Introduction

Spin ice state on a frustrated pyrochlore lattice has attracted numerous interests in condensed matter physics, due to the emergent magnetic monopole excitations from the manifold of degenerate ground states1,2. By introducing quantum fluctuations with Jeff = 1/2 moments, quantum spin ice (QSI) states can be stabilized, exhibiting quantum electrodynamics with extra excitations like photons and visons3. (In this paper, we adopt the naming convention that the magnetic monopoles refer to the spin-flip excitation, while the visons refer to the sources of emergent electric fields3.) Yb2Ti2O7, Tb2Ti2O7, and Pr2Zr2O7 are such promising QSI candidates3. On the other hand, iridates with 5d electrons have also drawn much attention in recent years owing to the various quantum phases and transitions therein, which originate from the competition between spin-orbit coupling and electron–electron correlation4,5. When these two aspects meet in the pyrochlore iridate Pr2Ir2O7, complex phenomena and exotic phases emerge6,7,8,9,10,11,12,13.

Pr2Ir2O7 is a metal with the antiferromagnetic RKKY interaction of about 20 K in non-Kramers Pr 4 f moments mediated by Ir 5d conduction electrons6. The Kondo effect leads to a partial screening of the Pr 4 f moments and gives a lower Weiss temperature of |θW| = 1.7 K (ref. 6). No long-range magnetic order was observed down to 70 mK evidenced by the magnetic susceptibility measurement, indicating a possible metallic spin liquid ground state, or even a U(1) QSI state6,13. A huge and anisotropic anomalous Hall effect (AHE) was probed under magnetic fields7,9, which may be the result of the spin chirality effect on the Ir sites from the noncoplanar spin texture of Pr 4 f moments. The observation of AHE in the absence of uniform magnetization at zero field further indicates a long-sought chiral spin liquid state in Pr2Ir2O7 (ref. 8). More interestingly, a zero-field quantum critical point (QCP) was uncovered from the diverging behavior and scaling law in the Grüneisen ratio measurement10. It was also theoretically investigated as a QCP between antiferromagnetic ordering and nodal non-Fermi liquid11. Apparently, multiple mechanisms govern the ground state of Pr2Ir2O7, which may induce rich phenomena beyond the spin-ice physics.

For such an exotic metallic spin-liquid candidate with quantum criticality, although various efforts have been made, two main issues remain to be solved. First, how do the electrons behave at the QCP? In other words, will the electrons still be well-defined Landau quasiparticles? Second, probably due to the large neutron absorption cross-section of the iridium ions and the very small size of its single crystals14, little information is known for possible exotic magnetic excitations in Pr2Ir2O7, from which knowledge about the role of multipolar interactions beyond the dipolar interactions and quantum fluctuations can be obtained.

Ultralow-temperature thermal conductivity measurement is an important technique to address the above two issues. For the former one, the verification of the Wiedemann–Franz (WF) law κ/σT = π2kB2/3e2 = L0 can be viewed as an evidence of the survival of Landau quasiparticles at the QCP. Anomalous reduction of the Lorenz ratio L(T)/L0 with L(T) = κ/σT has been observed in CeCoIn5 (ref. 15), YbRh2Si2 (under debate (refs. 16,17,18)), and YbAgGe (ref. 19), while in some other compounds such as CeNi2Ge2 (ref. 20) and Sr3Ru2O7 (ref. 21), the WF law is verified at the QCP. For the latter one, a sizable residual linear term of thermal conductivity indicates the presence of highly mobile gapless excitations in triangular organics EtMe3Sb[Pd(dmit)2]2 (ref. 22) (note that this result has been challenged by two recent reports23,24). Spinon thermal conductivity with a linear temperature dependence was also found in the ideal spin-1/2 antiferromagnetic Heisenberg chain copper benzoate25. No magnetic thermal conductivities were observed in other two QSL candidates κ-(BEDT-TTF)2Cu2(CN)3 and YbMgGaO4 (refs. 26,27).

