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ABSTRACT Exotic quantum phases, arising from a complex interplay of charge, spin, lattice and orbital degrees of freedom, are of immense interest to a wide research community. A well-known

example of such an entangled behavior is the Jahn-Teller effect, where the lifting of orbital degeneracy proceeds through lattice distortions. Here we demonstrate that a highly-symmetrical

5d1 double perovskite Ba2MgReO6, comprising a 3D array of isolated ReO6 octahedra, represents a rare example of a dynamic Jahn-Teller system in the strong spin-orbit coupling regime.

Thermodynamic and resonant inelastic x-ray scattering experiments, supported by quantum chemistry calculations, undoubtedly show that the Jahn-Teller instability leads to a ground-state

doublet, resolving a long-standing puzzle in this family of compounds. The dynamic state of ReO6 octahedra persists down to the lowest temperatures, where a multipolar order sets in,

allowing for investigations of the interplay between a dynamic JT effect and strongly correlated electron behavior. SIMILAR CONTENT BEING VIEWED BY OTHERS SPECTROSCOPIC SIGNATURES AND ORIGIN

OF HIDDEN ORDER IN BA2MGREO6 Article Open access 29 November 2024 MIXED ORBITAL STATES AND MODULATED CRYSTAL STRUCTURES IN LA1−_X_CA_X_MNO3 DEDUCED FROM SYNCHROTRON X-RAY DIFFRACTION

Article Open access 26 July 2023 UNVEILING NEW QUANTUM PHASES IN THE SHASTRY-SUTHERLAND COMPOUND SRCU2(BO3)2 UP TO THE SATURATION MAGNETIC FIELD Article Open access 24 June 2023 INTRODUCTION

The importance of the Jahn-Teller (JT) effect reaches across many areas of chemistry, physics, and applied sciences1. Examples range from localized systems like photo-excited methane

cations2,3, ultra-fast transients in molecular complexes4, and vacancy diffusion in graphene5 towards more cooperative phenomena, including high-temperature superconductivity in cuprates6,

colossal magneto-resistance in manganites7, metal-insulator transition in ruthenates8 and antiferroelectricity in lacunar spinels9. The cornerstone of the JT effect is the existence of an

electronic state with orbital degeneracy, leading to an overall decrease in the energy once the degeneracy is lifted. Typically, this occurs through ligand (_L_) displacement, which

consequently affects both the electronic and magnetic properties of materials. In strongly correlated systems, where ligands are shared between neighboring metal (_M_) sites, the distortions

at one site are correlated with distortions on neighboring sites, leading to a cooperative JT effect10. In the case of isolated, highly symmetric clusters, the system can choose among

equivalent directions of distortion, resulting in a potential energy surface (PES) with multiple valleys and energy barriers between them. In the simplest case of an _M__L_6 octahedron

having a single electron or a hole in a doubly-degenerate orbital state (well-known cases are Mn3+ and Cu2+ ions), the distortion can freeze along principal axes _x_, _y,_ and _z_, producing

a static JT effect. If the barriers are small enough so that the wave function is distributed over all valleys equally, the vibronic symmetry is restored, resulting in a dynamic JT effect.

Many aspects of solid-state physics have been theoretically investigated through the coupling to continuous symmetries in JT systems11,12,13, but the available materials are rather scarce.

The inter-cluster interaction needs to be very small to avoid a cooperative JT effect, while other couplings, like exchange interactions, should remain appreciable. Examples of materials

where a dynamic JT effect has been suggested to occur include an unconventional molecular superconductor Cs3C60, where, together with Mott localization, it drives the material across a

metal-to-insulator transition14,15,16. It has also been argued that the observed permittivity17 in a paraelectric phase of BaTiO3 stems from the resonating state between minima of the PES,

which, under the influence of an electric field, become nonequivalent and trap Ti ions along one direction18. Eventually, the dynamic JT effect is overcome by the interaction between the

centers, leading to a ferroelectric phase transition. These examples involve relatively light elements, including 3_d_ transition metal cations, where the influence of spin-orbit coupling

