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Masaki Gen, Hidemaro Suwa, Shusaku Imajo, Chao Dong, Hiroaki Ueda, [Makoto Tachibana](https://orcid.org/0000-0002-5907-5563), Akihiko Ikeda, Koichi Kindo, Yoshimitsu Kohama

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[Unified Description of Spin-Lattice Coupling and Thermodynamics in the Pyrochlore Heisenberg Antiferromagnet](https://mdr.nims.go.jp/datasets/02d882e9-54f4-44ca-8e4e-6b53e204e3e5)

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Unified Description of Spin-Lattice Coupling and Thermodynamics in the Pyrochlore Heisenberg AntiferromagnetUnified Description of Spin-Lattice Coupling and Thermodynamics in the Pyrochlore HeisenbergAntiferromagnetMasaki Gen,1, ∗ Hidemaro Suwa,2, † Shusaku Imajo,1 Chao Dong,1 Hiroaki Ueda,3Makoto Tachibana,4 Akihiko Ikeda,5 Koichi Kindo,1 and Yoshimitsu Kohama11Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan2Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan3Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044, Japan5Department of Engineering Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan(Dated: March 10, 2026)We study an extended model to describe the spin-lattice coupling, incorporating individual vibrations ofbonds and atomic sites alongside distance-dependent exchange interactions. The proposed spin Hamiltoniancan be effectively considered as an interpolation between two well-established minimum models, the bond-phonon model and the site-phonon model. The extended model, which treats bond phonons and site phonons oncomparable footing, well reproduces successive field-induced phase transitions as well as the thermodynamicproperties of a three-up–one-down state in the pyrochlore-lattice Heisenberg antiferromagnet, including negativethermal expansion, an enhanced magnetocaloric effect, and a sharp specific-heat peak. The present approachis broadly applicable to various spin models, providing a framework for identifying the primary phonon modesresponsible for spin-lattice coupling and for understanding complex magnetic phase diagrams.Introduction—The interplay between spin and lattice de-grees of freedom, namely the spin-lattice coupling (SLC), isubiquitous in magnetic compounds due to the variation of ex-change parameters against the relevant atomic displacements.A direct manifestation of the SLC effect is the exchangestriction, as represented by the observation of linear magne-tostriction proportional to the short-range spin-spin correla-tion 〈Si · S j〉 on a one-dimensional spin chain [1]. Further-more, the SLC can be a source of structural instability evenwith no orbital degrees of freedom, potentially leading to aspin-Peierls(-like) transition characterized by multimer for-mation [2–4] and/or a spin Jahn-Teller transition accompaniedby global lattice deformation [5–7].Two minimal models have been proposed to describe theSLC microscopically: the bond-phonon (BP) model [8] andthe site-phonon (SP) model [9]. The former assumes indepen-dent vibration of each bond, whereas the latter assumes inde-pendent displacement of each atomic site (Fig. 1). The BPmodel has been demonstrated as a powerful approach to re-produce a metamagnetic transition to a magnetization plateauand the associated magnetostriction in frustrated chromiumspinels [8, 10–16]. Nevertheless, it oversimplifies the phonon-mediated spin interactions because no further-neighbor inter-actions beyond the nearest-neighbor ones are considered. Incontrast, the SP model neglects the global lattice deforma-tion but effectively incorporates phonon-mediated three-bodyspin interactions, enabling the reproduction of magnetic long-range orders (LROs) [9, 17–21], complicated phase transitions[22–25], and molecular spin excitations [26–29]. Recently,the SP model has been applied to a variety of crystallographicsystems, such as the triangular lattice [25], kagome lattice[25, 30], breathing pyrochlore lattice [20–23], and honeycomb∗ gen@issp.u-tokyo.ac.jp† suwamaro@phys.s.u-tokyo.ac.jplattice [31], sparking theoretical predictions for the emergenceof diverse magnetic phases.As described above, both the BP and SP models have theirown advantages and limitations, thus complementing eachother. Researchers have so far chosen one of these mod-els arbitrarily to explain experimental data, highlighting theneed for a more comprehensive description of the SLC fora robust discussion. As exemplified by the first-principlescalculations on the honeycomb-lattice system [31], the pre-dominant optical phonon modes in real compounds shouldbe renormalized to either BP or SP terms. Here, we presentan extended SLC model by introducing a phenomenologi-cal parameter that characterizes a ratio of the SP to the BPcontribution, and investigate the magnetic field and tempera-ture dependence of various thermodynamic quantities of thepyrochlore-lattice Heisenberg antiferromagnet. For compar-ison, we perform magnetization, magnetostriction, magne-tocaloric effect (MCE), and specific heat measurements on amodel compound CdCr2O4 in pulsed high magnetic fields. WeFIG. 