Crystal-field magnetostriction of the spin ice under ultrahigh magnetic fields
Nan Tang Masaki Gen Martin Rotter Huiyuan Man Kazuyuki Matsuhira Akira Matsuo Koichi Kindo Akihiko Ikeda Yasuhiro H. Matsuda Philipp Gegenwart Satoru Nakatsuji Yoshimitsu Kohama




nan.tang@uni-a.de
Institute for Physics, University of Augsburg, Augsburg 86159, Germany


gen@issp.u-tokyo.ac.jp
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

*These authors contributed equally to this work.


McPhase Project, Dresden 01159, Germany


Geballe Laboratory for Advanced Materials, Stanford University, CA 94305, USA


Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu, Fukuoka 804-8550, Japan


Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan


Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan


Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Department of Engineering Science, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan


Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan


Institute for Physics, University of Augsburg, Augsburg 86159, Germany


Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
Institute for Quantum Matter and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
Trans-scale Quantum Science Institute, University of Tokyo, Tokyo  113-0033, Japan


Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan


§ ABSTRACT 
We present a comprehensive study of the magnetoelastic properties of the Ising pyrochlore oxide , known as spin ice, by means of high-field magnetostriction measurements and numerical calculations.
When a magnetic field is applied along the crystallographic ⟨ 111 ⟩ axis, the longitudinal magnetostriction exhibits a broad maximum in the low-field regime around 30 T, followed by a dramatic lattice contraction due to crystal-field (CF) level crossing at B_ cf∼ 65 T.
The transverse magnetostriction exhibits a contrasting behavior, highlighting the anisotropic nature of the CF striction.
We identify distinct timescales of spin dynamics and CF-phonon dynamics by applying a magnetic field with different field-sweep rates. 
Our mean-field calculations, based on a point-charge model, successfully reproduce the overall magnetostriction behavior, revealing the competition between the exchange striction and CF striction.
A signature of the CF level crossing is also observed through adiabatic magnetocaloric-effect measurements, consistent with our magnetostriction data.



Crystal-field magnetostriction of the spin ice under ultrahigh magnetic fields
    Yoshimitsu Kohama
    September 9, 2024
==============================================================================





§ INTRODUCTION

Magnetostriction refers to the distortion of a lattice when a material is subjected to an external magnetic field.
Certain magnetostrictive materials exhibit relative length change Δ L/L as large as several percent <cit.>.
These materials have been extensively studied for their practical applications in developing multifunctional devices, such as actuators, oscillators, and sensors <cit.>.
Beyond their applicative aspects, magnetostriction serves as a key physical quantity for probing phase transitions as well as underlying spin correlations.
For instance, magnetostriction measurements on the quantum spin ice  revealed a sharp first-order transition at extremely low temperatures, shedding light on a liquid-gas metamagnetic transition and offering insights into spin-orbital liquids <cit.>.
Magnetostriction measurements on the Shastry–Sutherland magnet SrCu_2(BO_3)_2 uncovered a series of spin superstructure phases under ultrahigh magnetic fields <cit.>, which are characterized by devil's-staircase-like magnetization plateaus <cit.>.

There are several well-known microscopic origins of magnetostriction <cit.>.
One such mechanism involves the dependence of an ionic radius on its electronic configuration.
A field-induced change in the ionic state can result in significant volume magnetostriction, as exemplified by a spin-state transition in LaCoO_3 <cit.> and valence transitions in f electron-based valence-fluctuating materials <cit.>.
In contrast to these on-site mechanisms, exchange striction refers to the change in distance between two magnetic ions to minimize exchange coupling energy [Fig. <ref>(a)].
Here, Δ L/L reflects changes in local spin correlations ⟨ S_i· S_j⟩ between neighboring sites i and j, and consequently correlates with the magnetization M <cit.>.
The crystal field (CF) also influences the arrangement of surrounding anions and the wave function of the magnetic site to minimize the Coulomb energy.
Applying a magnetic field alters the wave function through CF level hybridization, potentially causing anisotropic lattice deformation, known as CF striction [Fig. <ref>(b)].
Although magnetostriction is universally observed in all magnetic materials, its theoretical description is far from straightforward and is often simplified by phenomenological approaches <cit.> or effective magnetoelastic models <cit.>.

