Dichotomy in temporal and thermal spin correlations observed in the breathing pyrochlore LiGa1−xInxCr4O8

A breathing pyrochlore system is predicted to host a variety of quantum spin liquids. Despite tremendous experimental and theoretical efforts, such sought-after states remain elusive as perturbation terms and lattice distortions lead to magnetic order. Here, we utilize bond alternation and disorder to tune a magnetic ground state in the Cr-based breathing pyrochlore LiGa1−xInxCr4O8. By combining thermodynamic and magnetic resonance techniques, we provide experimental signatures of a spin-liquid-like state in x = 0.8, namely, a nearly T2-dependent magnetic specific heat and persistent spin dynamics by muon spin relaxation (μSR). Moreover, 7Li NMR, ZF-μSR, and ESR unveil the temporal and thermal dichotomy of spin correlations: a tetramer singlet on a slow time scale vs. a spin-liquid-like state on a fast time scale. Our results showcase that a bond disorder in the breathing pyrochlore offers a promising route to disclose exotic magnetic phases.


INTRODUCTION
A pyrochlore lattice provides a fertile ground to discover exotic quantum and topological phenomena since a three-dimensional (3D) network of corner-sharing tetrahedra features a macroscopic ground-state degeneracy [1][2][3][4][5][6] . When quantum fluctuations are sufficiently strong, the pyrochlore Heisenberg antiferromagnet is predicted to harbor a spin-liquid ground state, whose defining characteristics are fractionalized excitations, emergent gauge fields, and long-range entanglement [7][8][9][10] . However, such a quantum spin liquid (QSL) in pyrochlore materials has proven exceptionally difficult to achieve because a large coordination number suppresses quantum fluctuations and a range of perturbations such as spin-lattice couplings is apt to stabilize magnetic order 11-13 . In the context of a generalized pyrochlore lattice, a breathing pyrochlore has recently garnered much attention as a reservoir of exotic quantum states 14,15 . In the breathing pyrochlore, A and B tetrahedra alternate in size and, thus, exchange interactions (see Fig. 1a). As such, its spin Hamiltonian is given by Here, the breathing anisotropy B f = J B /J A quantifies the degree of the bond alternation. By gauging inter-tetrahedral coupling J B, decoupled tetrahedra (B f = 0) is tuned to a regular pyrochlore (B f = 1). At the endpoints, the ground states are tetramer singlet and QSL 9,16 . The breathing pyrochlore has an advantage over its uniform counterpart in searching for QSL thanks to its strong thermal stability 17 . Besides, the inclusion of magnetic anisotropies to a pyrochlore Heisenberg model leads to a realization of Weyl magnons-a bosonic analog of a Weyl fermion in electron systems and a fracton spin liquid whose elementary excitations mimic electric and magnetic fields having the form of rank-2 tensors 18,19 .
On the material side, the Cr-based breathing pyrochlore  20 . The lasting antiferromagnetic orders suggest that the bond alternation of the x = 0 and x = 1 compounds is not sufficient to repress the effect of spin-lattice coupling that is responsible for a cubic-totetragonal transition accompanying structural and magnetic phase segregation. In this situation, bond randomness can counteract the magnetoelastic transition by inhibiting cooperative lattice distortions. This raises the exciting prospect of unveiling concealed magnetic behaviors of the breathing pyrochlore lattice, not being interrupted by long-range magnetic order.
In this work, we investigate spin dynamics and low-energy excitations of the bond-disordered breathing pyrochlore LiGa 0.2 In 0.8 Cr 4 O 8 (marked by the inverse triangle in Fig. 1b). By combining multiple magnetic resonances with thermodynamic techniques, we find a thermal and temporal dichotomy between a coupled tetramer singlet and a spin-liquid-like state with no indication of the substitution-induced phase coexistence. Spin fluctuations probed on the different time scales evidence that the QSL-like state is pertinent to an emergent diluted pyrochlore lattice by a random bond alternation.

