Fractionalization of Spin in Cluster-Based Haldane State of Triangular Spin Tube

The S=1 Haldane state has attracted much attention due to its topological characters, e.g., the string order and the symmetry-protected topological phase, and its potential for a quantum memory in the measurement-based quantum computation [1]. This state emerges not only in the S=1 spin chain, but also in an S=1/2 two-leg spin ladder with a ferromagnetic rung interaction [2]. Additionally, our recent study on low-temperature magnetism in a naturally occurring mineral Fedotovite, has shown the Haldane state consisting of cluster-based S=1 spins, in collaboration with experimental groups [3]. In this compound, edge-shared tetrahedral spin clusters which include six spins in a cluster, are linked by a nearest-neighbor antiferromagnetic inter-cluster interaction in one dimension. The low-energy state in a cluster is a triplet and the inter-cluster interaction is perturbative as compared with the intra-cluster excitation energy, so that we can explain the low-temperature physics in an effective S=1 spin model obtained by projection into cluster's triplet states.
In present study, to develop our knowledge about conditions for emergence of the Haldane state, we extend the cluster-based Haldane state to spin-cluster systems which have three S=1/2 spins in a cluster, corresponding to a triangular tube. Since the odd-number spin system with time-reversal symmetry obeys the Kramer's theorem, even-number states degenerate as its ground states. Therefore, it is difficult to make an effective S=1 spin in a cluster. To overcome this difficulty, we consider scalar spin chirality to break the time-reversal symmetry and reproduce a cluster-based Haldane state. The scalar spin chirality is regarded as a pseudo spin together with a real spin of a cluster. If we suppose a Heisenberg-like interaction of these pseudo and real spins, a triplet state emerges as a ground state of a cluster. Based on this strategy, we compose a possible spin model exhibiting a cluster-based Haldane state without the magnetic field. In a spin triangular tube, we find a condition to reproduce the cluster-based Haldane state. Furthermore, we also find a magnetization degree of freedom, corresponding to a quarter (S=1/4) spin as an edge mode of the cluster-based Haldane state [4]. This is a new mechanism of quantum fractionalization. Our study is useful for strategical design of the Haldane state by using S=1/2 spins. In addition to the triangular tube, we will also present general spin-cluster chains exhibiting the cluster-based Haldane states [3,5].

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[3] M. Fujihala, T. Sugimoto, T. Tohyama, S. Mitsuda, et al., Phys. Rev. Lett. 120, 077201 (2018).
[4] T. Sugimoto and T. Tohyama, arXiv:2108.00681.
[5] T. Sugimoto, K. Morita, and T. Tohyama, Phys. Rev. Res. 2, 023420 (2020).