1. Two minimal models describing the SLC: (a) the bond-phonon model assuming independent bond-length change ρi j, and(b) the site-phonon model assuming independent site displacementui. The BP model does not effectively produce any further-neighborinteractions, while the SP model does not take into account magne-tostriction.arXiv:2508.13535v2  [cond-mat.str-el]  8 Mar 2026mailto:gen@issp.u-tokyo.ac.jpmailto:suwamaro@phys.s.u-tokyo.ac.jphttps://arxiv.org/abs/2508.13535v22show that many important magnetic and thermodynamic prop-erties can be simultaneously reproduced by a single parameterset, ensuring the validity of the extended SLC model.Theoretical model—First, we consider an effective spinHamiltonian that incorporates both the BP and SP contribu-tions. While we specifically focus on the classical Heisenbergmodel on a pyrochlore lattice in this Letter, the present for-malization can be applied to any lattice system. Let ρi j bethe BP displacement between the adjacent sites i and j, andlet ui be the SP displacement vector for each site i (Fig. 1).We set antiferromagnetic J ≡ J(|r0i j|) > 0 as the bare nearest-neighbor exchange coupling, where r0i j≡ r0i− r0jis an equi-librium bond-length without the SLC. Bearing in mind thatthe direct exchange interaction dominates in CdCr2O4, we as-sume that J depends on the distance between the sites i andj. Using two types of phonon modes, the Hamiltonian of thespin-lattice system is given byH =∑〈i j〉(J(|ri − r j|) Si · S j +kBP2ρ2i j)+∑ikSP2|ui|2− h∑iS zi,(1)J(|ri − r j|) = J(|r0i j + ρi jei j + ui − u j|) (2)≈ J − Jγ[ρi j + ei j · (ui − u j)], (3)where 〈i j〉 runs over all the nearest-neighbor bonds, Si is athree-dimensional vector spin with unit length at site i, kBP >0 (kSP > 0) is the spring constant for bond (site) phonons, h isthe magnetic field, ei j ≡ r0i j/|r0i j|, and γ ≡ −(1/J)(dJ/dr)|r=|r0i j|.For the SP coupling, we take into account a displacementcomponent projected to the bond direction ei j as the lowest-order contribution to J. Given that the lattice displacementis small enough compared to the lattice constant, we approx-imate the dependence of J up to the first order of the dis-placement in Eq. (3). As the lattice degrees of freedom arequadratic in the Hamiltonian, Eq. (1) is rewritten asH = J∑〈i j〉Si · S j +kBP2∑〈i j〉((ρi j − ρi j)2 − ρ2i j)+kSP2∑i((ui − ui)2 − u2i)− h∑iS zi,(4)where ρi j = (Jγ/kBP)Si·S j and ui = (Jγ/kSP)∑j∈N(i) ei j(Si·S j),with N(i) denoting the set of neighboring sites of i. By tracingout the BPs and SPs from the Boltzmann distribution throughthe Gaussian integral, Eq. (4) is exactly reduced to an effectivespin HamiltonianHeff = J∑〈i j〉[Si · S j − b(Si · S j)2]−Jb′2∑i∑j,k∈N(i)ei j · eik(Si · S j)(Si · Sk) − h∑iS zi,(5)where b = Jγ2 (1/2kBP + 1/kSP) and b′ = Jγ2/kSP, which arethe primary control parameters in our model.Hereafter, we define the ratio of the two SLC parametersas η = b′/b (0 ≤ η ≤ 1); η = 0 corresponds to the pure BPmodel, while η = 1 corresponds to the pure SP model. Al-though the above parametrization of the SLC was proposedin Ref. [9], the magnetic phase diagram and thermodynamicproperties of the extended SLC model [Eq. (5)] have yet tobe investigated. To study the thermodynamic properties ofEq. (5), we performed Monte Carlo (MC) simulations for asystem size of N = 16 × L3 sites with L = 4 under periodicboundary conditions. For simplicity, the local spin length wasfixed to unity, |Si| = 1. Further details of the simulation meth-ods are provided in the Supplemental Material [32].Magnetic properties of CdCr2O4—Chromium spinel ox-ides ACr2O4 (A = Mg, Zn, Cd, Hg), where Cr3+ ions withS = 3/2 form a pyrochlore lattice, have been the focus ofmuch attention for studying the effect of SLC in geometricallyfrustrated magnets [11–13, 17–21, 26–29, 34–48]. As the or-bital degree of freedom is quenched due to the 3d3 electronicconfiguration under the octahedral crystal field, the SLC actsas a major perturbation to resolve the magnetic frustration,leading to magnetostructural transitions at low temperatures[17, 27, 34, 35] and in high magnetic fields [11–13, 