The target compound in this study is , one of the most extensively studied 4f Ising pyrochlore magnet, known as a classical spin ice <cit.>.
The strong Ising anisotropy towards the ⟨ 111 ⟩ axes arises from a combination of dominant nearest-neighbor ferromagnetic interactions and the CF effect.
The strong geometrical frustration results in a highly degenerate ground state, characterized by short-range spin correlations where two spins point inward and two outward within each tetrahedron (“2-in–2-out") <cit.> [Fig. <ref>(c)].
When a magnetic field is applied along the [111] direction, a metamagnetic transition occurs at approximately 2 T, flipping one spin and resulting in a “3-in-1-out” (or “1-in-3-out”) configuration in the tetrahedra, representing a fully polarized state within the Ising limit [Fig. <ref>(c)].
In this field orientation, each tetrahedron has one magnetic site with the easy axis parallel to B (the “easy-axis (EA) site”) and three sites with axes not parallel to B (the “hard-axis (HA) sites”).



Given the strong spin-orbit coupling intertwining spin and orbital degrees of freedom, the CF-phonon interaction is crucial for the magnetoelastic properties.
However, as the easy-axis directions vary at the four magnetic sites in  , local lattice deformations induced by the CF tend to cancel out, making its effect on the crystal structure less evident.
To clarify this, the observation of the CF striction in high magnetic fields would be useful.
In , the energy gap between the lowest and the first-excited CF levels is approximately 260 K in zero field <cit.>.
The application of an external magnetic field along [111] can hybridize these CF levels at the HA sites, as shown in the CF level diagram in Fig. <ref>(d), eventually realizing a forced ferromagnetic (“all-up") state [Fig. <ref>(c)].
The CF level crossing is theoretically expected at B_ cf^ cal = 67 T based on the exact diagonalization of the CF Hamiltonian.
Indeed, Erfanifam et al. observed a signature of the CF level hybridization for B ∥ [111] through magnetization and ultrasound measurements up to 60 T <cit.>, and Opherden et al. observed a sequence of metamagnetic transitions accompanied by the CF level crossing for B ∥ [5513] up to 120 T <cit.>.

Here, we investigate the CF striction in  using magnetostriction measurements under ultrahigh magnetic fields up to 120 T, complemented by magnetization and magnetocaloric-effect (MCE) measurements.
We observe significant anisotropic CF striction, characterized by a lattice contraction amounting to Δ L/L ∼ -5 × 10^-4 along the field direction.
Our mean-field calculations, based on a point-charge model and incorporating CF, two-ion exchange interactions, phonons, and CF-phonon coupling, successfully reproduce the magnetostriction data.
Similar high-field magnetostriction is observed in another spin ice compound, , suggesting that the CF striction is a common feature in rare-earth-based spin-ice systems.



§ METHODS




 §.§ Experiments


Single crystals of  and  were grown from polycrystalline feed rods using a high-temperature xenon-type optical floating zone furnace.
Crystal orientations were checked using a backscattering Laue x-ray diffractometer.
Several pieces of as-grown crystals were cut and polished into rectangular parallelepipeds with dimensions of approximately 1 × 1 × 3 mm^3.



We performed magnetization and magnetostriction measurements in the field orientation B ∥ [111].
Magnetization up to 7 T was measured using a superconducting quantum interference device (MPMS, Quantum Design).
Magnetization up to 62 T and 130 T was measured by the induction method in a nondestructive short-pulsed magnet (4 ms duration) and in a horizontal single-turn-coil (HSTC) system (8 μs duration) <cit.>, respectively.
Longitudinal magnetostriction (Δ L/L ∥ [111]) up to 55 T and 118 T was measured by the fiber-Bragg-grating (FBG) method in a nondestructive long-pulsed magnet (36 ms duration) and in the HSTC system, respectively.
Transverse magnetostriction (Δ L/L ⊥ [111]) up to 55 T was also measured in the nondestructive long-pulsed magnet, where the optical fiber was bent by 90 degrees, as in Refs. <cit.>.
A relative sample-length change Δ L/L was detected by the optical filter method <cit.>.
The optical fiber was glued on the flat surface of the crystal using Stycast1266.

