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arXiv:2512.16173

Tunable Topological Phases in an Organic One-Dimensional Mott Chain: Odd-Haldane (S = 1/2) and Haldane (S = 1)

Khalid N. Anindya, Hong Guo

Published Dec 18, 2025• 0 citations
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Abstract

Establishing symmetry-protected topological (SPT) phases with interactions in chemically realistic systems remains an open challenge. We show that a single, synthetically plausible organic one-dimensional chain, tunable via chemical modification of its radical sites, hosts two such phases: an odd-Haldane phase of a dimerized $S=\tfrac{1}{2}$ Heisenberg chain and a Haldane phase of an $S=1$ chain realized when Hund coupling locks two $S=\tfrac{1}{2}$ spins per monomer into $S=1$. Density-functional theory places the active manifold deep in the Mott regime ($U/t\!\approx\!126$), justifying a spin-only Heisenberg description; a compact $(t,U)\!\to\!J$ mapping then fixes exchange couplings. Exact diagonalization and DMRG reveal a consistent SPT fingerprint across both phases, including a quantized many-body Zak phase, even-degenerate entanglement spectrum, protected edge spins, and characteristic triplon/Haldane features in $S^{+-}(q,ω)$. Our results identify a chemically programmable molecular platform for interacting SPT physics in one dimension and suggest concrete spectroscopic routes to organic Haldane spin chains for nanoscale quantum devices.

DOI
10.48550/arxiv.2512.16173

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Establishing symmetry-protected topological (SPT) phases with interactions in chemically realistic systems remains an open challenge. We show that a single, synthetically plausible organic one-dimensional chain, tunable via chemical modification of its radical sites, hosts two such phases: an odd-Haldane phase of a dimerized S=12S=\tfrac{1}{2}S=21​ Heisenberg chain and a Haldane phase of an S=1S=1S=1 chain realized when Hund coupling locks two S=12S=\tfrac{1}{2}S=21​ spins per monomer into S=1S=1S=1. Density-functional theory places the active manifold deep in the Mott regime (U/t ⁣≈ ⁣126U/t\!\approx\!126U/t≈126), justifying a spin-only Heisenberg description; a compact (t,U) ⁣→ ⁣J(t,U)\!\to\!J(t,U)→J mapping then fixes exchange couplings. Exact diagonalization and DMRG reveal a consistent SPT fingerprint across both phases, including a quantized many-body Zak phase, even-degenerate entanglement spectrum, protected edge spins, and characteristic triplon/Haldane features in S+−(q,ω)S^{+-}(q,ω)S+−(q,ω). Our results identify a chemically programmable molecular platform for interacting SPT physics in one dimension and suggest concrete spectroscopic routes to organic Haldane spin chains for nanoscale quantum devices.
Establishing symmetry-protected topological (SPT) phases with interactions in chemically realistic systems remains an open challenge. We show that a single, synthetically plausible organic one-dimensional chain, tunable via chemical modification of its radical sites, hosts two such phases: an even-Haldane phase of a dimerized S = 1 2 Heisenberg chain and a Haldane phase of an S = 1 chain realized when Hund coupling locks two S = 1 2 spins per monomer into S = 1. Density-functional theory places the active manifold deep in the Mott regime ( U / t > 100), justifying a spin-only Heisenberg description; a compact ( t, U ) → J mapping then fixes exchange couplings. Exact diagonalization and DMRG reveal a consistent SPT fingerprint across both phases, including a quantized many-body Zak phase, even-degenerate entanglement spectrum, protected edge spins, and characteristic triplon-Haldane features in S +– ( q,ω). Our results identify a chemically programmable molecular platform for interacting SPT physics in one dimension and suggest concrete spectroscopic routes to organic Haldane spin chains for nanoscale quantum devices.
Abstract Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain 1,2 . Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter 3,4 . More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement 5–7 . Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.
Under an appropriate symmetric extensive bipartition in a one-dimensional symmetry-protected topological (SPT) phase, a bulk critical-entanglement spectrum can be obtained, resembling the excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. For the S=1S=1S=1 SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical-entanglement spectrum has a residual entropy per lattice site sr=0.67602{s}_{r}=0.67602sr​=0.67602, showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted $c\ensuremath{\approx}1.01\ifmmode\pm\else\textpm\fi{}0.01$. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse-square long-range interaction.
We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional (Q1D) geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean-field universality of the critical coupling times the coordination number of the lattice. Inspired by recent developments in the characterization of symmetry protected topological (SPT) states, of which the Haldane phase of the spin-1 Heisenberg antiferromagnetic chain is a preeminent example, we provide direct evidence that the Q1D Haldane phase is indeed a nontrivial SPT state.
We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-SSS Haldane phase is a topologically nontrivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of $\ensuremath{\pi}$ rotations about the xxx, yyy, and zzz axes, (ii) time-reversal symmetry ${S}^{x,y,z}\ensuremath{\rightarrow}\ensuremath{-}{S}^{x,y,z}$, and (iii) link inversion symmetry (reflection about a bond center), consistent with previous results [Phys. Rev. B 81, 064439 (2010)]. On the other hand, an even-SSS Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.
By performing large-scale density-matrix renormalization group simulations, we investigate a one-dimensional correlated bosonic lattice model with a synthetic spin-orbit coupling realized in recent experiments. In the insulating regime, this model exhibits a symmetry-protected topological phase. This symmetry-protected topological phase is stabilized by time-reversal symmetry and it is identified as a Haldane phase. We confirm our conclusions further by analyzing the entanglement spectrum. In addition, we find four conventional phases: a Mott insulating phase with no long range order, a ferromagnetic superfluid phase, a ferromagnetic insulating phase, and a density-wave phase.