36–43].CdCr2O4 undergoes a magnetic transition at TN ≈ 7.6 Kin zero field [44], accompanied by a cubic-to-tetragonal struc-tural transition [34]. Figure 2(a) shows a magnetization pro-cess of CdCr2O4 at the initial temperature of Tini = 4.2 K,where a 1/2-magnetization plateau appears between µ0Hc1 =28 T and µ0Hc2 = 58 T, accompanied by a sharp metam-agnetic transition. A high-field neutron-diffraction study re-vealed the emergence of a three-up–one-down LRO aboveHc1 [18], which should be stabilized by the SLC favoringa collinear spin configuration via the biquadratic interaction−(Si · S j)2. With further increasing of a magnetic field aboveHc2, a double-peak anomaly appears in dM/dH around 80–88T, as indicated by asterisks in Fig. 2(a), and then spins are fullypolarized at µ0Hsat ≈ 90 T. The presence of an additional high-field (HF) phase immediately below saturation was also indi-cated by a previous magneto-optical spectroscopy of the d–dtransitions and the exciton-magnon-phonon transitions [39].The above-mentioned successive phase transitions are univer-sally observed for other ACr2O4 families [36–40].We consider CdCr2O4 as a suitable compound for verify-ing the extended SLC model because of the relatively lowsaturation field and the availability of high-quality singlecrystals. To comprehend the thermodynamic properties ofCdCr2O4, we measured magnetostriction, MCE, and specificheat in pulsed high magnetic fields by utilizing recently devel-oped experimental techniques [49–54] at ISSP, University ofTokyo. Details of the experimental methods are presented inthe Supplemental Material [32].Magnetic phase diagram of the extended SLC model—Figures 2(a)–2(d) compare the experimental magnetizationand magnetostriction curves of CdCr2O4 with simulations,where we use different η values with b = 0.2 (for results withother values of b, see Fig. S3 [32]). The corresponding the-oretical phase diagram as a function of magnetic field and ηis shown in Fig. 2(e). Throughout the Letter, ∆L/L denotesthe relative change in sample length, referred to as magne-3(a)(b)Calc. (b = 0.2, T/J = 0.01)(c)Exp. (CdCr2O4)Tini = 4.2 KHc1Hc2(d)(e)SitephononBondphononhc1hc2hc2hc1canted2:2canted3:13-up1-downFully-polarizedHFTini = 4.2 KFIG. 2. (a),(b) Experimental (a) magnetization and (b) magnetostric-tion curves of polycrystalline CdCr2O4 at Tini = 4.2 K obtained usingthe single-turn-coil technique, where the sample temperature is ex-pected to be under (quasi)adiabatic conditions. In panel (a), dM/dHis displayed by thin colors in the right axis. In panel (b), the dataup to 50 T (black) are obtained in a nondestructive pulsed magnet.(c),(d) Calculated (c) magnetization and (d) magnetostriction curvesfor the extended SLC model [Eq. (5)] with b = 0.2 and various valuesof η at T/J = 0.01. In (a) and (c), the field derivative of the magneti-zation (for η = 0.6) is plotted in the right axis. (e) Theoretical phasediagram of Eq. (5) with b = 0.2 as a function of magnetic field h andthe SP contribution η at T/J = 0.01.tostriction (thermal expansion) when plotted as a function offield (temperature). In the theoretical model, the saturationfield is h/J = 8 without the SLC. For both the BP and SPmodels, a first-order transition from a canted 2:2 to a three-up–one-down state occurs at a lower critical field of the 1/2-magnetization plateau (hc1), while a second-order transitionfrom the three-up–one-down to a canted 3:1 state occurs at anupper critical field (hc2). These trends agree with the experi-mental magnetization curve [Fig. 2(a)] [55]. Furthermore, thetheoretically predicted positive magnetostriction (except forthe case of η = 1), where the lattice expansion is rather en-hanced above hc2, also agrees with the experimental observa-tion [Fig. 2(b)] (see also Fig. S3 for the magnetostriction curveof HgCr2O4 [32]). In the BP model (η = 0), the plateau widthincreases with increasing b [8]. However, the plateau widthdecreases with increasing η; for b = 0.2, (hc2−hc1)/8J ≈ 0.31for η = 0 and 0.17 for η = 1 [Fig. 2(c)].Neither the conventional BP nor SP models can accountfor the double-peak anomaly in dM/dH observed in CdCr2O4[Fig. 2(a)], and the emergence of a spin-nematic phase drivenby quantum effects has been proposed [38, 41, 56]. Remark-ably, this high-field feature in CdCr2O4 can be explained byour extended SLC model, even under the assumption of clas-sical spins. As shown in Fig. 2(e), an additional HF phaseemerges immediately above the canted 3:1 phase for 0 < η <0.7. The field derivative of magnetization for η = 0.6 [bluecurve in Fig. 2(c)] successfully reproduces the