In this paper, we report ultralow-temperature thermal conductivity measurements on single crystals of Pr2Ir2O7. The WF law is verified at high fields and inferred at zero field, suggesting the normal behavior of electrons at the QCP and the absence of fermionic magnetic excitations. A giant magneto-thermal conductivity at finite temperature is found, which may result from the strong scattering of phonons by the transverse fluctuations. The thermal conductivity is isotropic in different magnetic field directions, which is contrary to specific heat. We shall discuss the implications of these results.

Results

Charge and heat transport

Figure 1a shows the temperature dependence of the resistivity ρ(T) at zero field for the Pr2Ir2O7 single crystal. The upturn behavior and the lnT dependence below 45 K where the resistivity displays a minimum, and the well fit to the Hamann’s equation are the evidences for Kondo effect, as shown in the inset of Fig. 1a. This is consistent with ref. 6. The magnetoresistance MR = (ρ(H) − ρ(0 T))/ρ(0 T) × 100% at T = 0.34 K is presented in Fig. 1b. It is quite small, less than 5% up to 9 T, indicating the little influence of magnetic field on the charge transport. Note that no anisotropy of resistivity is reported with respect to the electric current direction6, while the magnetoresistance is anisotropic in different magnetic field directions9. In the inset of Fig. 1b, ρ(T) below 1 K in µ0H = 0, 3, and 6 T are plotted. Since all the curves are very flat, we can safely extrapolate them to the zero-temperature limit and get the residual resistivity ρ0 = 776, 757, and 769 µΩ cm for µ0H = 0, 3, and 6 T, respectively.

Fig. 1: Charge transport results of Pr2Ir2O7.
figure 1

a Temperature dependence of the resistivity at zero field for Pr2Ir2O7 single crystal. Inset: zoomed view of the resistivity minimum at 45 K due to the Kondo effect. The solid line is the fit to Hamann equation between 3 K and 35 K. b The magnetoresistance at T = 0.34 K. The magnetic field is applied along the [111] direction. Inset: ρ(T) below 1 K in µ0H = 0, 3, and 6 T.

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The thermal conductivities of Pr2Ir2O7 single crystal up to 7 T are shown in Fig. 2a. The magnetic fields were applied perpendicular to the (111) plane. At high fields like 5 T and 7 T, the thermal conductivity data overlap with each other. With decreasing the field, while κ/T data still overlap with the high-field curves below a certain temperature Ts, they are suppressed more and more strongly above Ts. In Fig. 2b, the collapse region of κ/T is shown in detail. The arrows in Fig. 2b denote the suppression temperatures Ts at corresponding fields. (κ(H)-κ(5 T))/T start to deviate from 0 above Ts. At zero field, Ts is about 0.12 K. Similar behavior is also observed in another sample B (for details, see Supplementary Note 3). The field-dependence of Ts is plotted in the inset of Fig. 2b.

Fig. 2: Heat transport results of Pr2Ir2O7.
figure 2

a The thermal conductivity of Pr2Ir2O7 single crystal at various magnetic fields along the [111] direction. At zero field, the thermal conductivity is strongly suppressed above Ts ≈ 0.12 K. b The data of thermal conductivity in (a) after subtracting the data of 5 T. A collapse region is clearly shown below the temperature at which the arrows point. This suppression temperature is defined as Ts. Inset: field dependence of the suppression temperature Ts. The error bars reflect the uncertainty in determining the temperature above which the data deviate from 0. They are estimated as the possible suppression temperature range due to the discrete data points. c The magneto-thermal conductivity MTC = (κ(H)−κ(0 T))/κ(0 T) × 100% at various temperatures. Above 5 T, the thermal conductivity tends to saturate.

