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Experimental observation of Bethe strings


Almost a century ago, string states—complex bound states of magnetic excitations—were predicted to exist in one-dimensional quantum magnets1. However, despite many theoretical studies2,3,4,5,6,7,8,9,10,11, the experimental realization and identification of string states in a condensed-matter system have yet to be achieved. Here we use high-resolution terahertz spectroscopy to resolve string states in the antiferromagnetic Heisenberg–Ising chain SrCo2V2O8 in strong longitudinal magnetic fields. In the field-induced quantum-critical regime, we identify strings and fractional magnetic excitations that are accurately described by the Bethe ansatz1,3,4. Close to quantum criticality, the string excitations govern the quantum spin dynamics, whereas the fractional excitations, which are dominant at low energies, reflect the antiferromagnetic quantum fluctuations. Today, Bethe’s result1 is important not only in the field of quantum magnetism but also more broadly, including in the study of cold atoms and in string theory; hence, we anticipate that our work will shed light on the study of complex many-body systems in general.

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Figure 1: Quantum spin chain in SrCo2V2O8, psinon–(anti)psinon pairs and strings.
Figure 2: Softening of spinons and emergent magnetic excitations at the quantum phase transition in SrCo2V2O8.
Figure 3: Absorption spectra of psinon–psinon, psinon–antipsinon, two-string and three-string excitations for Bc < B < Bs and of magnons for B > Bs in SrCo2V2O8.
Figure 4: Magnetic excitations in the longitudinal-field Heisenberg–Ising chain SrCo2V2O8.


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We thank I. Bloch, M. Karbach, T. Lorenz and X. Zotos for discussions. We acknowledge partial support by the DFG via the Transregional Collaborative Research Center TRR 80, and by the HFML-RU/FOM and the HLD-HZDR, members of the European Magnetic Field Laboratory (EMFL). J.W., W.Y., S.X. and C.W. are supported by NSF grant number DMR-1410375 and AFOSR grant number FA9550-14-1-0168. C.W. also acknowledges partial support from the National Natural Science Foundation of China (grant number 11729402).

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Authors and Affiliations



Z.W. conceived and performed the optical experiments, analysed the data and coordinated the project. J.W., W.Y. and S.X. carried out the Bethe-ansatz calculations. A.K.B. and A.T.M.N.I. prepared and characterized the high-quality single crystals. A.K.B. and J.M.L. performed the high-field magnetization measurements. D.K. assisted with the high-field optical experiments. B.L., C.W. and A.L. supervised the project. Z.W., J.W., W.Y., C.W. and A.L. wrote the manuscript with input from all authors. All authors discussed the results.

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Correspondence to Zhe Wang.

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Reviewer Information Nature thanks M. Batchelor, J. van den Brink and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Figure 1 Crystal and magnetic structure of SrCo2V2O8.

a, The screw-chain structure consists of edge-shared CoO6 octahedra. Each chain has screw-axis symmetry with a period of four Co2+ ions (as numbered by the integers 1, 2, 3 and 4), corresponding to the lattice constant along the c axis. The Néel-ordered phase is illustrated by antiparallel arrows representing magnetic moments at the Co2+ sites. Intra-chain nearest-neighbour interaction is denoted by J. b, Viewing from the c axis, each unit cell contains four screw chains with left- or right-handed screw axes. The leading inter-chain coupling J is indicated, which is between the Co2+ ions in the same layer (denoted by the same integer as the Co site) and from chains with the same chirality. It is very small compared to the intra-chain interaction, J/J < 10−2 (refs 32, 33).

Extended Data Figure 2 High-field magnetization and magnetic susceptibility of SrCo2V2O8.

a, Magnetization M as a function of an applied longitudinal magnetic field B along the Ising axis (B c), measured at 1.7 K (circles). Theoretical magnetization of the Heisenberg–Ising chain model is shown by the dashed line. b, Magnetic susceptibility dM/dH as a function of the applied longitudinal field B. A quantum phase transition from the Néel-ordered phase to the critical phase is revealed by the onset of magnetization and the peak in the susceptibility curve at the critical field Bc = 4 T. Saturated magnetization is observed above the field Bs = 28.7 T and indicated by the sharp peak in the susceptibility. The small anomaly at Bhs = 25 T seen in the susceptibility is close to the field of half-saturated magnetization.

Extended Data Figure 3 Low-energy phonon spectrum of SrCo2V2O8.

The phonon spectra of SrCo2V2O8 measured for the polarization Eω a at 5 K. Strong reflectivity due to phonon excitations is observed in the spectral range 8–13.5 meV.

Extended Data Figure 4 Schematics of patterns of Bethe quantum numbers.

a, The ground state. b, One-pair psinon–psinon state 1ψψ. c, One-pair psinon–antipsinon state 1ψψ*. d, Length-two string state 1χ(2)R. The system size is taken as N = 32 and the magnetization is .

Extended Data Figure 5 DSFs.

a, b, S+−(q, ω) and S−+(q, ω), respectively, as functions of energy ħω/J (vertical axis) and momentum q/π (horizontal axis) for 2m = 0.4 and N = 200. The gapless continua are formed by real Bethe eigenstates (psinon–antipsinon pairs in S+− and psinon–psinon pairs in S−+). For S+− (a), the higher-energy continua correspond to excitations of two-string (ħω > 3J) and three-string (ħω > 5J) states.

Extended Data Figure 6 The momentum-integrated ratios.

a, b, ν+− for S+− and ν−+ for S−+, respectively, as functions of magnetization 2m. In a, the green line is the 1ψψ* contribution. The blue, red and the black lines are augmented by progressively taking into account the 2ψψ*, two-string and three-string contributions, respectively. In b, the blue and black lines represent the 1ψψ and 1ψψ + 2ψψ contributions, respectively.

Extended Data Figure 7 DSF of psinon–psinon pairs as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

Extended Data Figure 8 DSF of psinon–antipsinon pairs as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

Extended Data Figure 9 DSF factor of two-string states as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

Extended Data Figure 10 DSF of three-string states as a function of energy for 2m = 0.1–0.9.

a, q = 0; b, q = π/2; c, q = π.

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Wang, Z., Wu, J., Yang, W. et al. Experimental observation of Bethe strings. Nature 554, 219–223 (2018).

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