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Nakamura, M., [Hase, Masashi](https://orcid.org/0000-0003-2717-461X), Kajimoto, R., Morioka, T., Sakakibara, T., Frontzek, M., Reynolds, N., Hagiwara, M., Kimura, T., Yamaguchi, Y., Tomiyasu, K., White, J. S., Onoda, S., Masuda, T., Ohira-Kawamura, S., Soda, M., Ueda, H., Nakajima, K., Inamura, Y., Yasui, Y., Yoshizawa, D., Hagihala, M.

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[Emergent spin-1 Haldane gap and ferroelectricity in a frustrated spin-1/2 ladder](https://mdr.nims.go.jp/datasets/dcfa6b4c-5e08-4645-8adc-57a79d294cc3)

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PHYSICAL REVIEW B 101, 140408(R) (2020)Rapid Communications Editors’ SuggestionEmergent spin-1 Haldane gap and ferroelectricity in a frustrated spin-12 ladderH. Ueda ,1,2 S. Onoda,3,4 Y. Yamaguchi,5 T. Kimura,6 D. Yoshizawa,7 T. Morioka,7 M. Hagiwara,7 M. Hagihala,8 M. Soda,8T. Masuda,8 T. Sakakibara,8 K. Tomiyasu,9 S. Ohira-Kawamura,10 K. Nakajima,10 R. Kajimoto,10 M. Nakamura,10Y. Inamura,10 N. Reynolds,11 M. Frontzek,11,12 J. S. White,11 M. Hase,13 and Y. Yasui141Computational Materials Science Research Team, RIKEN Center for Computational Science (R-CCS), Kobe 650-0047, Japan2JST, PRESTO, Kawaguchi, Saitama 332-0012, Japan3Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan4Quantum Matter Theory Research Team, RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan5Division of Materials Physics, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan6Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan7Center for Advanced High Magnetic Field Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan8Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan9Department of Physics, Tohoku University, Sendai 980-8578, Japan10Materials and Life Science Division, J-PARC Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan11Laboratory for Neutron Scattering, Paul Scherrer Institut, CH-5232 Villigen, Switzerland12Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA13National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan14Department of Physics, School of Science and Technology, Meiji University, Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan(Received 19 March 2018; revised manuscript received 20 August 2018; accepted 31 March 2020;published 22 April 2020)We report experimental and theoretical evidence that Rb2Cu2Mo3O12 has a nonmagnetic tetramer ground stateof a two-leg ladder comprising antiferromagnetically coupled frustrated spin- 12 chains and exhibits a Haldanespin gap of emergent spin-1 pairs. Three spin excitations split from the spin-1 triplet by a Dzyaloshinskii-Moriyainteraction are identified in inelastic neutron-scattering and electron spin resonance spectra. A tiny magnetic fieldgenerates ferroelectricity without closing the spin gap, indicating a unique class of ferroelectricity induced by avector spin chirality order.DOI: 10.1103/PhysRevB.101.140408Quantum spin fluctuations offer a source of various non-trivial states including resonating valence bonds and quan-tum spin liquids [1]. In a one-dimensional (1D) antiferro-magnet having only the first-neighbor exchange couplingJ1 [Fig. 1(a)], the spin quantum number S critically de-termines the magnitude of quantum spin fluctuations of along-wavelength mode n(τ, x) around a short-range Néelorder. The topological Berry-phase term gives a contributionof SB1DHAF = i2πSQ to a nonlinear-σ model action for nwith a topological integer Q = 14π∫ 1/T0 dτ∫dx n · ( ∂n∂x × ∂n∂τ)and the temperature T . Thus, e−SB1DHAF can take −1 fora half-integer S, allowing for gapless excitations from adisordered ground state. On the other hand, it is alwaysunity for an integer S, leading to a so-called Haldane gap[2,3] in the S = 1 excitation spectrum from a nonmag-netic ground state [4,5], as experimentally evidenced inNi(C2H8N2)2NO2(ClO4) (NENP)[6–8].In the presence of an antiferromagnetic second-neighborexchange coupling J2, however, the above simple argumentsno longer hold. In particular, quasi-1D spin- 12 multifer-roic and/or magnetoelectric edge-sharing cuprates, such asLiCu2O2 [9,10], LiCuVO4 [11,12], PbCuSO4(OH)2 [13,14],and Rb2Cu2Mo3O12 [15–17], involve a ferromagnetic J1 be-cause of nearly 90◦ Cu-O-Cu bond angles, in addition toan antiferromagnetic second-neighbor exchange coupling J2.The J1-J2 frustrated spin- 12 Heisenberg chain accommodatesa dimerized spin-singlet short-range resonating valence bondground state [18,19]. This state is formed by emergent spin-1pairs [Fig. 1(b)] and has an extremely small Haldane gapand incommensurate short-range spin correlations. Weak butfinite easy-plane exchange magnetic anisotropy then inducesa quasi-long-range gapless incommensurate spin-spiral andlong-range vector spin chirality (∑�〈S� × S�+1〉) [20] orders[19,21]. A coexisting phase of the vector spin chirality or-der and the Haldane gap also appears in between the twophases [22]. These states are, however, readily driven to along-range spiral magnetic order by three-dimensional cou-plings. This scenario elucidates the ferroelectricity due to thecycloidal magnetism in LiCu2O2 [10], LiCuVO4 [12], andPbCuSO4(OH)2 [13].In fact, the ferroelectricity associated with the vector spinchirality order may appear robustly in the vector-chiral Hal-dane dimer phase without the long-range spiral magnetism,if the spin gap is enhanced [23] so that it dominates overthe interchain interactions. Indeed, Rb2Cu2Mo3O12 providesa unique example of a field-induced ferroelectricity hosted bya nonmagnetic ground state with a spin gap [16,17]. A recentmuon spin relaxation (μSR) study also indicates the formationof a spin-singlet state on cooling below ∼7 K and satura-tion at around 1–2 K [24]. In this Rapid Communication,2469-9950/2020/101(14)/140408(6) 140408-1 ©2020 American Physical SocietyH. UEDA et al. PHYSICAL REVIEW B 101, 140408(R) (2020)FIG. 1. Structures of spin chains. (a) Antiferromagnetic spinchain. (b) Frustrated spin- 12 chain with emergent spin-1 pairs (blueclouds). Solid (broken) lines represent antiferromagnetic (ferromag-netic) interactions. (c) Short-range resonating valence bond stateinvolving tetramers (yellow plaquettes) connecting emergent spin-1pairs. (d) Crystal structure of a pair of spin- 12 chains comprisingedge-sharing distorted CuO6 octahedra in Rb2Cu2Mo3O12. (e) Anideal centrosymmetric chain of edge-sharing regular CuO6 octahe-dra (black points), compared with the noncentrosymmetric one inRb2Cu2Mo3O12. Electric dipole moments due to ionic displacementsare shown on the first-neighbor Cu spin pairs by dark green arrows.(f) A unit cell of Rb2Cu2Mo3O12. Two-leg ladders are located intranslucent orange tubes.we report combined experimental and theoretical evidencethat in the quasi-1D cuprate Rb2Cu2Mo3O12, a Haldane-gapground state formed by emergent spin-1 pairs of S = 12 Cuspins [Fig. 1(c)] harbors a ferroelectricity stabilized by a tinymagnetic field.Figures 2 shows a temperature dependence of thermody-namic properties of polycrystalline Rb2Cu2Mo3O12 samples.The dielectric constant ε gradually increases on cooling.Then, as in most magnetically induced ferroelectrics, it ex-hibits a kink for B = 0.3 and 0.5 T or a peak for B = 1and 2 T at around 8 K [Fig. 2(a)], below which the electricpolarization P emerges at an even weaker magnetic fieldB = 0.05 T [Fig. 2(b)]. Thus, the anomaly in ε at B � 0.05 Tshould be ascribed to a ferroelectric transition at TFE ∼ 8 K. Itis natural to expect that the ferroelectric polarization persistsat T < 2 K because of no sign of a reentrant behavior in ε andP in the low-temperature range. Remarkably, ε does not showa significant decay on cooling down to 2 K for B � 0.5 T,while it does for B � 1 T. Furthermore, doping nonmagneticZn impurities into Cu sites by 2% [25] drastically suppressesε and removes the anomaly associated with the ferroelectrictransition [Fig. 2(a)]. Therefore, it is clear that the ferroelec-tricity is indeed triggered by a coherence in the spin degreesof freedom under the weak magnetic field.The signals of both ε and P below TFE are larger for theconfiguration of E, P ⊥ B than for E, P ‖ B at least at 2 T[Figs. 2(a) and 2(b)], as in many edge-sharing multiferroiccuprates showing a cycloidal magnetic order [10,12,13]. Thisimplies that the uniform vector spin chirality gives rise to adominant contribution to the ferroelectric polarization among0.00(a)(b)(c)(d)2 T1 T0.5 T0.3 T0 T2 T2 T0 T0.05 T0.3 T1 T2 T0.010.020.00.80.60.40.25.96.00.00.51.00 5 10 155.45.3210−1−2−3ExperimentalTheory0.020.010.000 100 200 300FIG. 2. Temperature dependence of thermodynamic propertiesof polycrystalline Rb2Cu2Mo3O12. (a) Dielectric constant ε/ε0.