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The magneto-thermal conductivity MTC = ∆κ(H)/κ(0 T) = (κ(H) − κ(0 T))/κ(0 T) × 100% at various temperatures is plotted in Fig. 2c. MTC tends to saturate above 5 T below 0.8 K, when the thermal conductivity curves start to overlap with each other. In contrast to the magnetoresistance of less than 5% in charge transport at 0.34 K, the MTC is as large as 100% at 0.25 K and even 650% at 0.7 K. For other QSL candidates such as κ-(BEDT-TTF)2Cu2(CN)3 (ref. 26) and YbMgGaO4 (ref. 27), there is also a positive MTC, but the magnitude is much smaller. Unexpectedly, a crossover from the positive MTCs at low temperatures to the negative MTCs at high temperatures is observed at θw energy scale (for details, see Supplementary Note 4). We will come back to discuss the origin of this giant MTC later.

We would like to emphasize that besides the electron thermal conductivities, κ above 5 T is entirely due to phonons without magnetic scatterings. First, a metamagnetic transition at Bc ~ 2.3 T can be induced only when applying the field along the [111] direction in the magnetization M(B) measurements8. This means that a sizeable fraction of the “2-in, 2-out” configurations are transformed into the “3-in, 1-out” configurations at the critical field. Therefore, it is natural to expect that the scatterings between the local spins and phonons will be reduced as the magnetization approaches saturation. In fact, the thermal conductivity of 3 T is quite close to the value of 5 T and 7 T, which coincides with the critical field observed in the magnetization measurements. Second, the thermal conductivities are field-independent between 5 T and 7 T. This indicates that even if the magnetic moments are not fully static in our experiment temperature range, they do not scatter phonons since the field has no effect on the thermal conductivities. This behavior strongly suggests that the high-field thermal conductivities are due solely to phonons without magnetic scatterings. Note that there may remain scatterings from structure disorder like stacking faults and grain boundaries.

Verification of Wiedemann–Franz Law

In Fig. 3, we fit the thermal conductivity data below 0.3 K for µ0H = 5 T to examine the WF law in Pr2Ir2O7. At ultra-low temperatures, thermal conductivity usually can be fitted to κ/T = a + bTα−1, where aT represents electrons and other fermionic quasiparticles such as spinons, while bTα represents phonons and other bosonic quasiparticles such as magnons28,29. For phonons, the power α is typically between 2 and 3, due to the specular reflections at the sample surfaces28,29. The fitting gives κ0/T ≡ a = 0.031 ± 0.008 mW K−2 cm−1 and α = 2.41 ± 0.13 for sample A. From Fig. 1b, ρ0(5 T) = 764 µΩ cm is estimated, giving the WF law expectation L0/ρ0(5 T) = 0.032 mW K−2 cm−1 with L0 = 2.45 × 10 − 8 W Ω K−2. Therefore, the WF law is verified nicely. In order to confirm this result, the data of another sample B are also plotted in Fig. 3. The fitting gives κ0/T = 0.033 ± 0.006 mW K−2 cm−1 and α = 2.62 ± 0.12. Since it has ρ0(5 T) = 755 µΩ cm, thus L0/ρ0(5 T) = 0.032 mW K−2 cm−1, the WF law is also verified in sample B. The verification of WF law above µ0H = 5 T is reasonable, because the thermal conductivity above µ0H = 5 T below 0.8 K is purely contributed from normal electrons and phonons, without other exotic excitations or magnetic scatterings.

Fig. 3: Verification of Wiedemann–Franz Law in Pr2Ir2O7.
figure 3

The thermal conductivity of two Pr2Ir2O7 single crystals at µ0H = 0 and 5 T along the [111] direction, respectively. Solid lines are the fits of the thermal conductivity data to κ/T = a + bTα−1 at 5 T below 0.3 K. The dashed line is the Wiedemann-Franz law expectation L0/ρ0(5 T) = 0.032 mW K−2 cm−1 for sample A at 5 T, which meets the extrapolated κ/T ≡ a = 0.031 mW K−2 cm−1 very well. The overlap of the 0 and 5 T curves below Ts ≈ 0.12 K suggests that the Wiedemann-Franz law is also satisfied in zero field.