(SOC) is considered negligible. Recently, a growing interest has been devoted to strong SOC systems, encompassing topological insulators19, Dirac and Weyl semimetals20, and Kitaev spin

liquids21. The JT effect, on the other hand, has seldom been investigated in the context of strong SOC, with several examples devoted to Ir4+ systems22,23. Some 4_f_ systems have been

investigated before, like PrO224, where the crystal field effect is much weaker compared to compounds based on _d_ orbitals. More generally, a strong SOC qualitatively changes the

ground-state wave functions and has been proposed to renormalize the associated distortion, or even completely remove it above a certain critical value25. In this study, we present

compelling evidence that Ba2MgReO6, a member of the 5_d_1 double-perovskite family, displays a distinctive set of properties that make it exceptionally well-suited for the exploration of the

role of the dynamic JT effect on strongly correlated electrons in the presence of a strong SOC. This is promoted by a high-symmetry environment of Re6+ ions, covalently coupled to six

oxygen ions, alternating with ionic Mg2+ ions and forming a three-dimensional array of nearly-independent, JT-active ReO6 octahedra, Fig. 1a. Strong SOC mixes the _t_2_g_ levels into a

_j_eff = 3/2 quartet and a _j_eff = 1/2 doublet, separated by ΔSO, as shown in Fig. 1b. The JT instability manifests itself as the splitting of the ground-state quartet into two doublets,

with the gap ΔJT directly related to the amplitude of the distortion. The correlated nature of 5_d_ electrons is reflected in the formation of a multipolar order26,27, encompassing a charge

quadrupole state below _T_Q = 33 K and a magnetic dipole state below _T_M = 18 K (Fig. 1c, d). Using spectroscopic and thermodynamic evidence, we show that the JT instability causes the

doublet-doublet splitting in Ba2MgReO6 at temperatures _T_ > > _T_M, _T_Q. We quantify the associated distortions through local cluster calculations and determine that the energy

barriers between the local minima are smaller than the zero-point motion energy, establishing Ba2MgReO6 as a dynamic JT system. Additionally, through a careful analysis of specific heat

results, we establish that the total entropy across both transitions reaches only \(R{{{\rm{\ln }}}}2\). RESULTS LOCAL _D-D_ EXCITATIONS To probe the localized states of Re ions, we employ

resonant inelastic x-ray scattering (RIXS). In Fig. 2a, we display the low temperature (_T_ = 8 K) energy dependence of the scattering intensity obtained at the Re _L_3 edge, displaying

multiple peaks of varying widths, which we assign as follows (see also the Supplementary Material). The peak around zero energy is an elastic line, where no energy transfer occurs during the

virtual excitation of core-level electrons. Further peaks reflect virtual processes where core-level electrons are excited to unoccupied states while the recombination of the core-level

hole occurs with electrons on occupied Re 5_d_ states. We find that the observed scattering intensity profile can be fully described by assuming that each peak is represented by a simple

Gaussian contribution. The positions of maxima correspond to crystal field levels of Re 5_d_-orbitals and can be readily assigned following the scheme presented in Fig. 1b. The first peak at

Δ_E_ > 0 represents the SOC-induced splitting between the ground state quartet and the higher-lying doublet state ΔSO ≅ 0.5 eV which allows the estimate of the SOC constant _λ_ ~ 335 

meV. Up to 4 eV, the line shape is featureless, with two additional peaks present around 4.5−5 eV. A notable tail coming from an even higher-lying peak can be observed, which can be assigned

to virtual processes involving the recombination with electrons originating from lower-lying, predominantly oxygen _p_-orbitals, located  ~ 6.3 eV below the elastic line. A recent report28

focusing on the comparison between Ba2MgReO6, Sr2MgReO6, and Ca2MgReO6 found a slightly asymmetric line-shape around ΔSO, ascribing it to the influence of phonon modes on spin-orbit

excitations. Nevertheless, their value for ΔSO in Ba2MgReO6, as well as the energies of higher-lying excitations, are in excellent agreement with our results. The two peaks, situated at 4.7 

eV and 4.9 eV, can be assigned to states belonging to _e__g_ orbitals, with four-fold degeneracy being split into two doublets, in accordance with the JT scenario. An equivalent split of the

ground state quartet is expected but not directly observed due to an insufficient energy resolution (~ 100 meV) of the experimental setup at the Re _L_3 edge. On the other hand, the results

obtained at the O _K_ edge (Fig. 2b), with its  ~ 50 meV resolution, show an additional peak at low energies reflecting the doublet-doublet splitting ΔJT ≅ 88 meV. A similar finding has

been reported very recently29 for the sister compound Ba2NaOsO6, with the corresponding excitation at 95 meV. It is interesting to note that both experiments at the O _K_ edge resulted in a

similarly asymmetric line shape around ΔJT and ΔSO, with possible faint features on the high-energy side. The description of those features has recently been attempted in terms of coupling

of phonon modes with spin-orbit excitations28, although the asymmetry itself is still not fully understood. It is clear that several additional modes are necessary to describe such an

asymmetric line shape, but if all the parameters are left independent, it quickly becomes unfeasible to perform the fitting procedure (see the Supplementary Material for more details). The

approach we used is to model the high-energy tail of each _d_ − _d_ excitation through a set of _n _dependent modes, whose energies and amplitudes are determined by simple relations,