(b) Electric polarization P at magnetic fields. Note that a powderaverage of the magnetic field direction broadens the transition.(c) Magnetic susceptibility χ (open circles). The impurity contribu-tion χimp, responsible for the upturn of χ below 0.5 K, was fittedby the Curie-Weiss law with the spin vacancy concentration of 0.5%and the Weiss temperature −0.5 K (blue dashed curve). Red pointsrepresent the data χspin subtracted by χimp. Also shown is dχ/dT(green points). The inset shows a high-temperature fitting of χ (blackcurve) with a powder average of the exact diagonalization results(red curve). (d) Specific heat C at B = 0. The solid curves in (c) and(d) are the fitting curves proportional to exp(−Eg/T ) with the energygap Eg = 1.7 K.many mechanisms [26]. On the other hand, no anomaly is ob-served in the magnetic susceptibility χ and dχ/dT [Fig. 2(c)],in contrast to the multiferroic cuprates [10,12,13]. Moreover,a spin gap Eg ∼ 1.7 K has been observed in both χ and thespecific heat C [16,17] [Figs. 2(c) and 2(d)].The emergence of this spin gap is also confirmed bythe measurements of the magnetization M. Figures 3(a) and3(b) present experimental results on M and on dM/dB andd2M/dB2, respectively. A subtraction of a small impuritycontribution as outlined in Fig. 2 caption reveals that M at T =0.08 K shows an activation by the threshold field Bc ∼ 2.0 Twhere d2M/dB2 exhibits a peak. On the other hand, at a much140408-2EMERGENT SPIN-1 HALDANE GAP AND … PHYSICAL REVIEW B 101, 140408(R) (2020)FIG. 3. Magnetic field dependence of thermodynamic propertiesof polycrystalline Rb2Cu2Mo3O12. (a) Magnetization M per Cuatom. (b) Derivatives of M with respect to B. (c) Ferroelectricpolarization P for both P ‖ B and P ⊥ B.lower temperature T = 2.0 K than TFE, P(⊥ B) steeply ap-pears at a much lower field, at least 0.03 T, than Bc [Fig. 3(c)].It exhibits a broad peak at around 0.2–0.3 T, and then grad-ually decays to a constant at higher fields up to 4 T. Thisobservation confirms that the ferroelectricity is stabilized by atiny magnetic field but not affected by a closing of the spin gapand an onset of the magnetization at Bc. Namely, at the energyscale associated with 0.03 T or less, there exists a low-energymode, which is magnetic-dipole inactive but electric-dipoleactive, and thus linearly coupled to the vector spin chirality.All the above thermodynamic properties provide evidenceof a spin-gapped ferroelectric behavior stabilized by thetiny applied magnetic field, most likely through the vectorspin chirality. It should also be possible to confirm thisfrom spectral properties. To probe S = 1 triplet excitationsfrom the nonmagnetic ground state, low-energy inelasticneutron-scattering experiments have been performed onpowder samples. Figure 4(a) represents the results at 1.5 Kmeasured on the AMATERAS spectrometer. Discrete excitedlevels are clearly seen at 0.2, 0.38, and 0.6 meV. Theperiodicity of these spin excitations along the chain can bedetermined from the onset wavenumber Q ∼ 0.3 Å−1 ofthe powder-averaged intensities, and roughly corresponds toeight spins. A natural interpretation will be that S = 1 tripletexcitations are split into the three by Dzyaloshinskii-Moriyainteractions. Note that cooling below TFE and applying themagnetic field do not alter the diffraction patterns (Fig. 4(f)and Ref. [28]): neither a superlattice peak nor any visibleadditional diffraction intensity appears. Note also that a clearlong-range magnetic order is absent at the incommensuratewavevector (0, Qb, 0) to an accuracy of 0.06μB. Actually,FIG. 4. Neutron-scattering results on polycrystallineRb2Cu2Mo3O12 at B = 0. (a) Experimental and (b) theoreticallow-energy powder-averaged spectra, measured at 1.6 K andcalculated at T = 0, respectively. (c) Theoretical low-energyspectra along the b axis without the powder average. Theresults obtained from a 28-site cluster by taking the parameterset for the thermodynamic limit [27] have been interpolated.The incommensurate wavevector Q = 0.3 Å−1 is marked byblack arrows in (a), (b), and (c). (d) Experimental