Magnetic phase diagram
As sketched in Fig. 1a, the A-site ordered LiACr 4 O 8 (A = Ga, In) spinels are a nearly perfect realization of the breathing pyrochlore lattice due to the significant difference between the Li + and A 3+ valence states. The Ga-for-In substitution disrupts a uniform arrangement of the two alternating tetrahedra and leads to the rich magnetic phase diagram of LiGa 1−x In x Cr 4 O 8 , as shown in Fig.  1b. The magnetic ordering vanishes for x > 0.1 or x < 0.95. A spinglass phase occupies a wide range of x = 0.1-0.625. For the Ga-rich compound LiGa 0.95 In 0.05 Cr 4 O 8 , the 7 Li NMR and neutron experiments revealed the spin nematic transition at T f = 11 K, driven by the small bond disorder. In the In-rich range of x = 0.75-0.95, LiGa 1−x In x Cr 4 O 8 exhibits a spin-gap-like behavior with the lack of magnetic ordering and spin freezing down to 2 K. However, little is known about an exact ground state and spin dynamics. Here, we have chosen the x = 0.8 compounds (marked by the inverse triangle in Fig. 1b) to address this issue. Figure 1c presents the T dependence of dc magnetic susceptibility χ dc (T) of x = 0.8. On cooling, χ dc (T) shows a broad hump at around 70 K, and a subsequent upturn below 10 K. Above 100 K, χ dc (T) follows the Curie-Weiss law with the Curie-Weiss temperature Θ CW = −386(1) K and the effective magnetic moment μ eff = 3.873 (4) μ B [see Supplementary Note 1] that are consistent with the previously reported values 21 . As the first step to single out the intrinsic χ intr (T), we attempted to fit the T < 5 K data to χ dc (T) = χ intr (T)+ χ orp (T), where χ orp (T) = C orp /(T-Θ orp ). We obtain Θ = −2.0 (2) K, alluding to the contribution of weakly interacting orphan spins. The orphan spin concentration is estimated to be 0.6% of the Cr 3+ spins. Defects and chemical disorders induced by the chemical substitution may be responsible for the appearance of a tiny fraction of the weakly interacting orphan spins. The resulting χ intr (T < 5 K), which is nearly T-independent, indicates the presence of abundant low-energy in-gap states. Besides, the ac magnetic susceptibility χ' ac (T) excludes the formation of spin freezing or spin glass state [see Supplementary Note 2]. Heat capacity In Fig. 2, we plot the heat capacity C tot of x = 0.8 as a function of T and B. We subtract lattice and nuclear Schottky contributions from C tot to isolate the magnetic specific heat C m [see Supplementary Note 3 for details]. As evident from Fig. 2a, C m shows a broad maximum around 70 K, indicative of the onset of short-range ordering, and is largely quenched above 150 K.

Magnetic susceptibilities and magnetization
From a log-log plot of C m /T vs. T in Fig. 2b, we observe a hump at around T = 0.6 K in addition to the high-T broad hump. On the application of an external magnetic field, the T = 0.6 K hump shifts to a higher temperature and is quickly suppressed at μ 0 H > 3 T. The B-dependence of the low-T hump is attributed to the weakly coupled orphan spins, which occupy the lowest m s level by thermal population. Note that the spin entropy S m has a value of 0.186 J·K −1 ·mol −1 ·Cr −1 at 0.6 K, which amounts to roughly 0.2% of Rln4 [see Supplementary Note 3 for details]. This value is somewhat smaller than the estimated value from χ dc (T) and M(T = 1.8 K, B).