In the nondestructive pulsed magnet, the magnetization and magnetostriction measurements were performed in either liquid-^4He or gas ^4He environment, i.e., nonadiabatic conditions.
In the HSTC system, the sample was cooled to approximately 5 K using a liquid-^4He flow-type cryostat, where the measurement condition should be quasi-adiabatic because of the short pulsed-field duration.
To monitor the temperature change of the sample during the field sweep, we measured the magnetocaloric effect (MCE) up to 55 T under both nonadiabatic and quasi-adiabatic conditions using the nondestructive long-pulsed magnet <cit.>.
A sensitive Au_16Ge_84 film thermometer was sputtered on the surface of the crystal, with temperature calibration performed using a commercial RuO_2 thermometer.

Table <ref> summarizes the measurement conditions for all experiments conducted under pulsed magnetic fields.
The typical magnetic-field profiles of the three types of magnets used in the experiments are shown in Fig. <ref> in Appendix <ref>.



 §.§ Theoretical calculations


To capture the CF level scheme in , we first consider a single-ion Hamiltonian that includes both the CF and Zeeman terms. The CF Hamiltonian can be expressed as follows:

    Ĥ_ cf =   ∑_lm B_lm O_lm

where B_lm denotes the CF parameters, and O_lm represents Stevens Operator equivalents.
The single-ion Hamiltonian is then written as

    Ĥ_ single-ion = Ĥ_ cf-g_Jμ_ BĴ_i· B,

where g_J is the Landé's g factor, μ_ B is the Bohr magneton, Ĵ_i is the magnetic moment at site i, and B is the external magnetic field.
We obtain the field evolution of the CF level energy at the HA site [Fig. <ref>(d)] by exact diagonalization of eq. (<ref>) <cit.>.

For detailed analyses of magnetization and magnetostriction, we additionally take into account magnetic interactions, phonons, and CF-phonon coupling.
The magnetic properties of  can be accurately described by the classical dipolar spin ice model, incorporating dominant long-range dipolar interaction J_ dipo and short-range nearest-neighbor exchange interaction J_ ex.
As the dipolar interaction J_ dipo can be effectively renormalized into the nearest-neighbor exchange interaction <cit.>, we consider a two-spin exchange coupling term described by

    Ĥ_ ex   =-1/2∑_ij J(R_ij)Ĵ_iĴ_j,

where the sum is taken over all nearest-neighbor pairs, J(R_ij) ≡ J_ dipo + J_ ex is the distance-dependent effective exchange coupling, and R_ij is the vector connecting sites i and j.
Assuming that J is linearly modulated by a slight bond-length change ϵ, we expand J(R_ij) as,

    J (R_ij) ≈ J (R_ij^0) + ∑ _α, β, γ∂ J/∂ R^α∂ϵ_αγR_ij^γ/∂ϵ_βϵ_β,

where R_ij^0 is the distance between the original positions of sites i and j, and the indices α, β, γ = x, y, z refer to an Euclidean coordinate system parallel to the crystal axes, i.e., x ∥ a, y ∥ b and z ∥ c.
For the lattice contribution, we consider a phonon term, Ĥ_ ph, and a CF-phonon term, Ĥ_ cfph, which represents the coupling between the CF and phonons.
The detailed formulas for these two terms are provided in Appendix <ref>.
Combining all these terms, we construct the total Hamiltonian as

    Ĥ_ total =  ∑ _iĤ_ single-ion + Ĥ_ ex + Ĥ_ ph + Ĥ_ cfph.