and (e) theoreticalpowder-averaged spectra, measured at 6.5 K and calculated at T = 0,respectively, in a wider energy range. Note that the incommensuratewavevector is shifted downwards from the maximum position of thepowder-averaged spectra to the onset in the panels (a) and (b). (f)Neutron powder diffraction patterns of Rb2Cu2Mo3O12 measuredat 9.89 K > TFE (black) and 1.58 K < TFE (red) in B = 0 T. Thefour peaks with ∗ symbols are derived from a nonmagnetic impurityphase Rb2Mo3O10. A cold neutron wavelength λ = 4.5 Å waschosen.the absence of clear muon spin precession or relaxation[24] precludes a long-range order of all the Cu spins withmoment amplitude �0.01μB and of dilute (>1%) Cu orimpurity spins with moment amplitude 1μB. The possibilityof having a tiny fraction (<1%) of magnetically ordereddomains in the polycrystalline samples can hardly be ruledout. However, such order is absolutely extrinsic and irrelevantto the observed magnetic and ferroelectric properties of thebulk, because the Weiss temperature −0.5 K is much lowerthan TFE and the exchange coupling constants obtained below.The overall experimental results on the magnetic propertiescan be elucidated theoretically from the following two-legladder model of frustrated J1-J2 spin- 12 chains [Fig. 1(c)] [27]:H =∑�∑σ=±⎡⎣ ∑j=1,2JjSσ,� · Sσ,�+ j + J ′S+,� · S−,�,+ σ ((−1)�Ds · Sσ,� × Sσ,�+1 + Du · Sσ,� × Sσ,�+1)− gμBB · Sσ,�⎤⎦ (1)140408-3H. UEDA et al. PHYSICAL REVIEW B 101, 140408(R) (2020)with the g factor g = 2.16 [27] and the applied magneticfield B, where Sσ,� stands for an S = 12 spin at the site� in the chain of edge-shared CuO6 octahedra [Fig. 1(d)]labeled by the index σ = ±. It has already been revealedthat the antiferromagnetic rung exchange coupling J ′between the nearest-neighbor spins in the adjacent J1-J2chains is required for enhancing the spin gap [29]. Duand Ds represent the uniform and staggered componentsof intrachain Dzyaloshinskii-Moriya vectors caused bytwo inequivalent first-neighbor Cu-Cu bonds involvingnoncollinear electric dipole moments, as shown by darkgreen arrows in Fig. 1(e). No crystal symmetry constrains thedirections of the Dzyaloshinskii-Moriya vectors. However,since the numerical results shown below are insensitive to anonzero value of Du · Ds, we take Du ⊥ Ds. Henceforth, weadopt J1 = −114 K, J2 = 35.1 K, J ′ = 20.5 K, Ds = 44.3 K,and Du = 24.4 K to explain overall results of the magneticsusceptibility and inelastic neutron-scattering spectra fromexact-diagonalization calculations on a 16-site cluster. [SeeSupplemental Material [27] for examinations of finite-sizeeffects by means of the density-matrix renormalization groupfor infinite systems (iDMRG).] Indeed, the numerical resultson χ for B = 0 reasonably agree with the experimental data[15], as shown in the inset of Fig. 2(c), and the iDMRG result2.15 T on the critical magnetic field agrees with the experi-mental one ∼2.0 T [Fig. 3(b)]. Furthermore, the experimentalresults on the low-energy powder-averaged inelastic neutron-scattering spectra [Fig. 4(a)] are nicely explained by thetheoretical results [Fig. 4(b)] [27]. From Figs. 4(a) and 4(b),the three low-energy excitations might look dispersionless.However, this is an artifact of powder averaging. Figure 4(c)shows the theoretical results of the dispersive spectra as afunction of the particular wavevector component Qb in thechain direction, the crystallographic b axis, with Qa = Qc =0. Actually, the agreement in the inelastic neutron-scatteringspectra extends to a much higher energy ∼10 meV, as isapparent by comparing the current experimental results in thehigh energy range measured at the 4SEASONS spectrometer[Fig. 4(d)], which are refined from the previous data [30],with the theoretical results [27] [Fig. 4(e)].The scenario of a splitting of the S = 1 excited statesdue to Dzyaloshinskii-Moriya interactions is also supportedby electron spin resonance (ESR) experiments on powdersamples. Figure 5(a) presents the temperature dependence ofthe ESR transmission spectra at a frequency f = 81 GHz ∼0.33 meV as a function of B. A paramagnetic resonanceis found as a significantly broad peak at 2.7 T for a muchlower temperature, 8.7 K, than J1, J2, and J ′, as indicatedby red arrows. It should appear as