In addition, C m (T)/T displays a moderate B dependence in the T range of T = 0.1-10 K. Overall, the C m (T)/T data are described by the power-law dependence C m /T~T n with n = 0.68-0.82 at T = 2-10 K and n = 0.67-0.86 below 1 K. The observed C m~T 1.68-1.82 is close to a quadratic dependence. We recall that the T 2 dependence of C m has been reported in a range of 3D frustrated magnets, which are commonly proximate to a disordered QSL with unconventional low-energy excitations [23][24][25] . A comparison between the C m /T data and the calculated curve of a s = 3/2 isolated tetrahedron unveils a large discrepancy for temperatures below 10 K [see the inset of Fig. 2b]. Furthermore, the spin gap is evaluated to be Δ/k B = 0.289(5) K from the low-T heat capacity data, suggesting that the excitation gap by tetramer singlets is filled by the in-gap magnetic states at low temperatures [see Supplementary Note 3 for details]. Thus, we conclude that the observed quadratic dependence of C m is associated with emergent gapless low-energy excitations of the bond-disordered breathing pyrochlore and that the orphan spin contribution is minor in forming a spin-liquid-like state, judging from the fact that the gapless excitations develop at temperature as high as 10 K, well above T = 0.6 K.

Nuclear magnetic resonance
We now examine temporal and thermal spin fluctuations by combining multiple magnetic resonance techniques.
The 7 Li NMR spectra and the spin-lattice (spin-spin) relaxation rate 1/T 1 (1/T 2 ) measured at μ 0 H = 11.99 T are plotted as a function of temperature in Fig. 3a, e. At high temperatures, the NMR spectrum exhibits a single sharp line with no quadrupolar splitting, similarly observed in LiInCr 4 O 8 20,26 . On cooling down to T = 2.7 K, the NMR spectra gradually broaden and shift to higher magnetic fields without developing additional structured peaks, further confirming the absence of long-range magnetic ordering and structural phase transition.
With decreasing temperature, 1/T 1 displays an exponential-like decrease down to 10 K, and then a subsequent upturn to a powerlaw growth. The activation behavior of 1/T 1 in the T range of 25-100 K is well fitted with the Arrhenius equation 1/T 1~e xp(−Δ/ k B T) [see the semi-log plot of 1/T 1 vs. 1/T in the inset of Fig. 3e]. We obtain the spin pseudogap Δ/k B = 35(1) K, somewhat larger than Δ M /k B~2 0 K extracted from M intr (B). Generally, the spin gap Δ/k B obtained from 1/T 1 is larger than the true spin gap Δ min (q) occurring at a specific point q [27][28][29] . This is because 1/T 1 maps out the q-average of the dynamical spin susceptibility, 1/T 1~T Σ q A 2 (q) χ"(q,ω 0 ). Here, A(q) is the form factor of hyperfine interactions, and ω 0 is the nuclear Larmor frequency.
The salient feature is that 1/T 1 follows a power law 1/T 1~T −n with the exponent of n = 0.46(3) below T * = 10 K. The drastically distinct behavior of 1/T 1 through T * = 10 K suggests the switching of a dominant relaxation mechanism: a high-T activated vs. a low-T fast fluctuating. Here the small exponent n is indicative of dynamically fluctuating spin excitations, defying the development of a λ-like critical slowing down. The spin-spin relaxation rate 1/T 2 shows a weak power-law decrease 1/T 2~T 0.071(4) as T → 0. A nearly T-independent 1/T 2 means that spin dynamics is in the fastfluctuating limit without critical slowing down, as observed in the QSL states of the triangular material 1T-TaS 2 30 .