We performed numerical simulations on Eq. (<ref>) using the  program <cit.>, which specializes in calculating the physical properties of 4f-based magnets within the framework of mean-field theory <cit.>. 
 employs a point-charge model to determine the CF wavefunctions and CF energy gap <cit.>.
To match the experimental energy gap of approximately 260 K <cit.>, we apply a scaling factor of 0.7 to the nominal charges Ho (3+), Zr (4-), O (2-). To set the initial state in zero field, we utilized the crystallographic parameters of  reported in Ref. <cit.>.
We set the dipolar interaction to J_ dipo = 2.4 K <cit.> and the nearest-neighbor exchange interaction to J_ ex = 0.1 K, yielding a total effective exchange coupling of J = 2.5 K.
This set of parameters successfully reproduces both the metamagnetic transition in the low-field regime and the CF level crossing at B_ cf∼ 65 T, as experimentally observed.
We note that even a small change of 0.1 K in J_ ex results in a 6 T shift in B_ cf^ cal.
In Eq. (<ref>), we assume dJ/dR ≃ dJ_ ex/dR, which is determined by the Bethe-Slater curve, as shown in Fig. <ref> in Appendix <ref>.

To calculate the magnetization, we solve the mean-field equations self-consistently to obtain the magnetization at each sublattice i, which is given by

    M_i = ∑_ng_J μ_ B⟨ n |Ĵ_̂î| n ⟩/Zexp(-E_n/k_ BT),

where Z = ∑_nexp(-E_n/k_ BT) is the partition function for sublattice i, and | n⟩ is the eigenstate corresponding to the eigenvalue E_n of Ĥ_ single-ion+Ĥ_ ex.
The total magnetization M is obtained by averaging the magnetization of all sublattices, in the unit of μ_ B/Ho.

For the magnetostriction, we calculate strains self-consistently based on Eq. (<ref>) and the following equation:

    ∑_β=1,...,6c^αβϵ_β   =∑_i,δ=1,2,3G^αδ_ mix(i)⟨ u^δ_i ⟩
       + ∑_i,lmG_ cfph^α lm(i)⟨ O_lm(Ĵ_̂î)⟩
       + 1/2∑_i,i',δ,δ',α',γ∂ J_δδ' ( R_ii')/∂ R^α'∂ϵ_α ' γ∂ R^γ_ii'/∂ϵ_α⟨Ĵ_iδĴ_i'δ'⟩,

where i and i' denote atomic site positions, while δ, δ', and α', γ run over the spatial coordinates x, y, and z, represented as the integers 1, 2, and 3, respectively.
c^αβ denotes the elastic constant, where α and β are indices in Voigt notaion, ranging from 1 to 6.
G_ mix^αδ represents the phonon-strain coupling, and G_ cfph^α lm represents the CF-phonon coupling.
For details of these parameters, see Appendix <ref>.
In Eq. (<ref>), the first and second terms on the right side contribute to the CF striction, and the third term is the exchange striction.
The total lattice strain, i.e., magnetostriction, can be obtained by

    Δ L/L = ∑_ijϵ_ijl_il_j,

where l denotes the unit vector in the direction of measurement.
Strain ϵ_ij is a 3 × 3 symmetric tensor, hence has 6 independent matrix elements: ϵ_xx, ϵ_yy, ϵ_zz, ϵ_xy,  ϵ_xz,  ϵ_yz.
We obtain the longitudinal magnetostriction (Δ L/L ∥ [111]) by calculating Δ L/L=(ϵ_xx+ϵ_yy+ϵ_zz+ 2(ϵ_xy+ϵ_xz+ϵ_yz))/3. In this study, we use Voigt notation, where ϵ_ij is relabeled as ϵ_1=ϵ_xx, ϵ_2=ϵ_yy, ϵ_3=ϵ_zz, ϵ_4=2ϵ_yz=2ϵ_zy,  ϵ_5=2ϵ_xz=2ϵ_zx, and ϵ_6=2ϵ_xy=2ϵ_yx.