\({E}_{n}^{\alpha }={E}_{\alpha }+n\cdot {E}_{mode}\) and \({A}_{n}^{\alpha }={A}_{0}^{\alpha }{e}^{-n\cdot B}\), with all the additional modes having the same width. Here _α_ stands for a

JT or SO _d_ − _d_ excitation (indicated by light colors in the inset of Fig. 2b), while additional modes are indicated by darker colors. Such an approach significantly reduces the number of

independent parameters and leads to a very satisfactory description of the observed line shape within the relevant energy range up to 1 eV. The central result of RIXS experiments comes from

the temperature evolution of the acquired spectra. In Fig. 2c, we compare the Re _L_3 RIXS spectra above and below _T_M and _T_Q. Within the resolution, there is no visible variation

between the two line profiles. Specifically, the asymmetric shape at 4.5−5 eV, associated with the splitting of _e__g_ orbitals, remains clearly unaffected. A more direct and profound

evidence comes from ΔSO(_T_) and ΔJT(_T_), extracted from O _K_ RIXS spectra and plotted in Panels 2d, e. Both are practically _T_-independent, with a small decrease of ΔJT below _T_Q. It is

not surprising that ΔSO shows very little temperature dependence. On the contrary, the persistence of ΔJT well above 200 K goes against the prevailing idea that the splitting of the

ground-state quartet is linked with the appearance of the quadrupolar order at _T_Q30,31. Moreover, we can infer that ΔJT > 0 to much higher temperatures since any order-parameter-like

decrease would be readily observed in our data, as indicated in Fig. 2e by a dashed line. POTENTIAL ENERGY SURFACE AND THE JAHN-TELLER EFFECT Now we turn our efforts to characterize the type

of distortion that accompanies the observed splitting. The relevant modes of an _M__L_6 octahedron are sketched in Fig. 3a–c. The magnitude of the isotropic mode mainly determines the

octahedral splitting ΔO which for Ba2MgReO6 is large enough to disregard the _t_2_g_ − _e__g_ mixing. Two other modes are JT-active, with _Q_3 representing a tetragonal distortion while _Q_2

describes the orthorhombic mode. For an isolated octahedron, with a linear coupling and in the limit of very small distortions, the PES is characterized by an infinite-degeneracy line, the

so-called ‘Mexican hat’ potential (Fig. 3d). The octahedron remains dynamic, exploring all possible linear combinations in the _Q_2 − _Q_3 parameter space, with ΔJT = _c__o__n__s__t_. > 

0. Increasing the amplitude of distortions leads to anharmonic effects and, together with the presence of surrounding charges, breaks the line degeneracy, resulting in a multi-valley PES. An

opposing effect has been argued to arise from strong SOC25, driving the PES back towards a Mexican-hat potential but also suppressing the JT distortion, as well as reducing the energy gain

due to the JT effect. For a more quantitative insight, we perform embedding multi-reference quantum chemistry calculations taking into account SOC (see “Methods”). We start from the

experimentally established structure at 40 K27 and calculate the PES of the ReO6 octahedron by parameterizing ligand displacements in terms of the above-described modes for a general

distortion _Τ_ = _α__Q_iso + _β__Q__3_ + _γ__Q__2_. In Fig. 3e, we show the calculated PES exhibiting three valleys, corresponding to tetragonal distortions along the three principal axes.

One of the minima is given by _α_ ≈ − 1%, _β_ ≈ − 1.5%, and _γ_ = 0, where _β_ < 0 indicates that the degeneracy lifting of the ground-state quartet occurs through a contraction of the Re

 − O_a__p__i__c__a__l_ distance. A similar result has been obtained for Ba2NaOsO629, while calculations performed for a 4_d_1 compound Ba2YMoO6 indicate the same sign and a similar magnitude

of distortions32, thus establishing a clear tendency of _d_1 octahedral systems towards a contraction, regardless of the strength of SOC. Importantly, both the sign and the magnitude of the

calculated JT distortion are in stark contrast with the reported shift of oxygen ions below _T_Q27. Detailed synchrotron x-ray diffraction analysis revealed27 the lowering of the global

symmetry from a cubic \(Fm\overline{3}m\) above _T_Q to a tetragonal _P_42/_m__n__m_ below _T_Q, with a tetragonal elongation amounting to  ~ 0.1% at low temperatures. This separation of

scales undoubtedly disentangles the formation of the quadrupolar order at _T_Q from the dominant, JT-induced splitting of the presumed ground-state quartet30,33,34. The validity of the

performed calculations can be assessed by comparing the calculated energy levels with experimental data, as listed in Table 1. The calculations for the experimentally established structure

at 40 K and the isotropically contracted one contain undistorted octahedra, resulting in ΔJT = 0 as well as four-fold degenerate _e__g_ states (see Fig. 1). The calculations for distortions

characterizing the minimum of a valley lead to ΔJT = 78 meV and the splitting of _e__g_ states, a hallmark of the JT effect. All the calculated observables are found to be in the range of