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Since the thermal conductivity data at low fields collapse on the high-field data below Ts (see Fig. 2b) and the MR is less than 2% for µ0H ≤ 5 T (see Fig. 1b), it would be inferred that the WF law is obeyed at all the applied fields, even at zero field. This result is of great help to characterize the quantum criticality in Pr2Ir2O7. Many efforts have been made to describe the QCP phenomena30, among which two formalisms are highlighted: The Hertz-Millis formalism31,32 and the Kondo breakdown formalism33. In the former one, the critical fluctuations are centered at a small part of the Fermi surface, called hot spots, leaving the majority unaffected and the electrons retaining as Landau quasiparticles. The WF law will be satisfied in this type of QCP due to the integrity of the electrons. In the latter one, the hot spots cover the whole Fermi surface and the critical fluctuations reconstruct the Fermi surface abruptly. The WF law will be violated in this type of QCP due to the breakdown of quasiparticles. The verification of the WF law in Pr2Ir2O7 at its QCP unambiguously excludes the possibility of the breakdown of Landau quasiparticles, and is incompatible with the Kondo breakdown formalism. However, this does not immediately indicate a Hertz–Millis type quantum criticality in Pr2Ir2O7, because the scaling exponents in the magnetic Grüneisen ratio experiment differ from the expectations within the Hertz–Millis theory10. Therefore, new kinds of quantum criticality where the quasiparticle picture is applicable may be realized in Pr2Ir2O7, and the confirmation of the WF law puts strong constraints on the description of such QCP.

Absence of positive contributions to κ from magnetic excitations

Also, the verification of WF law in Pr2Ir2O7 demonstrates that there is no additional contribution to the thermal conductivity from mobile fermionic magnetic excitations. Furthermore, since the phonon thermal conductivity in high fields defines the upper boundary of κ in Pr2Ir2O7, there is also no positive contribution to κ from other bosonic magnetic excitations. For pyrochlore Pr2Ir2O7, one scenario to describe its possible spin liquid state is the QSI (refs. 3,13). Three topological excitations, including photon, vison, and magnetic monopole, may emerge from the ground state3. The gapless photons have a rather narrow bandwidth, about 1/1000 of the nearest-neighbor coupling Jzz (ref. 34). Since Jzz of Pr2Ir2O7 is only 1.4 K (ref. 8), the photons are definitely beyond the accessible temperature regime of our experiment. The gap of visons is about J3⊥/J2zz, in which J⊥ is the transverse exchange coupling between local moments3. Taking J⊥/Jzz = 0.3, a typical value for QSI materials34,35,36, the vison gap is estimated as 40 mK for Pr2Ir2O7, which is also likely beyond the experiment temperature range. For magnetic monopoles, however, with the gap comparable to Jzz, the thermally excited magnetic monopoles should be detectable in our temperature range. But we do not observe their positive contributions here. One possibility is that the velocity and/or mean free path of these excitations may be too small so that their contribution to κ is negligible comparing to that of phonons. Another possibility is that the ground state of Pr2Ir2O7 is not a QSI, thus the above-mentioned magnetic excitations do not exist.

Giant isotropic MTC and its origin

Now let us discuss the origin of the giant MTC in Pr2Ir2O7. In Fig. 2a, starting from the phonon thermal conductivity in high fields, the strong suppression of κ at low fields apparently comes from the scattering of phonons by the spin system through the spin-lattice coupling, either by well-defined excitations like magnetic monopoles, which accompany flips of the magnetic dipole moments pointing along the local <111> axes, or by transverse quantum fluctuations, possibly including the quadrupole moments of Pr3+. Since the Ir 5d spins are Pauli paramagnetic and polarized due to the Kondo coupling6,7, the Ir moments are not responsible for the scattering process. In order to examine these possibilities, we measure the thermal conductivity in other two different field directions and compare to the [111] direction, as shown in Fig. 4a and b. One can see that the curves in all three field directions overlap with each other. It shows that the MTC is isotropic, i.e., insensitive to the direction of field.