double-peakstructure in CdCr2O4. Note that, in our calculations, the two-step magnetization jump is rapidly smeared out and eventuallydisappears with increasing temperature (see Fig. S4 [32]). Inthe HF phase between the two dM/dH peaks, we identify theemergence of a magnetic LRO state characterized by a peri-odic stacking along the 〈111〉 axis, consisting of a triangularlayer with a 120◦ spin configuration, an all-up kagome layer,all-up triangular layer, and another all-up kagome layer (seeFig. S6 for the detailed magnetic structure and lattice distor-tion [32]). The existence of a phase transition into the HFphase is further supported by temperature-dependent simu-lations at fixed fields in the range h/J = 6.6–6.75, wherethe transition temperature is found to be TN/J ≈ 0.01 ath/J = 6.75 (see Fig. S5 [32]).Thermodynamic properties in the 1/2-plateau phase—Figures 3(a)–3(c) show the calculated temperature depen-dence of magnetization, thermal expansion, and specific heatfor various η values with b = 0.2 at h/J = 3.7. Figure 3(d)shows the corresponding phase diagram as a function of tem-perature and η. In the BP limit (η = 0), the specific heatexhibits a broad peak at T ∗/J ≈ 0.26, indicating a crossoverfrom the paramagnetic state to the spin-liquid plateau state, asreported in Refs. [10, 21]. Below the crossover temperatureT ∗, each tetrahedron adopts a three-up–one-down configura-tion with local T2 symmetry, while macroscopic degeneracypersists across the entire system. As η increases, T ∗ graduallydecreases, and another sharp specific-heat peak emerges at alower temperature Tp for η ≈ 0.3, signaling a first-order tran-sition to a three-up–one-down LRO state. The transition tem-perature Tp increases with increasing η and eventually con-verges with T ∗ for η ≈ 0.6. Then, Tp gradually decreases asη further increases, resulting in Tp/J ≈ 0.14 in the SP limit(η = 1), consistent with Ref. [21].We now discuss the behavior of the specific heat in appliedmagnetic fields. Figure 4(a) shows the specific heat data ofCdCr2O4 at 24 T and 34 T for H ‖ [111] (see Fig. S7 for thedata obtained from polycrystalline samples [32]). At 24 T(< Hc1), the transition temperature shifts to a lower valuecompared to TN = 7.6 K at zero field. Remarkably, a morepronounced peak is observed at Tp = 9.7 K at 34 T (> Hc1), in-dicating the appearance of three-up–one-down LRO below Tp.A similar behavior has also been reported for polycrystallineHgCr2O4 [43]. In the SP model, the three-body quadraticterms of the form−(Si·S j)(Si·Sk) in Eq. (5) effectively producesecond- and third-nearest-neighbor antiferromagnetic interac-tions, Jeff2and Jeff3[Fig. 1(b)]. Under the three-up–one-downconstraint, Jeff3dominates over Jeff2, as Jeff3= 2Jeff2, leading toa three-up–one-down LRO with cubic P4332 symmetry ratherthan rhombohedral R3m symmetry [9]. This is consistent withexperimental observations in HgCr2O4 and CdCr2O4 based onhigh-field neutron diffraction [17, 18]. As mentioned above,4(a)Calc. (b = 0.2, h /J = 3.7)(b)(c)(d)Para-magneticSpin-liquidplateau3-up-1-downLROTpFIG. 3. [(a)–(c)] Temperature dependence of (a) magnetization, (b)thermal expansion, and (c) specific heat for the extended SLC model[Eq. (5)] with b = 0.2 and various η values in the low-field side ofthe 1/2-magnetization plateau at h/J = 3.7. (d) Theoretical phasediagram of Eq. (5) with b = 0.2 as a function of temperature T andthe site-phonon contribution η at h/J = 3.7. A phase transition toa three-up–one-down LRO state (crossover to a spin-liquid plateaustate) is characterized by a sharp (broad) specific-heat peak, which isindicated by red circles (open blue diamonds).our calculations reveal that, in the extended SLC model, ηmust be at least 0.3 to induce the three-up–one-down LRO.Moreover, our specific heat data at 34 T does not exhibit ahump structure above Tp, which is consistent with the calcu-lated one for η ' 0.6 [Fig. 3(c)].The transition temperature to the 1/2-plateau phase at 34 T,Tp, is higher than TN at zero field, suggesting the stabiliza-tion of collinear spin configuration by thermal fluctuations[13]. This trend cannot be reproduced by the pure SP model(η = 1), as shown in Fig. 6 in Ref. [21]. This discrepancycan be resolved by considering the extended model that in-corporates the additional BP contributions. Figure 4(b) showsthe calculated specific heat at several h values for Eq. (5) withb = 0.2 and η = 0.6. The transition temperature to the three-up–one-down LRO phase at h/J = 4.0 is higher than that tothe (canted) 2:2 LRO phase at h/J = 0 and 2.2. The corre-sponding specific-heat peak