Cluster-based Haldane states in spin-1/2 cluster chains

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Cluster-based Haldane states in spin-1/2 cluster chains

Takanori Sugimoto, Katsuhiro Morita, and Takami Tohyama

Phys. Rev. Research 2, 023420 – Published 30 June 2020

Abstract

The Haldane state is a typical quantum and topological state of matter, which exhibits an edge state corresponding to symmetry-protected topological order in a one-dimensional integer spin chain. Here we propose a concept to design the Haldane state with one-half spins by making use of a chain composed of one-half spin clusters. If the clusters contain two spins, the ground state of a chain corresponds to the Affleck-Kennedy-Lieb-Tasaki state. We thus extend this knowledge to general clusters consisting of not only even, but odd numbers of one-half spins. In the case of an odd number of spins in the clusters, we present a concrete procedure to construct a field-induced Haldane state, and demonstrate the Haldane state in a five-spin cluster chain. Our concept is useful to understand quantum magnetism in real materials such as Fedotovite, consisting of six-spin clusters aligned in one dimension. Furthermore, the edge state designed by our flexible scheme can be utilized for a processing unit of quantum computation with recent progress on intended synthesis of organic materials, quantum dots, and optical lattices.

Received 23 December 2019
Revised 7 June 2020
Accepted 8 June 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.023420

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Magnetic phase transitions Symmetry protected topological states Topological phase transition Topological phases of matter Topological quantum computing

Quantum spin models

Quantum spin chains

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Takanori Sugimoto^1,2,*, Katsuhiro Morita^1,†, and Takami Tohyama ¹

¹Department of Applied Physics, Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan
²Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan

^*sugimoto.takanori@rs.tus.ac.jp
^†Present address: Department of Basic Science, University of Tokyo, Meguro, Tokyo 153-8902, Japan.

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Vol. 2, Iss. 2 — June - August 2020

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Images

Figure 1
Schematic ground state of a spin cluster chain (SCC). One-half spins ( $S = \frac{1}{2}$ ) are represented by arrows. We assume that spins in a transparent ball (cluster) strongly interact with each other, and that the spins weakly interact with spins in the next balls. The detailed behaviors in three temperature regions are explained in the main text.
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Figure 2
(a) Spin ladder as a two-spin cluster chain (an $N = 2$ SCC). The chain is elongated in the $a$ axis. Blue balls and colored segments denote the spin sites and interaction bonds, respectively. The interaction $J$ ( $J^{'}$ ) corresponds to the intra- (inter)cluster interaction. $(i, j)$ denotes the spin site index, e.g., (1,2) means the first site in the second cluster. (b) Energy spectrum of a cluster (rung) for $J < 0$ , where the ground state is a triplet. If the intracluster interaction is much smaller than the intercluster interaction $| J^{'} | ≪ | J |$ , only the triplet states of a cluster contribute to the low-energy physics as an effective $S = 1$ spin. (c) Schematic ground state of a SCC for $J^{'} > 0$ . The triplet state of an effective $S = 1$ spin is symmetric, and two spins of neighboring rungs are connected with the singlet (antisymmetric) configuration. Thus, this ground state corresponds to the AKLT state.
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Figure 3
(a) Four-spin cluster chain (an $N = 4$ SCC). Notations are the same as Fig. 2. (b) Energy of a cluster as a function of $θ$ that controls $J_{i}$ with fixed $ϕ = π / 4$ . The lowest-energy level of total spin ( $S = 0$ , 1, and 2) is denoted by $ε_{S}$ . The energy levels of singlet and triplet states cross at $θ_{1} = \frac{3}{4} π$ and $θ_{2} = \frac{7}{4} π$ . (c) Absolute value of string and spin correlation functions for $J_{inter} / J_{intra} = 0.2, θ = \frac{5}{4} π$ , and $ϕ = \frac{π}{4}$ in a 60-cluster chain ( $L = 60$ ) with open boundary condition. The correlation functions in (c) are calculated at $M = 0$ .
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Figure 4
(a) Schematic of condition (i). If $ε_{S_{0} + 2} - ε_{S_{0} + 1} = ε_{S_{0} + 1} - ε_{S_{0}} = Δ$ without magnetic fields, a field-induced triplet (FIT) consisting of $| S^{z} 〉 = | \frac{1}{2} 〉, | \frac{3}{2} 〉$ , and $| \frac{5}{2} 〉$ for $S_{0} = 1 / 2$ emerges at the critical field $h_{c}^{z}$ . (b) Two weakly coupled subsystems denoted by orange and blue circles in a five-spin cluster. The two subsystems are connected by an exchange interaction $α J$ ( $J > 0$ is the energy unit of the cluster). (c) $δ E_{S_{0}} = ε_{S_{0} + 2} - 2 ε_{S_{0} + 1} + ε_{S_{0}}$ as a function of $J$ and a deformation parameter $β$ independent of $α$ . The $β = 0$ case corresponds to (b). The white (purple) region denotes the (positive) negative $δ E_{S_{0}}$ . Since $δ E_{S_{0}}$ changes its sign at $α = 0$ without deformation ( $β = 0$ ), we can deform the model parameters along the $δ E_{S_{0}} = 0$ line, on which condition (i) is satisfied. (d) An example of deformation of a five-spin cluster. The point of $(α, β) = (0, 0)$ corresponds to the trivial case represented in (b). We can reach a simpler model, a bowtie cluster, in Fig. 5, as we trace the boundary.
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Figure 5
(a) Five-spin cluster chain (an $N = 5$ SCC). Notations are the same as Fig. 2. (b) Energy of a cluster as a function of $λ$ that controls $J_{i}$ with fixed $η = π / 4$ . Here, $λ_{1} = \frac{π}{4} - \frac{1}{2} {sin}^{- 1} (\frac{1}{3})$ and $λ_{2} = \frac{3}{2} π + {sin}^{- 1} (\frac{1}{3})$ . (c) Magnetization curve normalized by $M_{sat}$ and (d) absolute value for string and spin correlation functions for $J_{inter} / J_{intra} = 0.3, λ = \frac{π}{8}$ , and $η = \frac{π}{4}$ in a 48-cluster chain ( $L = 48$ ) with open boundary condition. The inset of (c) is an enlarged view around $M = \frac{3}{5} M_{sat}$ , where we find the magnetization plateau at $M = \frac{3}{5} M_{sat} + 1$ denoted by a circle. The correlation functions in (d) are calculated at $M = \frac{3}{5} M_{sat} + 1$ .
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