§ RESULTS AND DISCUSSION




 §.§ High-field magnetization and magnetostriction


Figure <ref>(a) shows the magnetization curve of  for B ∥ [111] measured up to 130 T at an initial temperature of T_ ini = 5 K using the HSTC system. The M–B curve measured up to 62 T at T_ ini = 4.2 K using the nondestructive pulsed magnet is also displayed, which reproduces Ref. <cit.>. Both M–B curves exhibit a plateau-like feature between 20 and 60 T, followed by a metamagnetic increase, where the field derivative of magnetization dM/dB exhibits a broad peak at B_ cf∼ 65 T for the field-down sweep [see the inset of Fig. <ref>(a)].
This metamagnetic behavior suggests the CF level crossing at the HA sites, consistent with the CF scheme shown in Fig. <ref>(d). Above B_ cf, M gradually approaches the expected full moment, 10 μ_ B/Ho, suggesting the realization of the “all-up” state.
We note that the sensitivity of magnetization detection decreases as dB/dt approaches zero, which may introduce errors in the absolute value of M near the maximum field.



Figure <ref>(b) shows the longitudinal magnetostriction curve for B ∥ [111] measured up to 120 T at T_ ini = 5 K using the HSTC system, along with the curve measured up to 55 T at T_ ini = 4.2 K using the nondestructive pulsed magnet. Initially, Δ L/L increases and reaches a maximum around 30 T. Subsequently, Δ L/L reverses and dramatically decreases from 30 to 80 T, where the field derivative of the magnetostriction d(Δ L/L)/dB exhibits a peak at B_ cf∼ 65 T for the field-down sweep [see the inset of Fig. <ref>(b)].
Finally, Δ L/L is nearly saturated at ∼80 T.
The observed change in sample length across B_ cf, amounting to Δ L/L ∼ 5 × 10^-4, highlights the strong CF effect on the crystal structure of .



 §.§ Magnetocaloric effect (MCE)


Comparing the three M–B curves shown in Fig. <ref>(a), it is evident that the initial rise in magnetization at low magnetic fields becomes more gradual as the field sweep becomes faster.
This trend could be attributed to two factors: sample heating and slow spin relaxation.
We will discuss the spin dynamics in Sec. <ref>.
To verify the sample heating, we performed MCE measurements for B ∥ [111] up to 55 T in the nondestructive pulsed magnet.
The field dependence of the sample temperature T(B), measured at various initial temperatures (T_ ini's) under quasi-adiabatic conditions, are shown by the blue curves in Fig. <ref>.
The T(B) curves for field-up and field-down sweeps overlap for all the measured T_ ini's.
The absence of hysteresis ensures that nearly adiabatic conditions were achieved.
These T(B) curves can hence be regarded as isentropic, meaning that the total entropy, comprising both lattice entropy and magnetic entropy, remains constant.

For T_ ini = 5 K, T(B) approaches 30 K at 30 T, indicating a significant reduction in magnetic entropy <cit.>.
This occurs because the highly degenerate “2-in–2-out" states are destroyed by a magnetic field as weak as 2 T, stabilizing the nondegenerate “3(1)-in–1(3)-out" state.
At higher initial temperatures, the increase in T(B) is less pronounced due to greater thermal fluctuations and larger lattice heat capacity.
After T(B) reaches its maximum around 35 T, the sample gradually cools as the magnetic field increases up to 55 T.
This cooling behavior indicates an increase in magnetic entropy, suggesting the onset of CF level hybridization.

We also measured the sample temperature changes at T_ ini = 4.2 K in a liquid-^4He environment, i.e., under nonadiabatic conditions, as shown by the red curve in Fig. <ref>.
Surprisingly, T(B) reaches 28 K during the field-up sweep, which is only slightly lower than in the quasi-adiabatic case.
These MCE results indicate that sample heating effects are unavoidable when measuring  under pulsed high magnetic fields.
We infer that, in the magnetization and magnetostriction data presented in Fig. <ref>, the sample temperature is likely between 20 and 30 K at fields above 10 T.





 §.§ Spin dynamics


Our high-field data also provide valuable insights into the dynamical properties of the magnetic state.
As for the experimental data obtained in the millisecond-duration pulsed magnetic fields [Figs. <ref>(a) and <ref>(b)], both the M–B curve up to 62 T and the Δ L/L–B curve up to 55 T exhibit small hysteresis (Fig. <ref>).
These hysteresis are likely due to the lower sample temperature during the field-down sweep compared to the field-up sweep, as suggested by our nonadiabatic MCE data (Fig. <ref>).
We note that the hysteresis loop in the Δ L/L–B curve cannot be attributed to slow spin dynamics because the sign of the hysteresis is opposite to the delayed response.
This suggests that the timescale of the spin dynamics is faster than the millisecond range.