10−20 % from the experimental values. The minimum of the PES does not directly reflect the amplitude of the distortion of an octahedron. We use the calculated PES to obtain the wave function

_Ψ_ of the quantum-mechanical oscillator and its ground-state energy _E_0. As shown in Fig. 3e, the wave function reflects the three-fold symmetry of the PES, extending towards each valley

but still maintaining the absolute maximum at the center of the PES. More importantly, the energy _E_0 ≈ 56 meV is found well above the barrier between valleys _E__B_ ≈ 35 meV (Fig. 3f),

implying that the zero-point motion prevents the freezing of vibronic degrees of freedom, which leaves each octahedron in a dynamic state. The characteristic time is typically expressed in

the form \({({t}_{0})}^{-1}=({E}_{1}-{E}_{0})/h\), where _E_1 ≈ 102 meV is the first excited state35. With _t_0 ≈ 9 ⋅ 10−14 s, only the scattering processes of x-rays and neutrons are fast

enough to observe an instantaneous state of the system, all other experimental techniques obtain a dynamically averaged response. We can calculate the probability distribution of ΔJT using

_S_(ΔJT) = ∫ ΔJT(_Q__i_)∣_Ψ_(_Q__i_)∣2_d__Q__i_, where _d__Q__i_ designates an integration over the _Q_2 − _Q_3 space. The result presented in Fig. 3g reveals an asymmetric shape with a

maximum occurring around 60 meV. We should emphasize that despite numerous approximations that underlie calculations, the results are directly comparable to experimentally obtained values.

For comparison, the last column in Table 1 contains the results for a fictitious tetragonal _I_4/_m__m__m_ structure, where a small 0.15 % tetragonal elongation is implemented to reflect the

observed cubic-to-tetragonal transition at _T_Q27. The calculated ΔJT is two orders of magnitude too small, unambiguously assigning the observed doublet-doublet splitting to a JT

instability. These results can be naturally extended to other members of the double perovskite family, where bonding with ligands alternates regularly between an ionic and a covalent type.

Many similarities exist with Ba2NaOsO6, which exhibits much smaller _T_M = 7.5 K and _T_Q = 9.5 K31,36,37. Another example is Cs2TaCl6, where a tetragonal contraction of  < 1 % has also

been associated with the quadrupolar order38, although a later single crystal study indicated a different value for _T_Q39. Several _d_2 compounds seem to retain high local symmetry40, as

well as a _d_3 Ba2YOsO6 system41, offering a variety of configurations where the interplay of a (dynamic) JT effect and strong SOC can be tested25. ENTROPY AND THE GROUND STATE DOUBLET The

JT-splitting of the ground-state quartet has profound consequences for the thermodynamic properties of 5_d_1 systems. Already, the first report on Ba2NaOsO6 indicated a ‘missing entropy’36,

extracted from electronic specific heat (_C_el) measurements _S_el = ∫(_C_el/_T_)_d__T_ reaches only \(\lesssim R{{{\rm{\ln }}}}2\), significantly smaller than the expected \(R{{{\rm{\ln

}}}}4\). Namely, the total recovered entropy is expected to reflect the ground-state degeneracy at high temperatures, released across two successive transitions _T_M and _T_Q, each

contributing one \(R{{{\rm{\ln }}}}2\)30,33,34. Experimentally, the main uncertainty comes from the determination of a proper phonon background, which leads to a wide array of recovered

values of _S_el26,37,42,43,44. We approach the problem of the phonon background determination by measuring specific heat _C__P_ in magnetic fields, which, by affecting the electronic levels,

causes a redistribution of measured _C__P_. Since the _g_-factor is strongly renormalized due to the partial cancellation of spin and orbital moments30, it is necessary to apply large

magnetic fields to see those effects around _T_Q and above. In Fig. 4c, we plot the _C__P_(_T_) of Ba2MgReO6 in magnetic fields up to 35 T. For _B_ = 0, a sharp peak is seen at _T_M and a

broad shoulder around _T_Q, in agreement with previously published results26,44. For _B_ = 35 T, a broad feature is seen around _T_Q, with the tail merging with _C__P_(_B_ = 0 T) above  ~ 45