at h/J = 4.0 is also the sharpest,supporting the validity of the choice η = 0.6 for reproducingthe experimentally observed specific-heat behavior.Finally, we turn to the negative thermal expansion (NTE)that emerges in the 1/2-plateau phase. A previous thermal ex-pansion measurement in CdCr2O4 reported the NTE on thelow-field side of the 1/2-plateau phase at 30 T, accompaniedby a discontinuous jump in thermal expansion at the transi-tion to the paramagnetic phase [13]. These behaviors are wellreproduced by our extended SLC model for intermediate ηvalues with b = 0.2 at h/J = 3.7 (close to hc1) [Fig. 3(b)].In Ref. [13], the NTE is attributed to a negative change inmagnetization with temperature, i.e., ∂m/∂T < 0, arisingfrom a nearly localized band of spin excitations in the 1/2-plateau phase. Our MC simulations indicate that the temper-E��� �����2O4) E��� �����2O4)A��� ��� ��E(c)(a)(b)Hc1Hc2HtopH   
�1�124 T0 T34 T��C��H   
�1�1FIG. 4. (a) Temperature dependence of specific heat at 24 T and 34 Tfor H ‖ [111] in CdCr2O4, obtained using a flat-top nondestructivelong pulsed magnet [51–54]. (b) Temperature dependence of specificheat at several h values for the extended SLC model [Eq. (5)] withb = 0.2 and η = 0.6. (c) Magnetocaloric effect (MCE) for H ‖ [111]in CdCr2O4, measured under adiabatic conditions in a nondestruc-tive pulsed magnet. The inset shows a magnified view of the T (H)curve around Hc2 measured using a different setup. All the data cor-respond to the field-increasing process. The thick gray lines indicatethe phase boundaries separating the canted 2:2, three-up–one-down,and paramagnetic phases.ature range exhibiting NTE approximately corresponds to theregion where either the three-up–one-down LRO phase or thespin-liquid plateau phase emerges. We also theoretically pre-dict that, on the high-field side of the 1/2-plateau phase, thesystem exhibits positive thermal expansion down to the low-est temperature due to the spin-gap closing on approachinghc2 (see Fig. S8 [32]), although this behavior has not yet beenobserved experimentally.From a thermodynamic point of view, the NTE behavior isclosely related to an enhanced MCE, as proposed in Ref. [13].According to thermodynamic relations, the field derivative ofthe sample temperature T (H) under adiabatic conditions is ex-pressed as(∂T∂H)S= −TCp(∂M∂T)H. (6)Figure 4(c) shows the adiabatic MCE data of CdCr2O4 duringthe field-up sweep for H ‖ [111], measured at various initialtemperatures. Below TN, the T (H) curve exhibits a dip at Hc1,indicating the increase in magnetic entropy. Upon enteringthe 1/2-plateau phase, the T (H) curves develop a domelikestructure with a maximum near µ0Htop ≈ 50 T, where T (H)increases by up to 5 K. The upper phase boundary at Hc2 isalso visible as a kink in the T (H) curve [inset of Fig. 4(c)],beyond which sample cooling persists in the higher field re-gion, at least up to 65 T. These MCE data suggest that thesign change in ∂M/∂T from negative to positive occurs ata magnetic field higher than the midpoint of the 1/2-plateauphase, i.e., µ0Htop > µ0(Hc1 + Hc2)/2 ≈ 43 T [57]. This5trend is consistent with the experimental magnetization curves[36], and is also reproduced by our extended SLC model (seeFig. S4 in the SM [32]). The comparison with these MCE re-sults demonstrates that our extended SLC model consistentlycaptures all the thermodynamic properties in the 1/2-plateauphase of CdCr2O4.Conclusion—We have validated the extended SLC modelfor the pyrochlore-lattice Heisenberg antiferromagnet bydemonstrating its consistency with the thermodynamic prop-erties of CdCr2O4. The extended SLC model interpolates be-tween the BP and SP models, characterized by a phenomeno-logical parameter η, which describes the ratio of SP modes.Our MC simulations reveal that introducing both the BP andSP modes on comparable footing, e.g., η = 0.6, providesthe best agreement with experimental observations, includingnegative thermal expansion, an enhanced MCE, and a sharpspecific heat peak in the 1/2-plateau phase, as well as a two-step phase transition just below the saturation field. This the-oretical framework offers a practical approach for testing theprimary phonon modes responsible for SLC without the needfor sophisticated techniques such as first-principles calcula-tions. Furthermore, applying the extended SLC model to otherlattice systems with strong SLC may provide new insights intothe complex phase diagrams [58].Acknowledgments—Calculations were performed usingcomputational resources from the Supercomputer Center atthe Institute for Solid State Physics, the University of Tokyo.This work was financially supported by the JSPS KAK-ENHI Grants-In-Aid for Scientific Research (No. 20J10988,No. 24H01609, and No. 24H01633). The authors thank A.Zampa for his kind support with the specific heat measure-ments, and K. Penc, N. Shannon, and A. Samanta for fruitfuldiscussions.