--charge factorization at weak Hund coupling and to rung-triplet binding with reduced pair mobility at strong Hund coupling. Our results show how mobile carriers reorganize the Haldane spin structure while preserving its characteristic signatures.
We systematically investigate the stability of the symmetry-protected topological (SPT) Haldane phase in spin- 1 2 Heisenberg and half-filled Hubbard ladders coupled by ferromagnetic Hund's interactions. Using the density-matrix renormalization group method, we analyze key signatures of the Haldane phase: long-range string order, finite spin gap, and characteristic entanglement spectrum degeneracies. In spin-only Heisenberg ladders, we find immediate onset and continuous strengthening of the Haldane phase with increasing Hund's coupling. In contrast, the inclusion of charge fluctuations in Hubbard ladders leads to a nontrivial stability regime, revealing a robust yet bounded region where SPT order persists despite significant charge fluctuations. We identify distinct boundaries separating a trivial insulating phase from the Haldane SPT phase, governed by both Coulomb repulsion and Hund's coupling. Our results highlight the subtle interplay of spin and charge degrees of freedom in correlated itinerant systems and establish essential criteria for observing Haldane physics experimentally in fermionic ladder materials.
Different models for doping of two-orbital chains with mobile S=1/2S=1/2S=1/2 fermions and strong, ferromagnetic (FM) Hund's rule couplings stabilizing the S=1 spins are investigated by density matrix renormalization group (DMRG) methods. The competition between antiferromagnetic (AF) and FM order leads to a rich phase diagram with a narrow FM region for weak AF couplings and strongly enhanced triplet pairing correlations. Without a level difference between the orbitals, the spin gap persists upon doping, whereas gapless spin excitations are generated by interactions among itinerant polarons in the presence of a level difference. In the charge sector we find dominant singlet pairing correlations without a level difference, whereas upon the inclusion of a Coulomb repulsion between the orbitals or with a level difference, charge density wave (CDW) correlations decay slowest. The string correlation functions remain finite upon doping for all models.
We investigate the pairing tendencies in the hole-doped Haldane spin-1 chain. To allow for doping, we extend the original spin chain Hamiltonian into a fermionic model involving a two-orbital Hubbard chain at intermediate or strong repulsive interaction strengths UUU and for degenerate orbitals. At half filling and large UUU, the ferromagnetic Hund's coupling, JH{J}_{\mathrm{H}}JH​, generates effective spin-1 moments, with antiferromagnetic correlations between sites. Using large-scale density matrix renormalization group calculations, we accurately study the system's behavior under light hole-doping. For U=1.6U=1.6U=1.6 in units of the noninteracting bandwidth and for ${J}_{\mathrm{H}}/U\ensuremath{\gtrsim}0.275$, we find that singlet pairing dominates the long-distance physics, establishing this system as a promising platform for repulsively mediated superconductivity. We provide concrete examples of materials that could realize the physics described here. We also provide evidence that the system approaches a Luther-Emery liquid state at large system sizes, reminiscent of the behavior of doped one-orbital two-leg ladders at weak coupling, which also have superconducting tendencies. The numerically calculated central charge approaches one in the thermodynamic limit, indicating a single gapless mode as is expected for the Luther-Emery state. Exponents characterizing the power-law decays of singlet pair-pair and charge density-density correlations are determined, and found to approximately satisfy the Luther-Emery identity.
We report the numerical discovery of a Luther-Emery liquid with pair-density-wave (PDW) correlations from doping either (i) a spin-one Haldane chain or (ii) a two-leg ladder in the rung-singlet phase in which the doped charges occupy a single leg. We model these systems using a generalized Kondo model. The itinerant electrons are correlated and described by the $t\text{\ensuremath{-}}J$ model, and are further coupled to a spin-1/21/21/2 Heisenberg model through the Kondo coupling JK{J}_{K}JK​. When the density of electrons xxx is one, the Mott insulator is in a Haldane phase or in a rung-singlet phase depending on whether JK{J}_{K}JK​ is negative or positive. Upon doping, a pair-density-wave with $\mathbf{Q}=\ensuremath{\pi}$ can emerge for both signs of JK{J}_{K}JK​. In the ${J}_{K}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ limit, the model reduces to the recently proposed type II $t\ensuremath{-}J$ model. We also identify a composite order parameter for the superconductor, which can be understood as a Cooper pair formed by two nearby fermionic spin-polarons. Our model and the predicted PDW phase can be experimentally realized by doping S=1S=1S=1 chains formed by Ni2+{\mathrm{Ni}}^{2+}Ni2+ in a solid-state system or a two-leg ladder of fermionic cold atoms with a potential bias between legs, which preferentially dopes carriers into a single leg. Long-range order of the PDW can be achieved in quasi-one-dimensional system with finite interwire coupling.
Many quantum phases are described by the Landau theory, which is based on the spontaneous symmetry breaking.However, the symmetry-protected topological (SPT) phases have been discovered as new quantum phases that cannot be described by the Landau theory.One of the examples is the Haldane phase in the one-dimensional spin-1 Heisenberg chain.This concept of the SPT phase is developed for gapped states.However, many researchers are attempting to extend this concept to gapless states.Such as SPT phase in a gapless system is called a gapless SPT phase.To calculate the ground state of a spin system, we perform the numerical calculation based on the density-matrix renormalization group (DMRG) method.All the quantum spin states on a lattice is written in the form of a matrix product state (MPS).The DMRG method is one of the variational methods by which we optimize the MPS.In addition, an MPS characterizes SPT phases.An entanglement spectrum is defined using singular values of an MPS and becomes even-fold degenerate when the state is in the Haldane state.We can also calculate an entanglement entropy that characterizes the gapless system.In this thesis, we investigate a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions.We call this model an anisotropic triangular-strip (ATS) model.According to the quantum field theory and renormalization group theory, when the rung coupling is sufficiently small, the frustrated interchain interaction almost isolates the second chain in the middle from the other two chains.In addition, at the same time, the first and third chains effectively mediate the ferromagnetic interaction.As a result, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions.By calculating the entanglement spectrum and the entanglement entropy, we show that the ground state of the ATS model with weak rung coupling is the gapless SPT state.Moreover, the ground state of the ATS model with sufficiently large rung coupling becomes ferrimagnetic state.We also investigate the other phases of the ATS model by the DMRG method.Through this study, we found the gapless SPT state which is not trivially decoupled into the gapless system and the nontrivial gapped SPT system in the ATS model, and we determine the phase diagram of the ATS model.This doctoral thesis would not have been completed without the support of many people.I am grateful to Shunsuke C. Furuya for proposing the basic concept of this thesis and supporting my investigation.He also gives me much fruitful advice especially when I got stuck in my research.I also would like to thank Atsushi Ueda especially for teaching me how to use the supercomputer in ISSP.Thanks to Yasuhiro Tada for teaching me great advice which is valuable to my study and for polishing this thesis.This study is performed with their cooperation, and some results are published in the journal Physical Review B.I am grateful to my supervisor, Prof. Masaki Oshikawa, for supporting many things about my research.He introduced the gapless SPT phase to me.I had meaningful discussions with him and was given much fruitful advice for completing this doctoral thesis.Oshikawa Group members gave me great and meaningful advice when I presented in the group seminars
We reexamine the one-dimensional spin-1 XXZXXZXXZ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-state phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and N\'eel phases, appearing at large single-ion and spin-exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.