Muon spin relaxation (μSR)
In Fig. 3b, we present ZF-μSR spectra at selected temperatures. At high temperatures, the ZF-μSR spectra show a Gaussian-like shape and gradually change to an exponential-like form. With decreasing temperature, the muon spin polarization rapidly relaxes in the initial time interval (t = 0-2 μs). We observe neither an oscillating muon signal down to T = 25 mK nor a recovery to 1/3 of the initial polarization, evidencing the formation of a dynamically fluctuating state 31 . No loss of the initial polarization is not compatible with inhomogeneous magnetism. For quantitative analysis, we fitted the ZF-μSR spectra to the sum of a stretched exponential function and an exponential function, P z (t) = a fast exp[−(λ f t) β ] + a slow exp[−λ s t]. Here, a fast (a slow ) is the fraction of the fast (slow) relaxation component, and λ f (λ s ) is the muon spin relaxation rate for the fast (slow) relaxing component. We summarize the obtained parameters in Fig. 3f and Supplementary Note 4. As the temperature decreases, the slow relaxation component λ s (T) suddenly appears below T * = 10 K, at which 7 Li 1/T 1 shows an upturn to the power law. Furthermore, λ s (T) shows a T-independent behavior (λ s~0 .042 μs −1 ) below 1 K. Figure 3f is a log-log plot of the fast relaxation rate vs. T. With lowering the temperature, λ f (T) exhibits a weak power-law increase T −0.11 (2) down to 10 K, and then a steep power-law increase with the exponent of n = −0.71 (6). Noteworthy is that μSR probes weakly correlated spins in the T > 10 K range where NMR senses gapped spin excitations, implying the variation of spin dynamics with a detecting time scale. We further note that the steep increment of λ f (T) concurs with the leveling-off of χ intr (T) in Fig. 1c and the upturn of 1/T 1 in Fig. 3e. Upon further cooling toward T = 25 mK, λ f (T) flattens out below 2 K, entering a persistent-spin-dynamical regime. Such a leveling-off of the muon spin relaxation rate is often observed in an assortment of QSL candidates [32][33][34] . With decreasing temperature, the stretched exponent β gradually decreases from β = 2 (a Gaussian-like decay) to β = 1 (an exponential decay) below T * = 10 K [see Supplementary Note 4, Fig. S4]. The low-T exponential relaxation suggests the Shown in Fig. 3c and g are the LF-μSR spectra and the LF dependence of λ f (H LF ) measured at T = 25 mK. Remarkably, the LF-μSR spectra exhibit substantial relaxation even at H LF = 10 kG, lending further support to the dynamic ground state. In the Redfield model, λ LF ðHÞ ¼ 2γ 2 μ hH 2 loc iν=ðν 2 þ γ 2 μ H 2 Þ, the LF dependence of λ f (H LF ) is related to the fluctuation frequency ν, and the fluctuating time-averaged local field H 2 loc 35 . Here, γ μ is the muon gyromagnetic ratio. Fitting to the λ f (H LF ) data yields ν = 1.6(1) GHz and H loc = 296(11) G [see Fig. 3g]. We comment that the fluctuation frequency is faster than the value reported thus far for QSL materials while the magnitude of the local field is rather large, possibly due to a large spin number s = 3/2 32,33 .

Electron spin resonance
To probe spin fluctuations on the GHz time scale, we turn to highfrequency ESR. As shown in Fig. 3d, with decreasing temperature, the ESR spectra broaden continuously down to 2.3 K and shift to a lower field. We observe no impurity signals, excluding the possibility of the Ga 3+ /In 3+ substitution-induced phase separation. All the ESR spectra are fitted with a single Lorentzian profile,  7 Li NMR, ZF-μSR, and ESR. a 7 Li NMR spectra at selected temperatures T = 2.7 and 250 K. b ZF-μSR spectra at selected temperatures. c Longitudinal-field dependence of the μSR spectra measured at T = 25 mK. The solid lines denote the fit to the data using the sum of two simple exponential functions. d Derivative of the ESR absorption spectra measured at various temperatures. Spectra are vertically shifted for clarity. e T dependence of 7 Li spin-lattice relaxation rate 1/T 1 and spin-spin relaxation rate 1/T 2 . The red solid line denotes the Arrhenius equation, yielding the spin pseudogap of Δ/k B = 35(1) K. The solid black line represents a power-law behavior of 1/T 1 and 1/T 2 . f Muon spin relaxation rate as a function of temperature λ f (T). The solid magenta line represents the power-law fit to the data. g Muon spin relaxation as a function of longitudinal field λ LF (H), along with fitting to the Redfield formula. h T dependence of the peak-to-peak ESR linewidth ΔH pp (T). The solid orange lines indicate a power-law behavior of ΔH pp (T) in two different temperature ranges marked by the shaded region at T * = 10 K. Error bars on the data points indicate a standard error. suggesting that they originate from one type of the correlated Cr spins. The extracted parameters vs. temperature are plotted in Fig. 3h and Supplementary Note 5, Fig. S5. The peak-to-peak linewidth ΔH pp is associated with the development of spin-spin correlations, while the resonance field H res emulates the build-up of internal magnetic fields.