In contrast, the M–B curve up to 130 T, obtained in the microsecond-duration pulsed magnetic fields [Fig. <ref>(c)], exhibits significant hysteresis [Fig. <ref>(a)].
Given that no hysteresis is observed in the quasi-adiabatic MCE (Fig. <ref>), the observed hysteresis can be attributed to slow spin relaxation, which originates from nonequilibrium processes of monopole formation or annihilation <cit.>.
Notably, the hysteresis observed in the Δ L/L–B curve up to 120 T is much less pronounced compared to that in the M–B curve (Fig. <ref>).
This suggests that the relaxation time associated with CF striction, which is dominant in the high-field magnetostriction, is likely faster than the microsecond range.
This timescale is consistent with previous studies based on AC susceptibility, neutron spin echo, and μSR experiments <cit.>. 



 §.§ Temperature dependence and anisotropic magnetostriction


We also investigate the anisotropy and temperature dependence of the magnetostriction.
Figures <ref>(a) and <ref>(b) show the longitudinal and transverse magnetostriction of , respectively, for B ∥ [111] measured at various T_ ini's in the nondestructive pulsed magnet.
As T_ ini increases, the position of the hump in the longitudinal magnetostriction curve shifts to a lower field, and eventually, the hump disappears at T_ ini = 60 K.

In contrast to the longitudinal magnetostriction, the transverse magnetostriction at T_ ini = 4.2 K exhibits a broad dip around 30 T.
As the temperature increases to T_ ini = 40 K, this dip structure disappears, and Δ L/L begins to increase steadily with the magnetic field.
The anisotropic behavior between the longitudinal and transverse magnetostricition at high magnetic fields is the characteristic of the CF striction.



 §.§ Numerical calculations


To theoretically understand the behavior of M and Δ L/L in high magnetic fields, we performed numerical calculations based on the mean-field approach for the Hamiltonian in Eq. (<ref>).
Figures <ref>(a) and <ref>(b) show the calculated M–B and Δ L/L–B curves, respectively, for B ∥ [111] at various temperatures.
By adopting the “2-in–2-out” spin-ice structure as the initial state in zero field, the metamagnetic transition to the “3(1)-in–1(3)-out” state is reproduced at 5 K, as shown in the inset of Fig. <ref>(a).
At low temperatures below 20 K, a plateau-like feature appears between 10 and 50 T, followed by another metamagnetic transition at approximately 65 T induced by the CF level crossing.
These trends are qualitatively consistent with the experimentally observed M–B curve [Fig. <ref>(a)].
The less pronounced plateau-like feature and broader metamagnetic behavior in the experimental curve can be attributed to the MCE and slow spin relaxation, as mentioned in Secs. <ref> and <ref>.



Having Confirmed that our simulations accurately describe the magnetization of  across the entire field range, we now turn to the longitudinal magnetostriction.
According to Eq. (<ref>), the total magnetostriction is composed of the CF striction (Δ L_ cf/L) and exchange striction (Δ L_ ex/L).
The field dependence of Δ L_ cf/L and Δ L_ ex/L at 5 K is plotted in the inset of Fig. <ref>(b).
The combination of a positive change in Δ L_ ex/L in the low-field regime and a negative change in Δ L_ cf/L in the high-field regime results in a hump structure in the total magnetostriction curve.
Notably, the rounded shape of the experimentally observed hump in the Δ L/L–B curve at T_ ini = 5 K [Fig. <ref>(b)] is presumably due to sample heating, which is consistent with the calculated Δ L/L–B curve at 20 K.
Our calculations predict that the hump in Δ L/L shifts to a lower magnetic field with increasing temperature and eventually disappears above 60 K.
This behavior also agrees well with the experimental observations, as shown in Fig. <ref>(a).