 K. We take this as an estimate up to which temperature the electronic contribution is appreciable and consider only the phonon contribution above it. Instead of an analytical form, we aim

to utilize the results of first-principle calculations, which were recently used as an approximate description of the phonon specific heat in Ba2MgReO644. We use a functional rescaling of

the calculated results along both _T_ − and _C__P_ − axes to match the experimental data in the range 60–100 K (see the Supplementary Material for more details). The result of this procedure

is presented in Fig. 4a as a black line, with the relative discrepancy remaining below 0.5% across the relevant range (see Fig. 4b). The electronic contribution _C_el extracted in this

manner is displayed in Fig. 4c. Both features at _T_M and _T_Q are more clearly seen, as well as the magnetic field dependence. Initially _T_M becomes rounded and shifts to higher

temperatures, while there is very little effect on _T_Q. With _B_ = 35 T, however, we see the two features practically merged, with a sharp high-temperature tail. Qualitatively similar

behavior has been observed in CeB645,46 and assigned to field-induced octupolar moments on Ce ions47,48. Similarly, the NMR study31 revealed a shift to higher temperatures of the

order-disorder boundary in Ba2NaOsO6. The total electronic entropy _S_el extracted from Fig. 4c is presented in Fig. 4d. Within an error, the recovered values for all magnetic fields

saturate very close to \(R{{{\rm{\ln }}}}2\), with very little change above 40 K. This result is rather robust against small changes of the exact functional form of the phonon background and

therefore, together with spectroscopic evidence obtained by RIXS and supported by quantum chemistry calculations, unequivocally establishes that the low energy physics in Ba2MgReO6 is

dominated by a ground state doublet. A similar conclusion has been reached by a recent study on Ba2NaOsO6 where a somewhat larger doublet-doublet splitting has been found (95 meV) based on O

_K_ edge RIXS and x-ray magnetic circular dichroism29. We note that these two 5_d_1 systems exhibit qualitatively very similar behavior in terms of multi-polar ordering tendencies, with the

advantage that Re6+ orders at sufficiently larger temperatures than Os7+ to allow a better insight into the peculiar interplay between magnetic dipole and charge quadrupole orders.

DISCUSSION The calculated ground-state energy suggests a dynamic type of distortion, where ReO6 octahedra fluctuate with  ~ 10 THz between elongations along three equivalent principal axes.

The scattering of x-ray photons, which occurs at a time scale much faster than these oscillations, reflects a statistical average of all the distortions distributed across the sample at a

given time and, therefore, cannot be used to distinguish it from a random distribution of frozen static distortion, which might still preserve the global cubic symmetry, despite the breaking

of the local one. On the contrary, the static version of the JT effect can be reasonably argued against using several lines of reasoning (see Fig. 5 for details). First, there is a clear

signature of order-parameter-like increase of intensity associated with charge quadrupolar order composed of two components, an antiferro \({{{{\mathcal{T}}}}}_{x}\) and a ferro

\({{{{\mathcal{T}}}}}_{z}\)27,49. Concomitantly, it is observed that oxygen ions move away from their high-symmetry positions accordingly, reflecting a strong coupling between local

distortions and \({{{{\mathcal{T}}}}}_{z}\) charge quadrupoles30,50. Those distortions are, however, 5 times smaller than calculated for the Jahn-Teller effect, and of an opposite sign. The

frozen landscape of static JT distortions would inevitably lead to a glassy behavior of a charge quadrupolar and magnetic dipole orders since the direction of magnetic dipoles below _T__Q_

is determined by the local orientation of \({{{{\mathcal{T}}}}}_{z}\) quadrupolar moments50. The second argument for the dynamic JT state is related to the local symmetry of Re ions. As has

been demonstrated in 23Na NMR experiments on Ba2NaOsO6, a single resonance line at high temperatures splits into three lines below _T__Q_, indicating a lowering of the local symmetry31. This

has been associated with changes in local charge distribution around Na ions, which are surrounded by six OsO6 octahedra and, therefore, directly reflect shifts of oxygen equilibrium

positions. Since the time scale of NMR experiments is much longer than _t_0, in the dynamic JT scenario, the local symmetry breaks only below _T__Q_, while in the static scenario, the

triplet line would have been observed at all temperatures, which is in clear contradiction with experimental results31. We note that the deformation due to the JT effect is significantly