Fig. 4: The giant isotopic magneto-thermal conductivity and anisotropic magneto-specific heat in Pr2Ir2O7.
figure 4

a The thermal conductivities of Pr2Ir2O7 single crystal in magnetic fields H // [111] and H ⊥ Q (empty symbols) are compared with those in H ⊥ [111] and H ⊥ Q (filled symbols), where Q is the heat current. b They are also compared with those in H ⊥ [111] and H // Q (half-filled symbols). The schematic illustrations of these three geometries are also shown. The overlap of these curves in all three field directions show that the MTCs are isotropic. c The specific heats of Pr2Ir2O7 in magnetic fields H // [111] (empty symbols) are also compared with those in H ⊥ [111] (filled symbols). Contrary to the magneto-thermal conductivities, anisotropy of magneto-specific heats is clearly seen in these two field directions.

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However, it has been well established that different magnetic structures may emerge in different field directions in the dipole spin-ice formalism37. For example, threefold degenerate kagome-ice states can be stabilized for H // [111], while fields along [110] can align the spins on two independent chains. These different magnetic structures induce highly anisotropic specific heat in Dy2Ti2O7 (refs. 38,39). Especially, the different field directions allow the Dy moments to develop phase transitions by decoupling some of the Ising moments40,41. For Pr2Ir2O7, anisotropic specific heats at μ0H = 2 T and 5 T in two field directions are also observed, as shown in Fig. 4c. The low-temperature upturn is attributed to the Schottky anomaly from Pr nuclei. Since specific heats of phonons and electrons are isotropic in response to magnetic fields, the anisotropic part must be contributed from the spin system, likely the dipole moments as in Dy2Ti2O7. As a result, the scattering rate between phonons and monopoles Γp-m should be anisotropic, if one only considers the classical spin ice scenario originating from the local dipole moments. This leads to an anisotropic phonon thermal conductivity κp = 1/3Cpv2p/(Γp-m + Γother), where Cp, vp, Γother, are the phonon specific heat, phonon velocity and scattering rate from other mechanism like defects, respectively. Therefore, the isotropic MTC in Pr2Ir2O7 indicates that the scattering of phonons is unlikely from magnetic monopoles, the quasiparticles emerging from spin-ice physics due to flips of dipole moments.

Now that the well-defined magnetic excitations such as magnetic monopoles neither contribute to κ nor suppress κ in Pr2Ir2O7, the strong scattering of phonons at low fields may be associated with the transverse quantum fluctuations. Pr3+ can be represented by a 1/2-pseudospin. The z component along the local <111> direction carries a magnetic dipole moment, while the transverse component of the pseudospin corresponds to a quadrupole moment. Transverse fluctuations of moments away from the quantization axis, characteristics of the quantum spin ice, can induce quantum dynamics, and may play an important role in this system. Intuitively, the transverse fluctuations are weakened with lowering the temperature and increasing the field. For Pr2Ir2O7, a bifurcation of the field-cooled and zero-field cooled magnetic susceptibility curves at about 0.12 K suggests that a partial fraction of spins freezes6,8, which coincides with Ts of zero field in our thermal conductivity measurements. Therefore, this scenario may explain the temperature and field dependence of thermal conductivity in Pr2Ir2O7. In each field, the fluctuations are very slow below Ts, due to the partially frozen spins, so that they do not scatter phonons. The field will further weaken the phonon scatterings, thus the Ts increases with increasing field. Such a simple scenario was used to interpret the thermal conductivity of YbMgGaO4 (ref. 27). One may consider whether it can also apply to the heat transport behavior of other QSI candidates such as Pr2Zr2O7 (ref. 36), as recently pointed out by Rau and Gingras in ref. 42. Nevertheless, unlike the anisotropic fluctuations found in other spin ice materials, the fluctuations in Pr2Ir2O7 should be isotropic in response to the applied field.