[1] V. S. Zapf, V. F. Correa, P. Sengupta, C. D. Batista, M.Tsukamoto, N. Kawashima, P. Egan, C. Pantea, A. Migliori,J. B. Betts, M. Jaime, and A. Paduan-Filho, Direct measure-ment of spin correlations using magnetostriction, Phys. Rev. B77, 020404(R) (2008).[2] J. W. Bray, H. R. Hart, Jr., L. V. Interrante, I. S. Jacobs, J. S.Kasper, G. D. Watkins, and S. H. 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B 112, 214415 (2025).Supplemental Material for“Unified Description of Spin-Lattice Coupling and Thermodynamics in the Pyrochlore HeisenbergAntiferromagnet”Masaki Gen,1, ∗ Hidemaro Suwa,2, † Shusaku Imajo,1 Chao Dong,1 Hiroaki Ueda,3Makoto Tachibana,4 Akihiko Ikeda,5 Koichi Kindo,1 and Yoshimitsu Kohama11Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan2Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan3Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan4Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044, Japan5Department of Engineering Science, University of Electro-Communications, Chofu, Tokyo 182-8585, JapanThis Supplemental Material includes contents as listed below:Note 1. Monte Carlo simulationsNote 2. Experimental methodsNote 3. Choice of the SLC parameter bNote 4. System-size dependence of calculated magnetization curvesNote 5. Magnetostriction curve of HgCr2O4 up to 50 TNote 6. Temperature dependence of calculated magnetization and magnetostriction curvesNote 7. Magnetic structure of the HF phaseNote 8. Specific heat data for polycrystalline CdCr2O4 in high magnetic fieldsNote 9. Thermodynamic properties on the high-field side of the 1/2-plateau phase∗ gen@issp.u-tokyo.ac.jp † suwamaro@phys.s.u-tokyo.ac.jpmailto:gen@issp.u-tokyo.ac.jpmailto:suwamaro@phys.s.u-tokyo.ac.jp2Note 1. Monte Carlo simulationsWe performed Monte Carlo (MC) simulations for the extended spin-lattice coupling (SLC) model,Heff = J∑〈i j〉[Si · S j − b(Si · S j)2]−Jb′2∑i∑j,k∈N(i)ei j · eik(Si · S j)(Si · Sk) − h∑iS zi, (S1)which is equivalent to Eq. (5) in the main text, in order to compute several physical quantities: magnetization m, magnetostric-tion/thermal expansion ∆L/L, and specific heat C. Each physical quantity is calculated bym =1N∑i〈Si · h〉, (S2)∆LL∝1Nb∑〈i j〉〈ρi j〉 =JγNbkBP∑〈i j〉〈Si · S j〉, (S3)C =〈H2eff〉 − 〈Heff〉2NT 2, (S4)where h is the magnetic field, and N and Nb are the number of sites and bonds, respectively. In Eq. (S3), ∆L/L is proportionalto the average displacement of bond phonons ρi j, as verified in several spinel compounds [1]. The absolute values of the latticedisplacements, ρi j and ui, are arbitrary in our model because their coefficient can be absorbed by the renormalization of theparameters, γ, kBP, and kSP, keeping the ratios γ2/kBP and γ2/kSP.Following Ref. [2], we retained the lattice degrees of freedom in the simulation and introduced microcanonical updates toaccelerate the thermalization process and shorten the autocorrelation time. The random or the over-relaxation-like update stepalternately follows five microcanonical update MC steps. The single MC step is composed of N local spin and site-phononupdates and Nb local bond-phonon updates. More than 222 MC steps were run for each parameter set, and the latter half wasused to calculate the averages of the physical quantities. We also combined the replica-exchange MC method [3], or paralleltempering, to avoid trapping in local minima at low temperatures. The error bars of the Monte Carlo simulations are smallerthan the symbol sizes in all plots, as estimated by standard binning (blocking) analysis.Note 2. Experimental methodsSingle crystals of CdCr2O4 were synthesized by a flux method using PbO as the flux [4]. Polycrystalline powder samples ofCdCr2O4 were synthesized by conventional solid-state reactions [5]. Polycrystalline powder samples of HgCr2O4 were preparedby a thermal decomposition of Hg2CrO4 in an evacuated silica tube [5].The longitudinal magnetostriction up to 50 T and 100 T was measured by the optical fiber-Bragg-grating (FBG) method inthe nondestructive pulsed magnet (36 ms duration) and the vertical single-turn-coil (STC) system (8 µs duration), respectively.A relative sample-length change ∆L/L was detected by the optical filter method [6]. The magnetocaloric effect (MCE) up to59 T and 65 T was measured under quasiadiabatic