larger than what occurs below _T__Q_, therefore the splitting of the NMR line would have been clearly seen. Finally, an argument against the random distribution of locally deformed octahedra

comes from the consideration of the effect on the local crystal environment. As revealed by high-resolution synchrotron experiments27, below _T__Q_ the equilibrium of apical oxygen ions

shifts by  ~ + 0.36 %, accompanied by a unit cell tetragonal elongation of  ~ + 0.15 %. Within the static JT scenario, the random orientation of local distortions would lead to a significant

local strain on the surrounding cage of eight Ba ions (almost 1 % of local compression) in order to maintain the observed high-symmetry, cubic unit cell environment above _T__Q_. Since the

quadrupolar moments are randomly frozen, and with such a high lattice strain, there is a clear lack of a feasible mechanism which would lead to the observed coherent distortions below _T__Q_

along the _c_-axis for both apical oxygen ions and the unit cell as a whole. In addition, if the local distortions are static, the system could remove the strain by lowering the unit cell

symmetry from cubic to tetragonal, which is the scenario observed in a series of JT-active _d_9 double perovskites _A_2Cu\({B}^{{\prime} }\)O6 (_A_ = Ba, Sr; \({B}^{{\prime} }=\) W, Te)51.

The dynamic JT effect ensures the local and global cubic symmetry above _T_Q, in accordance with experimental observations in both Ba2MgReO627 and Ba2NaOsO631, as well as with ab initio

results reported for Ba2NaOsO652. Without the shared ligand between ReO6 octahedra, a feature characteristic for double perovskites, the dynamic distortions on one cluster are not correlated

with its neighbors. Following the development of the charge quadrupolar order, where a coherent charge redistribution occurs on Re ions, the PES becomes slightly renormalized, which leads

to a slight, but coherent shift of the equilibrium position of oxygen ions, giving rise to the signal observed in REXS experiments27,49. Crucially, the dynamic state of ReO6 octahedra

continues to dominate the local Re environment even within the multi-polar ordered state. Even with a complete model of spin-orbit-vibronic coupling, the qualitative picture remains the

same34. The revelation that the ground-state doublet dominates the low-temperature physics in this class of materials establishes a new paradigm from the thermodynamic point of view. With

two phase transitions and two order parameters, it is puzzling how to reconcile \(R{{{\rm{\ln }}}}2\) across both _T_M and _T_Q. Concurrently, specific heat at _T_Q does not exhibit a

typical _λ_-type profile, and, as we have shown, no lifting of degeneracy accompanies the quadrupolar order. Some of the entropy (\(\sim 1/3R{{{\rm{\ln }}}}2\)) is released above _T_M,

pointing to a reduction from a three-dimensional towards a two-dimensional phase space for fluctuating magnetic moments50. The plane of fluctuation below _T_Q is then dictated by the

orientation of quadrupolar moments \({{{{\mathcal{T}}}}}_{x}\) and \({{{{\mathcal{T}}}}}_{z}\), being perpendicular to the \({{{{\mathcal{T}}}}}_{z}\)-induced elongation. At _T_M the

directional locking within that plane leads to magnetic dipole order and removes the remaining entropy. We infer that the magnetic and quadrupolar order parameters are part of a single,

entangled object that goes through a two-step transition53. Spin-orbit entanglement plays another crucial role that relates to the dynamical state of ReO6 octahedra. In the weak SOC limit,

the available states forming a Mexican hat potential are composed of purely orbital contributions, each state allowing for two spin projections. For the strong SOC case involving 5_d_

orbitals, the entanglement entails a varying spin projection along the degeneracy line, resulting in a dynamically renormalized directional coupling between neighboring ReO6 octahedra.

Similar to Kitaev physics21, and aided by the frustration on an FCC lattice, it could be envisaged that such a renormalization could lead to a spin-liquid state or other exotic quantum

phases. METHODS SAMPLE PREPARATION Single crystals were grown by the flux method. A stoichiometric mixture of BaO, MgO, and ReO3 powders was mixed with a flux composed of 36 wt % BaCl2 and

64 wt % MgCl2 in an argon-filled glove box. The mixture was sealed in a platinum tube of 20 mm length and 6 mm diameter. The tube was heated at 1300 0C and then slowly cooled to 900 0C at a

rate of 5 0C/h, followed by furnace cooling to room temperature. After the residual flux was washed away with distilled water, several crystals were obtained, The crystals exhibit octahedral

morphology, forming flat triangles along [111] and equivalent directions. Specific heat measurements up to 14 T were performed on as-grown crystals (up to 1 mm in size), while the _B_ = 35 