ΔH pp (T) exhibits a power-law increase ΔH pp (T)~T -n . We can identify a small change of the exponent from n = 0.37(1) to 0.30 (2) through T * = 10 K. This weak anomaly in the exponent is in sharp contrast to the drastically changing character of the magnetic correlations on the MHz time scale. A critical-like line broadening in the paramagnetic state is due to the development of local spin correlations below Θ CW = −386(1) K and is generic to frustrated magnets subject to critical spin fluctuations. We note that the observed critical exponent of n = 0.30-0.37 is much smaller than n = 0.6-0.7 for the uniform pyrochlore MCr 2 O 4 (for M = Zn, Mg, Cd) 36 , as shown in Fig. 3h. Compared to the uniform pyrochlore, the reduced exponent in x = 0.8 means that the random bond alternation weakens critical spin correlations. As such, the emergent low-T state in the GHz time scale bears a resemblance to a diluted pyrochlore lattice.

Raman spectroscopy
Next, we discuss the structural evolution of LiGa 1−x In x Cr 4 O 8 with Raman spectroscopy. We present the composition dependence of the Raman spectra and phonon parameters in Fig. 4. Herein, we focus the T 2g (1) and T' 2g (1) phonon modes, which are sensitive to the structural environments of the A and B tetrahedra, respectively [see Supplementary Note 7 and Fig. S7 for the phonon assignments].
As shown in Fig. 4a, b, the 314 cm −1 T' 2g (1) mode slightly softens with increasing x to 0.5 and then little changes with further In 3+ substitution. Its linewidth decreases linearly with x despite the introduction of Ga 3+ /In 3+ chemical disorders. This suggests that a single type of A tetrahedra (surrounded by four Li + ions) is present for all x. In sharp contrast, the T 2g (1) mode at 216 cm −1 is split into three peaks for x = 0.5 and 0.8. For a random occupation of the In 3+ and Ga 3+ ions, five types of the B tetrahedra surrounded by a combination of the In 3+ and Ga 3+ ions are possible in LiGa 0.2 In 0.8 Cr 4 O 8 : four In 3+ (4I), four Ga 3+ (4G), three In 3+ /one Ga 3+ (3I1G), 2G2I, or 1G3I. For x = 0.8, the three 118, 160, and 195 cm −1 modes have their scattering intensity in a relative ratio of 1:1.3:0.7. Given a low probability of occupying the 3G1I and the 4 G configurations, the three peaks are assigned to 4I, 3I1G, and 2I2G, respectively. From the composition-dependent Raman data, we infer that the In 3+ /Ga 3+ substitution engenders a single type of the A tetrahedra and three types of the B tetrahedra. However, we admit that the chemical substitution can give rise to diverse effects in addition to the primary chemical pressure, including crystal-electric-field distortion and inhomogeneous charge distribution near the Cr 3+ ions. Based solely on the Raman data, it is difficult to rationalize their impact on structure and magnetism. Future high-resolution x-ray studies are required to address this issue.