Discussion

As a topical system, thermal conductivities have been measured in various spin ice materials36,43,44,45. Highly anisotropic MTCs were discovered both in Dy2Ti2O7 and in another QSI candidate with non-Kramers doublets Tb2Ti2O7 (refs. 43,44). The anisotropic MTC in Dy2Ti2O7 was considered as a consequence of the mobility of magnetic monopole excitations in spin ice43. The anisotropic MTC with respect to H // [111] and H ⊥ [111] in Tb2Ti2O7 was interpreted as a result of the scattering with phonons by anisotropic fluctuating spins44. These behaviors are quite contrary to the isotropic MTC in Pr2Ir2O7. The isotropic MTC in Pr2Ir2O7 indicates that it is not fluctuations from Ising dipole moments but fluctuations from transverse part, likely quadrupole moments, that scatter phonons strongly above Ts. This implication is important since the transverse interactions in addition to the classical Ising interactions is a key ingredient to drive the classical spin ice state into the U(1) QSI state3. For a simplest example, when introducing a transverse coupling J⊥ into a classical spin ice model, the resulting Hamiltonian

$$H = \mathop {\sum }\limits_{ < i,j > } \left[ {J_{zz}\tau _i^z\tau _j^z - J_ \bot \left( {\tau _i^ + \tau _j^ - + \tau _i^ - \tau _j^ + } \right)} \right]$$
(1)

can result in quantum fluctuations and capture the universal properties of U(1) QSI states, where pseudospin-1/2 component τz is along the local <111> direction carrying a magnetic dipole moment, and τx,y represent the quadrupole moments giving τ± = τx ± iτy (ref. 3). The possible observation of the quadrupole moment scatterings in Pr2Ir2O7 suggests the presence of quantum fluctuations, which is a characteristic of QSI state. Indeed, due to the relatively small magnitude of the Pr3+ moments which reduces the dipolar interactions, quadrupolar interactions are expected to be important in Pr-based pyrochlore compounds, and quantum fluctuations therein can melt classical spin ice states46. This has been experimentally evidenced both in Pr2Sn2O7 and Pr2Zr2O7 (refs. 47,48). To our knowledge, the roles of quadrupole moments and quantum fluctuations in Pr2Ir2O7 have not been explored experimentally. Our results present the evidence of quantum effect in this spin liquid candidate, and put its 4 f moments in line with those in other Pr-based QSI candidates.

The magnetic excitation is another characteristic besides the quantum fluctuation that determines whether a material lies in the QSI ground state. It has been claimed that the thermally excited magnetic monopoles contribute to the thermal conductivity in the QSI candidate Yb2Ti2O7, with the mean free path of such monopoles as long as 100 nm (ref. 45). More exotically, a two-gap behavior and an enhancement below 200 mK are observed in the thermal conductivity of another QSI candidate Pr2Zr2O7, which are suggested as the signatures of the three emergent excitations predicted in QSI materials36. However, here in Pr2Ir2O7, neither positive contributions nor negative contributions to κ from well-defined magnetic excitations are detected. This may be a deviation from the QSI scenario despite the presence of quantum fluctuations. However, we should note that although no contributions from magnetic excitations are detected in another QSI candidate Tb2Ti2O7 by the longitudinal heat transport study44, large thermal Hall effect is observed and indicates the existence of possible neutral spin excitations49. Therefore, thermal Hall measurements would be intriguing as a tool to determine the true ground state and emergent excitations in Pr2Ir2O7.