conditions for H ‖ [111] using the nondestructive pulsed magnet with thefield duration of 36 ms and 11 ms, respectively. Here, the quasiadiabatic conditions were achieved by removing helium gas inthe sample space. A sensitive Au16Ge84 film thermometer was sputtered on the (111) plane of the CdCr2O4 single crystal [7].The specific heat Cp of CdCr2O4 up to 34 T was measured using the quasiadiabatic method under a flat-top long pulsed field[8, 9]. All of the experiments were performed at the Institute for Solid State Physics, University of Tokyo, Japan.3Note 3. Choice of the SLC parameter bWe selected b = 0.2 as a typical value of the spin-lattice coupling (SLC) parameter in this study because it provides the bestoverall agreement with the key experimental observations in CdCr2O4 and HgCr2O4. Bergman et al. reported calculations atb = 0.1 for η = 0–0.5 [10]. However, the relative field range of the 1/2-magnetization plateau for b = 0.1 is rather narrower thanobserved in CdCr2O4 and HgCr2O4. Conversely, increasing the SLC parameter to b = 0.25 improves the width of the 1/2-plateaubut overenhances the magnetization anomaly just below the saturation field, as shown in Fig. S1. Balancing these two aspects,we conclude that b = 0.2 yields the most faithful overall reproduction of the magnetization curves for CdCr2O4 and HgCr2O4.b = 0.25, T/J = 0.01FIG. S1. Calculated magnetization curves for the extended SLC model [Eq. (S1)] with b = 0.25 and various values of η at T/J = 0.01.Note 4. System-size dependence of calculated magnetization curvesIn this study, we mainly performed MC simulations for a system size of N = 16×L3 with L = 4. We confirmed that the resultsremain essentially unchanged for L = 6. For example, Fig. S2 compares the calculated magnetization curves with b = 0.2 andη = 0.6 at T/J = 0.01 for L = 4 and L = 6, showing that both the width of the 1/2-magnetization plateau and the presence of theHF phase are reproduced in both cases.FIG. S2. Calculated magnetization curves for the extended SLC model [Eq. (S1)] with b = 0.2, η = 0.6 at T/J = 0.01 for L = 4 (red) andL = 6 (blue).4Note 5. Magnetostriction curve of HgCr2O4 up to 50 TFigure S3 shows the longitudinal magnetostriction curves of polycrystalline HgCr2O4 measured below and above TN = 5.8 K.At T = 1.4 and 4.2 K, a downward convex behavior is observed below µ0Hc1 = 10 T, where a metamagnetic transition to a1/2-plateau phase takes place [5]. Subsequently, a dramatic increase in ∆L/L is observed at each phase transition field up to thesaturation field, µ0Hsat ≈ 46 T, namely at µ0Hc2 = 27 T, µ0Hc3 = 36 T. Notably, the total change in ∆L/L between Hc2 and Hsatis approximately twice as large as that below Hc2. This enhanced magnetostriction above Hc2 is consistent with our theoreticalcalculation, as shown in Fig. 2(d) of the main text.FIG. S3. Magnetostriction curves of polycrystalline HgCr2O4 at several temperatures obtained using the nondestructive pulsed magnet.Note 6. Temperature dependence of calculated magnetization and magnetostriction curvesFigure S4 shows the calculated magnetization and magnetostriction curves with b = 0.2 and η = 0.6 at various temperatures.The double-peak structure in dM/dH just below saturation is relatively broadened for CdCr2O4 (TN = 7.8 K) at Tini = 4.2 K[Fig. 2(a)], compared to the calculated result at T/J = 0.01 [Fig. 2(c)]. This difference can be attributed to thermal fluctuations,given that in our calculations the two-step metamagnetic anomaly already disappears upon increasing the temperature aboveT/J = 0.05. In addition, Fig. S4(a) shows that the sign of ∂M/∂T changes from negative to positive around h/J = 4.5, which ishigher than the midpoint of the 1/2-plateau phase, h/J ≈ 4.1. This behavior is consistent with the trend of the distorted dome-likestructure of the MCE curves (see Fig. 4(a) and the related discussion in the main text for details).(a) (b)FIG. S4. Calculated (a) magnetization and (b) magnetostriction curves for the extended SLC model [Eq. (S1)] with b = 0.2 and η = 0.6 atvarious temperatures.5Note 7. Magnetic structure of the HF phaseHere, we focus on the HF phase, which is absent in the pure bond-phonon or site-phonon models, but appears in the extendedSLC model [Eq. (S1)], as shown in the phase diagram in Fig. 2(e) of the main text. The HF phase emerges between the canted3:1 phase and the fully-polarized phase. Figure S5 shows the calculated temperature dependence of magnetization and specificheat at high magnetic fields in the range h/J = 6.60–6.75 with b = 0.2 and η = 0.6. A clear signature of a magnetic transitionfrom the