In Fig. 5a, we zoom the x evolution of the T 2g (3) phonon mode, which involves the asymmetric bending of the CrO 6 octahedra. As such, the T 2g (3) mode is sensitive to the substitution-induced alteration of the Cr-O bonding geometry, allowing tracing the randomness in exchange interactions. As plotted in Fig. 5b, the phonon frequencies of the T 2g (2), T 2g (3), and A 1g modes show a linear decrease with increasing x. A close look at the T 2g (2) and T 2g (3) modes reveals the splitting of the T 2g (2) and T 2g (3) modes on the In 3+ /Ga 3+ substitution [see the decomposition of the phonon peaks into four Lorentzian profiles in Fig. 5a and the pink triangles in Fig. 5b]. The linewidths of the main peaks show little variation with x. This suggests that In 3+ /Ga 3+ substitution does not bring about strong disorders such as vacancies, defects, and interstitial cations. Rather, the distinct distortion field induced by the In 3+ /Ga 3+ substitution leads to the differentiation of Cr-O bonding environments. From the observation that most of the high-frequency phonons are split into two peaks, we conclude a differentiation of the intra-tetrahedral J A , weak inter-tetrahedral J B1 , and strong inter-tetrahedral exchange interaction J B2 . That is, the bond randomness creates at least exchange interactions J A , J' A , J B2 , J' B2 , J B1 , and J' B1 with an energy hierarchy J A , J' A > J B2 , J' B2 > J B1 , J' B1 . Assuming the variations of δJ A = J A − J' A, δJ B1, and δJ B2 are not bigger than the J A and J B2, the energy hierarchy is largely preserved.

DISCUSSION
With the aid of diverse thermodynamic and magnetic resonance techniques (0 Hz-1 THz), we are able to disclose the temporal and thermal structures of magnetic correlations.
We identify the characteristic temperature T * = 10 K, below which highly correlated spins develop a quantum paramagnetic state. The emergent low-energy state features nearly T-squared C m , as well as correlated spin fluctuations over a wide time scale of 10 −12 − 10 −4 s, as evident from a power-law law 1/T 1~T −0.46(3) , λ f (T)~T −0.71 (6) , and ΔH pp (T)~T −0.30 . The exponents extracted from different resonance techniques cannot be related to each other since the breathing pyrochlore model is not analytically solvable. Nonetheless, the common power-law behaviors advocate criticallike spin fluctuations. Here we stress that the energy scale of T * = 10 K is smaller than the spin pseudogap of Δ/k B = 20-35 K, deduced from M(B) and 1/T 1 . Yet, it is bigger than a few kelvins of weakly correlated orphan spins. The latter contribution is quenched by applying the external field of µ 0 H > 3 T. As such, the contribution of defect spins to the low-T correlated state is minute.
Next turn to the temporal spin dynamics. The thermodynamic data show the spin-gap behavior with the in-gap state. On the slow time scale (1-10 4 Hz), the NMR data demonstrate dichotomic spin dynamics, namely, high-T gapped spin correlations and low-T fast fluctuations. On the fast time scale (10 4 -10 12 Hz), the µSR and ESR data give no signature of the spin-singlet correlations. Instead, highly correlated spin dynamics prevails over the whole measured temperature range. It seems that the singlet fluctuations average out beyond the time scale of the NMR technique at low temperatures.
The dichotomic magnetic correlations are apt to be correlated with structural and magnetic inhomogeneities. First, we mention that the spectral shape and the relaxation rates of the magnetic resonance data rule out the coexistence of magnetically phasesegregated regions. Unlike the end members (x = 0, 1), 1/T 1 (T), λ f (T), and ΔH pp (T) of x = 0.8 show no notable anomalies such as critical divergence, abrupt drop, or kink 20,26 .