The above discussion is restricted in the dipole spin ice or QSI categories. However, due to the unique metallic nature of Pr2Ir2O7, which is distinct from any other insulating spin ice candidates, the interactions between Pr 4 f moments and Ir 5d itinerant electrons do complicate the microscopic description and induce further correlations besides the spin–ice correlations6,7,8,9,10. When considering the effect of conduction d electrons, another possible origin of giant isotropic MTC may be the Kondo coupling between conduction electrons and local moments. The complex f-d coupling between Pr moments and Ir electrons may well give rise to a magneto-thermal response which is itself isotropic. The Kondo coupling is also suggested as the ingredient to produce the divergence in the magnetic Grüneisen ratio measurements50,51, which is also independent of field directions. Interestingly, a scaling behavior of phonon thermal conductivities is observed, similar to the magnetic Grüneisen ratio10 (for details, see Supplementary Note 2). To check whether the f-d interactions result in the isotropic MTC, it is meaningful to measure the field orientation dependence of κ in Pr2Zr2O7, where both quantum fluctuations and three exotic magnetic excitations were claimed to be observed without the contamination of conduction electrons36,48. Our results raise the question how interactions between the local moment system of a spin-ice state and conduction electrons of a non-Fermi liquid state affect the heat transport properties and the ground state of Pr2Ir2O7.

In summary, we have measured the ultralow-temperature thermal conductivity of Pr2Ir2O7 single crystals. The Wiedemann–Franz law is verified at high magnetic fields and inferred at zero field, suggesting the normal behavior of electrons at the zero-field quantum critical point and the absence of mobile fermionic magnetic excitations. This result puts strong constraints on the description of the quantum criticality in Pr2Ir2O7. A giant isotropic magneto-thermal conductivity is found at finite temperatures, indicating that the quadrupolar interactions and quantum fluctuations may play important roles. Although further experimental work is required to investigate Pr2Ir2O7 comprehensively, our results suggest the importance of the quantum effect that deviates from the dipole spin-ice physics, and shed light on the future theoretical and experimental studies on determining the true ground state and magnetic excitations in Pr2Ir2O7.

Methods

Sample preparation

We prepare the high-quality single crystals of Pr2Ir2O7 using the KF flux methods, as described in ref. 14. A polycrystalline sample of Pr2Ir2O7 was first prepared by the solid-state reaction using stoichiometric starting materials Pr2O3(99.9%) and IrO2(99.9%). Then, single crystals were grown by combing the polycrystalline Pr2Ir2O7 with potassium fluoride (KF) flux in a ratio of 1:200. After being heated at 1100 °C for 36 h, the as-grown sample was slowly cooled down to 800 °C at a rate of 1 °C/h. The single crystals were obtained from the flux by dissolving the flux in water. The x-ray diffraction (XRD) measurement was performed on a typical Pr2Ir2O7 sample by using an x-ray diffractometer (D8 Advance, Bruker), and determined the largest surface to be the (111) plane (see Supplementary Note 1).

Specific heat measurements

The low-temperature specific heat was measured by the relaxation method in a physical property measurement system (PPMS, Quantum Design) equipped with a dilution refrigerator.

Charge and heat transport measurements

One Pr2Ir2O7 single crystal (sample A) for electric and thermal conductivity measurements was cut and polished into a rectangular shape of dimensions 0.69 × 0.38 mm2 in the (111) plane, with a thickness of 0.20 mm. Another Pr2Ir2O7 single crystal (Sample B) with the dimension of 0.72 × 0.62 × 0.20 mm3 was also measured to check the reproducibility of the transport results (for details, see Supplementary Note 3). If not specified, the data in the main text were taken on Sample A. The electric resistivities were measured using four-contact geometry in a 3He cryostat. The thermal conductivities at sub-Kelvin temperatures were measured in a dilution refrigerator, using a standard four-wire steady-state method with two RuO2 chip thermometers, calibrated in situ against a reference RuO2 thermometer. The thermal conductivities above 5 K were measured in a 4He cryostat, using a Chromel-Constantan thermocouple as the thermometer to detect the temperature difference. The electric current and heat current were applied in the (111) plane. The charge and heat transport data of the same sample were collected on the same geometry so that the geometry factor could cancel out when checking the Wiedemann-Franz law.