T experiment was performed on a 0.3 mm polished crystal. For the Re _L_3 edge RIXS experiment the sample was glued on a copper sample holder with the [111] direction perpendicular to the

surface. An in situ cleaving has been utilized for the O _K_ edge RIXS experiment. RESONANT INELASTIC X-RAY SCATTERING The experiments at the Re _L_3 edge were performed at BL11XU of

SPring-8, Japan synchrotron facility. Incident x-rays were monochromatized by a Si(111) double-crystal monochromator and a two-bounce channel-cut monochromator Si(444), and _π_-polarized

x-rays were irradiated on the samples. Horizontally scattered photons were energy-analyzed by a Ge(733) analyzer. The total energy resolution was 140 meV (full width at half maximum). The

experiments at the O _K_ edge were performed at the ADRESS beamline of the Swiss light source (SLS) at the Paul Scherrer Institute (PSI), Switzerland in both the normal and grazing

configuration with 50° between incoming and outgoing beams54. EMBEDDING QUANTUM CHEMISTRY CALCULATIONS Many-body wavefunction calculations on electrostatically embedded finite-size clusters

were performed using the Molpro package55. The model involved a 21-atom cluster consisting of the ReO6 octahedron together with the nearest 6 Mg and 8 Ba atoms. Re atoms were represented

with core potentials and triple-zeta plus two polarization _f_-functions basis functions56. All-electron triple-zeta basis functions were used for O atoms57. Both Mg and Ba atoms in the

cluster were described by effective core potential and supplemented with a single _s_ basis function58. The effect of the crystal lattice surrounding the cluster was treated at the level of

Madelung ionic potential, as described in ref. 59. For the complete active space self-consistent field (CASSCF) calculations, an active space of 3 _t_2_g_ + 2, _e__g_ orbitals, was used. The

optimization was carried out for an average of 5 states, each of unit weight, of the scalar relativistic Hamiltonian. Multireference configuration interaction (MRCI) treatment was performed

with single and double substitutions with respect to the CASSCF reference, as described in refs. 60,61. The spin-orbit coupling correction was added as described in ref. 62. The PES was

mapped out with respect to isotropic mode _Q__iso_ = (_x_1 − _x_2 + _y_3 − _y_4 + _z_5 − _z_6), and _e__g_-type distortions _Q__2_ = (− _x_1 + _x_2 + _y_3 − _y_4) and _Q__3_ = (− _x_1 + _x_2

 − _y_3 + _y_4 + 2_z_5 − 2_z_6), with superscripts 1,2 referring to the _x_-axis O atoms, 3,4 to the _y_-axis O atoms and 5,6 to the _z_-axis O atoms. In order to assess the PES,

calculations were performed on a grid that varies isotropic distortions were varied from 0.5% to  − 1.5% with 0.5% steps, with amplitudes referring to non-normalized _Q__iso_ = [1, 1, 1] and

the percentage being relative to the undistorted 1.926 Å Re − O interatomic distance. A 60∘ sector of the _Q_2 − _Q_3 plane was simulated, with the angle being varied with steps of 15∘ and

radius being varied from 0 to 3.5%, with the step of 0.25%, with amplitudes referring to normalized \({{{{\boldsymbol{Q}}}}}_{{{{\boldsymbol{2}}}}}=\frac{1}{\sqrt{2}}[-1,1,0]\) and

\({{{{\boldsymbol{Q}}}}}_{{{{\boldsymbol{3}}}}}=\frac{1}{\sqrt{6}}[1,1,-2]\). The stability of the minimum with respect to the _t_2_g_-type distortions _Q__4_ = (_y_1 − _y_2 + _x_3 − _x_4)

was confirmed. In addition, an attempt was made to find a minimum with a non-zero amplitude of the _t_2_g_-type distortions by alternating _Q__i__s__o_, _Q_2, _Q_3, and _Q_4 as energy

optimization coordinates, starting at the cubic configuration. No such minimum was found, with all paths considered converging back to the minimum described in the article. SPECIFIC HEAT For

magnetic fields up to 14 T we used a PPMS system (Quantum Design) that utilizes a relaxation method. The measurement at 35 T was performed at LNCMI, Grenoble, France using a Cernox

thermometer as a platform and a semi-adiabatic method. In both cases the addenda has been measured separately and subtracted to obtain specific heat of the sample. Zero-field measurements

have been used to normalize the results from two setups. DATA AVAILABILITY All the data that support the findings of this study are available in the Zenodo repository

(https://zenodo.org/doi/10.5281/zenodo.13765588). CODE AVAILABILITY The code for calculations presented in this study is available from the corresponding author upon request. REFERENCES *