Second, the Raman spectra exhibit no phonon anomalies pertaining to magnetostructural transitions [see Supplementary Note 7 for details]. Instead, as sketched in Fig. 6, a statistical distribution of In 3+ and Ga 3+ generates mainly a single type of A tetrahedra and three types of tetrahedra: four-neighbor A tetrahedra are connected by (i) four weak inter-tetrahedral couplings J B1 s, (ii) three weak J B1 s and one strong J B2 , and (iii) two weak J B1 s and two strong J B2 s. Given the magnetic energy hierarchy J A > J B2 > J B1 with J B1 /J A = 0.3 and J B2 /J A = 0.6, two magnetic subsystems emerge, namely (i) coupled tetramer singlets (red shaded circle) and (ii) diluted pyrochlore lattices (blue dashed region). Here, we note that J B1 /J A increases from 0.1 to 0.3 in LiInCr 4 O 8 through the magnetostructural transition with decreasing temperature 37 . The tetramer singlets forming an entangled state of four spins are coupled through J B1 and tetrahedral clusters. This bond randomness leads to a broadening and distribution of the discrete energy levels, yet still maintaining a singlet character. Taken together, the coupled tetramer singlets have longer temporal magnetic correlations than the pyrochlorelike subsystem that are governed by two-spin correlations. This raises the possibility that the quantum paramagnetic ground state bears a QSL-like characteristic pertinent in the diluted pyrochlore system. Here, we stress that despite the In 3+ /Ga 3+ substitutioninduced bond disorders, our thermodynamic and resonance data are not compatible with a spin glass, spin freezing, and random singlets (valence bonds). Of particular, we could find no scalings of thermodynamic quantities expected for random singlets [38][39][40] . Rather, LiGa 0.2 In 0.8 Cr 4 O 8 features a quantum disordered state with gapless low-lying excitations. We, thus, speculate that a degree of randomness is insufficient to engender random singlets or spin glass due to the presence of the tetramer-singlet-like state.
To conclude, we have investigated the temporal and thermal spin dynamics of a bond-disordered breathing pyrochlore compound. We find that a dichotomy of magnetic correlations is linked to two emergent subsystems. LiGa 0.2 In 0.8 Cr 4 O 8 occupies a special position in QSL candidate materials because quenched disorders do not lead to a spin glass and random singlet. The observed low-T QSL-like state opens up a venue for discovering an unknown state in a diluted breathing pyrochlore lattice.

Sample synthesis
Polycrystalline samples of LiGa 0.2 In 0.8 Cr 4 O 8 were synthesized by a solidstate reaction method. Stoichiometric amounts of Li 2 CO 3 , Cr 2 O 3 , Ga 2 O 3 , and In 2 O 3 were mixed in a 1:4:0.2:0.8 molar ratio and thoroughly ground in a mortar. The mixture was pelletized and sintered at 800°C in the air for 12 h. The substance was ground, pressured into pellets, and finally sintered at 1000°C for 24 h and 1100°C for 72 h. We checked the quality of the samples by X-ray diffraction measurements.

Magnetic properties characterization
dc and ac magnetic susceptibilities were measured using a superconducting quantum interference device magnetometer and vibrating sample magnetometer (Quantum Design MPMS and VSM). High-field magnetization experiments were performed at the Dresden High Magnetic Field Laboratory using a pulsed-field magnet (20 ms duration). The magnetic moment was detected by a standard inductive method with a pick-up coil system in the field range of μ 0 H = 0-60 T.

Heat capacity
Heat capacity measurements were carried out in the temperature and field range of T = 0.03-300 K and B = 0-9 T using a commercial set-up of Quantum Design PPMS with a thermal relaxation method. The raw heat capacity data C tot /T follow a T −3 power-law behavior at extremely low temperatures, corresponding to a nuclear Schottky contribution (see Fig. 6 Illustration of the emergent magnetic lattices in a bond-disordered breathing pyrochlore. Schematics of single-A tetrahedra (orange squares) and three different B tetrahedra (green squares) created by the Ga-for-In substitution. The thick and thin green lines denote J B2 and J B1 , respectively. Given the energy hierarchy of magnetic interactions J A > J B2 > J B1 , the breathing pyrochlore effectively comprises coupled tetramer singlets (red shaded circle) and diluted pyrochlore lattices (blue dashed region).