 §.§ Prevalence of CF striction in spin-ice compounds


We have revealed the CF striction in  through both experimental and theoretical approaches.
Applying a magnetic field along the [111] direction leads to the hybridization of the lowest and first-excited CF levels around B_ cf, which is associated with negative longitudinal magnetostriction and positive transverse magnetostriction.
To explore whether the CF striction is a common feature of spin-ice systems, we investigated high-field magnetostriction in another spin ice compound, .
The introduction of rare-earth ions with smaller magnetic moments, such as Pr (∼3 μ_ B), can modify the spin-ice state, making it less Ising-like and more susceptible to quantum effects.
In particular, Pr_2(Zr, Sn, Hf, Ir)_2O_7 are proposed as quantum spin ices <cit.>.
The lowest CF level of  consists of 90–96% of |J=± 4 ⟩ <cit.> (smaller compared to 98% of |J=± 8 ⟩ in ), indicating the presence of substantial transverse fluctuations.

Figures <ref>(a) and <ref>(b) display the M–B and Δ L/L–B curves for  at T_ ini = 5 K (or 4.2 K), respectively.
No hysteresis is observed, consistent with the reported fast spin dynamics on the picosecond timescale <cit.>.
Unlike , the magnetization shows a monotonic increase without any plateau-like features, suggesting that  does not exhibit a well-defined Ising limit state, and the low-energy CF levels gradually hybridize as the magnetic field increases.

Notably, the longitudinal magnetostriction in  decreases monotonically, while the transverse magnetostriction increases.
This anisotropic lattice deformation is similar to that observed in  (Fig. <ref>), suggesting that the similar CF environments in  and  lead to qualitatively identical CF striction effects.
We note that the sign of the magnetostriction in the low-field regime for  is opposite to that for .
This difference is likely due to the opposite signs of dJ/dR for  (dJ/dR > 0) and  (dJ/dR < 0).



§ CONCLUSION


We have thoroughly investigated the static and dynamic magnetoelastic properties in the spin-ice compound Ho_2Ti_2O_7 through magnetostriction measurements under ultrahigh magnetic fields.
Our observations reveal a large anisotropic magnetostriction occurring at the CF level crossing around B_ cf∼ 65 T, where the longitudinal magnetostriction amounts to Δ L/L ∼ -5 × 10^-4. This is an order of magnitude larger than the exchange striction observed in the low-field regime.
Complementary MCE measurements reveal a dramatic sample-heating effect exceeding 20 K in pulsed high-field experiments, and also highlight the onset of CF level hybridization above 30 T, which would correlate with the turnaround behavior of the magnetostriction curve. 
By varying the magnetic field sweep rates, we distinguish between the fast dynamics of the CF-phonon coupling and the slow dynamics of spin-spin correlations.

To model the CF striction, we developed a comprehensive Hamiltonian based on a point-charge model that includes CF-phonon coupling.
Our numerical calculations based on the mean-field approach successfully account for the experimental magnetostriction data across the entire field range.
This study paves the way for understanding the intricate role of the CF-phonon coupling in the crystal structure of a broad range of 4f-based magnets.



§ ACKNOWLEDGMENTS

This work was partly supported by the JSPS KAKENHI Grants-In-Aid for Scientific Research (No. 20J10988 and No. 24H01633), JST-ASPIRE Program (JPMJAP2317), and JST-MIRAI Program (JPMJMI20A1). N.T. was supported by the Alexander von Homboldt foundation. M.G. was supported by the JSPS through a Grant-in-Aid for JSPS Fellows. We appreciate the stimulating discussions with R. Moessner, R. Klingeler, M. Gingras, T. Fennell, and M. Udagawa.





§ WAVEFORM OF PULSED HIGH MAGNETIC FIELDS


Figure <ref> shows the typical magnetic-field profiles generated by three types of pulsed magnets used in this study.





§ PHONON AND CF-PHONON COUPLING TERMS




In this section, we provide the formulas for the phonon term Ĥ_ ph and the CF-phonon coupling term Ĥ_ cfph in the Hamiltonian of Eq. (<ref>).
Ĥ_ ph is written as follows:

    Ĥ_ ph =   ∑_i p_i^2/2m_i
    +1/2∑_ijk_ij/2|R_ij|^2( U_j R_ij- U_i R_ij)^2.