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experiments were supported by “Advanced Research Infrastructure for Materials and Nanotechnology in Japan (ARIM)” of the Ministry of Education, Culture, Sports, Science and Technology

(MEXT), Japan (Proposal No. JPMXP1222QS0107) and were performed at the QST experimental station at BL11XU of SPring-8, with the approval of the Japan Synchrotron Radiation Research Institute

(JASRI) (SPring-8 Proposal No. 2022B3596). The O _K_-edge XAS and RIXS experiments were performed at the ADRESS beamline of the Swiss Light Source at the Paul Scherrer Institut (PSI). We

acknowledge the support from LNCMI-CNRS, a member of the European Magnetic Field Laboratory (EMFL). This work was supported by the Swiss National Science Foundation (SNF) Quantum Magnetism

grant (No. 200020-188648 and 200021-228473) (H.M.R.), the European Union’s Horizon 2020 research and innovation program projects HERO (Grant No. 810451) (H.M.R., J.R.S., I.Ž.). The

experimental work at PSI is supported by the Swiss National Science Foundation through project nos. 178867 and 207904. T.Y. is funded by a CROSS project of PSI. Y.W. and G.C.W. acknowledge

funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 884104 (PSI-FELLOW-II-3i program). AUTHOR INFORMATION

AUTHORS AND AFFILIATIONS * Laboratory for Quantum Magnetism, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Ivica Živković, Jian-Rui Soh, Oleg

Malanyuk, Ravi Yadav, Federico Pisani, Davor Tolj, Jana Pasztorova & Henrik M. Rønnow * Department of Materials, ETH Zurich, Zurich, Switzerland Aria M. Tehrani * Department of Applied

Physics, Nagoya University, Nagoya, Japan Daigorou Hirai * Paul Scherrer Institute, Villigen PSI, Switzerland Yuan Wei, Wenliang Zhang, Carlos Galdino, Tianlun Yu & Thorsten Schmitt *

Synchrotron Radiation Research Center, National Institutes for Quantum Science and Technology, Sayo, Hyogo, Japan Kenji Ishii * Université Grenoble Alpes, INSA Toulouse, Université Toulouse

Paul Sabatier, CNRS, LNCMI, Grenoble, France Albin Demuer * Chair of Computational Condensed Matter Physics, Institute of Physics, École Polytechnique Fédérale de Lausanne, Lausanne,

Switzerland Oleg V. Yazyev Authors * Ivica Živković View author publications You can also search for this author inPubMed Google Scholar * Jian-Rui Soh View author publications You can also

search for this author inPubMed Google Scholar * Oleg Malanyuk View author publications You can also search for this author inPubMed Google Scholar * Ravi Yadav View author publications You

can also search for this author inPubMed Google Scholar * Federico Pisani View author publications You can also search for this author inPubMed Google Scholar * Aria M. Tehrani View author

publications You can also search for this author inPubMed Google Scholar * Davor Tolj View author publications You can also search for this author inPubMed Google Scholar * Jana Pasztorova

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Yuan Wei View author publications You can also search for this author inPubMed Google Scholar * Wenliang Zhang View author publications You can also search for this author inPubMed Google

Scholar * Carlos Galdino View author publications You can also search for this author inPubMed Google Scholar * Tianlun Yu View author publications You can also search for this author

inPubMed Google Scholar * Kenji Ishii View author publications You can also search for this author inPubMed Google Scholar * Albin Demuer View author publications You can also search for

this author inPubMed Google Scholar * Oleg V. Yazyev View author publications You can also search for this author inPubMed Google Scholar * Thorsten Schmitt View author publications You can

also search for this author inPubMed Google Scholar * Henrik M. Rønnow View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS Single crystals were

grown by D.H., D.T., and J.P. RIXS data was measured and analyzed by I.Ž., J-R.S., F.P., K.I., Y.W., W.Z., C.G., T.Y., T.S., and H.M.R. Embedding quantum chemistry calculations were

performed by O.M. and R.Y. under guidance of O.V.Y. and H.M.R. Specific heat was measured and analyzed by I.Ž. and A.D. The phonon contribution to specific heat was calculated by A.M.T. The

manuscript was written by I.Ž. with contributions from all co-authors. CORRESPONDING AUTHOR Correspondence to Ivica Živković. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no

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ABOUT THIS ARTICLE CITE THIS ARTICLE Živković, I., Soh, JR., Malanyuk, O. _et al._ Dynamic Jahn-Teller effect in the strong spin-orbit coupling regime. _Nat Commun_ 15, 8587 (2024).

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