paramagnetic phase to a magnetically ordered phase is indicated by a sharp peak in the specific heat. The transitiontemperature gradually decreases with increasing magnetic field, reaching TN/J ≈ 0.01 at h = 6.75.Figures S6(a) and S6(c) show MC snapshots of the spin configurations and lattice distortions, respectively, in the HF phase.Figures S6(b) and S6(d) present the corresponding layer-resolved plots for four consecutive layers: two triangular layers (T1 andT2) and two kagome (K1 and K2) layers alternately stacked along the 〈111〉 direction. These results reveal the emergence of amagnetic LRO state characterized by a periodic stacking sequence of a triangular layer with a 120◦ spin configuration (T1), an all-up kagome layer (K1), an all-up triangular layer (T2), and another all-up kagome layer (K2). Here, the out-of-plane componentof the spins in the T1 layer takes a negative value, oriented opposite to the magnetic field direction. Interestingly, in the T1layer, the in-plane components of the spins form a 120◦ configuration, exhibiting an antiferrochiral order in which neighboringT1 layer possess opposite vector spin chirality. On the site-phonon side, breathing-type in-plane lattice displacements appear inthe kagome layers, while both adjacent kagome layers are displaced toward the T1 layer along the out-of-plane direction.FIG. S5. Temperature dependence of (a) magnetization and (b) specific heat for the extended SLC model [Eq. (S1)] with b = 0.2 and η = 0.6at various magnetic fields in the HF phase.6(a)(c)(b)(d)T1T2K1K2T1T2K1K2T1K1K2T2T1K1K2T2FIG. S6. MC snapshots of the [(a)(b)] spin configurations and [(c)(d)] lattice-distortion patterns obtained at h/J = 6.65 and T/J = 0.005 forthe extended SLC model [Eq. (S1)] with b = 0.2 and η = 0.6. For clarity, the magnetic field is applied along the 〈111〉 direction instead ofthe z axis. Panels (b) and (d) show four consecutive triangular (T1 and T2) and kagome (K1 and K2) layers perpendicular to the 〈111〉 axis,as indicated in panels (a) and (c). In panels (a) [(c)], the arrow directions indicate the spins (lattice displacements) in the pyrochlore lattice;in panels (b) [(d)], the arrow directions indicate the in-plane components, and the arrow colors represent the out-of-plane components, S · n̂(u · n̂), where n̂ is the unit vector in the 〈111〉 axis. In panel (b) [(d)], the out-of-plane component of the vector spin chirality, κ · n̂, (the scalarchirality of lattice displacements, χsca) is visualized by color for each triangle. The layer-resolved averages of the spin and lattice-displacementcomponents parallel to the field direction are also displayed as m̄ and ū, respectively.7Note 8. Specific heat data for polycrystalline CdCr2O4 in high magnetic fieldsFigure S7 shows the specific heat data measured on polycrystalline sintered samples of CdCr2O4 at 16 T and 34 T, obtainedusing a flat-top long pulsed magnet [8, 9]. Compared to the single-crystal data shown in Fig. 4(c) of the main text, the specificheat peak associated with the phase transition is notably broadened, particularly at lower fields. However, the peak correspondingto the transition to the 3-up–1-down LRO phase at 34 T is remarkably sharp, as observed on HgCr2O4 [11]. !!"!! !#$%&' ()*' + !"!"#$&+,-,./01#$2)*3+####!#4# 5#4#61#4#FIG. S7. Temperature dependence of specific heat at 16 T and 34 T for polycrystalline CdCr2O4.Note 9. Thermodynamic properties on the high-field side of the 1/2-plateau phaseFigure S8 shows the calculated temperature dependence of magnetization, thermal expansion, and specific heat for the ex-tended SLC model [Eq. (S1)] with b = 0.2 and various η values in the high-field side of the 1/2-magnetization plateau ath/J = 4.7. In contrast to the low-field side of the 1/2-plateau, the thermal expansion exhibits a positive temperature dependence.(a) (b) (c)FIG. S8. Temperature dependence of (a) magnetization, (b) thermal expansion, and (c) specific heat for the extended SLC model [Eq. (S1)]with b = 0.2 and various η values at h/J = 4.7.8[1] A. Miyata, H. Suwa, T. Nomura, L. Prodan, V. Felea, Y. Skourski, J. Deisenhofer, H.-A. Krug von Nidda, O. Portugall, S. Zherlitsyn, V.Tsurkan, J. Wosnitza, and A. Loidl, Spin-lattice coupling in a ferrimagnetic spinel: Exotic H-T phase diagram of MnCr2S4 up to 110 T,Phys. Rev. B 101, 054432 (2020).[2] M. Gen and H. Suwa, Nematicity and fractional magnetization plateaus induced by spin-lattice coupling in the classical kagome-latticeHeisenberg antiferromagnet, Phys. Rev. B 105, 174424 (2022).[3] K. Hukushima and K. 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