Here, p_i and m_i denote the momentum and mass, respectively, of the atom at position i.
The displacement vector of the atom at position i is represented by U_i.
The difference between the undisplaced lattice positions is given by R_ij≡ R_j- R_i, where R_j (R_i) represents the position of atom at site j (i).
k_ij is the longitudinal atomic spring constants, which is determined based on the Born-von Karman model, assuming simple distance-dependent elastic constant (see Appendix <ref> for details).
Note that the sum over indices i and j counts each spring twice, thus a factor of 1/2 is included before the sum.
The elastic constants are then used to define the phonon-strain coupling constant G_ mix as follows:

    G_ mix^αβγ (i)=   -a_0∑_i k_ij/| R_ij|^2 R_ij^α R_ij^β R_ij^γ,

where a_0 is the Bohr radius, and the indices α, β, γ = x, y, z refer to an Euclidean coordinate system parallel to the crystal axes, i.e., x ∥ a, y ∥ b and z ∥ c.

The CF-phonon coupling is expressed by

    Ĥ_ cfph =  ∑_lm∑_ij(i<j)∇_ U_iB_lm(j) U_iO_lm(Ĵ_j),

where the index i denotes all atomic positions, B_lm denotes the CF parameters, and O_lm represents Stevens Operator equivalents.
The CF-phonon coupling constant G_ cfph can be derived from Eq. (<ref>) by taking the displacement derivative of the CF parameters as

    G_ cfph^αβ lm (j)= -1/2∑_i (R_i^β∂ B_lm(j)/∂U_i^α
    +R_i^α∂ B_lm(j)/∂U_i^β).

Importantly, G _cfph affects the magnitude of Δ L/L.
We scaled G _cfph obtained from McPhase by a factor of 0.16 to match the calculated Δ L/L with the experimental one.
Although the elastic constants c^αβ also affect the magnitude of Δ L/L, their values were fixed based on prior experimental data, as detailed in Appendix <ref>.



In order to obtain the positional derivative of the exchange interaction J_ ex, the Bethe-Slater curve is employed, as shown in Fig. <ref>: 

    J_ ex(R)=A[-(R/D)^2+(R/D)^4]exp^-(R/D)^2,

where A and D are set to 0.4 and 3.49, respectively, resulting in J_ ex=0.1 K and dJ_ ex/dr=1.03 K/Å at the nearest neighbour distance R_ nn=3.54 Å <cit.>.
It is important to constrain dJ_ ex/dr to a positive value for  in order to explain the positive longitudinal magnetostriction in the low-field regime.



§ ELASTIC CONSTANT




 computes the spring constants k_ij in units of N/m based on the Born-von Karman model:

    k =25 ×exp[-0.1 (r/a)^2],

where r represents the distance and a denotes the nearest-neighbor bond length.
Based on these spring constants, the elastic constant tensor 
c^αβ can be derived.
The resulting tensor closely matches those obtained from sound velocity experiments <cit.> and successfully reproduces both the slope of low-energy acoustic phonons as a function of wave vectors and the highest-energy optical phonon mode, which reaches 105 meV in DFT calculations <cit.>.
Equation (<ref>) produces a table of longitudinal spring constants, which are then converted to an elastic tensor with three independent elements in the units of eV/primitive unit cell: c_11=585.656, c_12=276.629, and c_44=276.629, reflecting the highly symmetric cubic crystal structure.
These values are in good agreement with those obtained from ultrasound measurements <cit.>, after performing a unit conversion of 1 meV/primitive unit cell volume = 0.000621 GPa, where the primitive unit cell volume is 257.835 Å^3 <cit.>. 

99
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notes For the exact diagonalization of Eq. (<ref>), we employed local coordinates using a Mathematica code. The corresponding CF parameters were taken from Ref. <cit.>. For the numerical simulations on Eq. (<ref>),  employs global coordinates to obtain the CF level scheme, where the application of a rotation matrix is necessary.
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