a much sharper peakin the absence of moderately large Dzyaloshinskii-Moriyainteractions [31]. On cooling, the peak becomes even morebroadened, and eventually bifurcates below 5 K. In the fre-quency dependence of the ESR spectra at 1.6 K [Fig. 5(b)],this new low-energy mode (green arrows) has been identified,as well as another lower-energy mode (blue arrows). Thesetwo modes are plotted with  and � in the B- f diagramof Fig. 5(c), in favorable comparison with a density plot ofthe theoretical results [27] on the optical absorption power[32] at the same temperature. The dominant contributions tothe two series originate from thermally activated transitions.f (GHz)(a))d()c((b)0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.00.0 0.5 1.0 1.5 2.0B (T)050100150200050100150200FIG. 5. Electron spin resonance spectra of polycrystallineRb2Cu2Mo3O12. (a) Temperature dependence of the experimentaltransmission spectra at 81 GHz. Arrows represent the resonancefields. (b) Experimental transmission spectra at 1.6 K for desig-nated frequencies. Green and blue arrows denote two sequencesof resonance fields. (c) Theoretical optical absorption power at1.6 K. Experimentally observed resonance fields indicated by blueand green arrows in (a) are plotted by  and �, respectively, forcomparison. (d) Energy levels at Qb = 0 (left) and Qb = 1/4 r.l.u.(right) computed under B applied along the x, y, and z directions,where Du ‖ z and Ds ‖ x. Transitions denoted by the arrows in theleft (right) panel produce resonance spectra shown by dashed (solid)curves in (c).Theoretically, the second-lowest-energy mode (�) is ascribedto transitions from the first excited state to the third at thewavevector Qb = 1/4 r.l.u. [the right panel of Fig. 5(d)] andfrom the first excited state to the second at Qb = 0 [the leftpanel of Fig. 5(d)], as shown by solid and dashed curvesin Fig. 5(c), respectively. The lowest-energy mode () isascribed to transitions from the first excited state to the secondand from the second to the third at Qb = 1/4 r.l.u. [the rightpanel of Fig. 5(d)], as shown by two solid curves in Fig. 5(c).A significantly large dependence of the excitation energieson the field direction in the theoretical calculations shown inFig. 5(d) also elucidates the unusually broad spectral featuresidentified in the powder ESR experiments.The current frustrated spin- 12 ladder model, that has repro-duced overall experimental results on Rb2Cu2Mo3O12, actu-ally has a tetramer-singlet ground state formed by emergentS = 1 spins with a Haldane gap. [See Fig. 1(c).] This groundstate is adiabatically connected to the limit of the two decou-pled chains with J ′ = 0, each of which has a singlet Haldanedimer ground state [22], and then to the two decoupled spin-1Haldane chains, as in an antiferromagnetic spin-1 ladder [33].At present, it remains open to explain the ferroelectricity sta-bilized by a tiny magnetic field. Nevertheless, it is clear fromthe symmetry that it is accompanied by a genuine long-rangevector spin chirality order, which is not parasitic to a (quasi-)long-range spiral magnetic order. This ground state has longbeen sought since the proposal by Villain [20]. Thus, thecurrent study uncovers a unique class of magnetically inducedferroelectricity in the absence of a long-range magnetic order,in contrast to many multiferroic magnets due to a cycloidalmagnetism. A quest for additional microscopic propertiesof this ferroelectric (vector-spin-chirality ordered) emergent140408-4EMERGENT SPIN-1 HALDANE GAP AND … PHYSICAL REVIEW B 101, 140408(R) (2020)Haldane-gap state will require experiments on single crystalsand the associated microscopic theoretical analyses.The authors acknowledge I. Terasaki for his support ofthe work and helpful discussion and K. Kaneko for pre-liminary neutron scattering experiments on the triple-axisspectrometer LTAS installed at the JRR-3 reactor, Japan.Numerical calculations were partially performed by using theRIKEN Integrated Cluster of Clusters and the RIKEN HOKU-SAI supercomputers. The time-of-flight neutron-scatteringexperiments were performed using the chopper spectrom-eters AMATERAS and 4SEASONS at J-PARC (Propos-als No. 2012P0202 and No. 2009A0093). 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