Giant isotropic magneto-thermal conductivity of metallic spin liquid candidate Pr2Ir2O7 with quantum criticality

Spin liquids are exotic states with no spontaneous symmetry breaking down to zero-temperature because of the highly entangled and fluctuating spins in frustrated systems. Exotic excitations like magnetic monopoles, visons, and photons may emerge from quantum spin ice states, a special kind of spin liquids in pyrochlore lattices. These materials usually are insulators, with an exception of the pyrochlore iridate Pr2Ir2O7, which was proposed as a metallic spin liquid located at a zero-field quantum critical point. Here we report the ultralow-temperature thermal conductivity measurements on Pr2Ir2O7. The Wiedemann–Franz law is verified at high fields and inferred at zero field, suggesting no breakdown of Landau quasiparticles at the quantum critical point, and the absence of mobile fermionic excitations. This result puts strong constraints on the description of the quantum criticality in Pr2Ir2O7. Unexpectedly, although the specific heats are anisotropic with respect to magnetic field directions, the thermal conductivities display the giant but isotropic response. This indicates that quadrupolar interactions and quantum fluctuations are important, which will help determine the true ground state of this material.


Supplementary Note 2: Scaling of thermal conductivities
Interestingly, an unusual scaling behavior κ -κe ∼ H 3 F(T/H 4/3 ) is observed, as shown in Supplementary   Fig. 2. κe = L0T/ρ(H) is the electron thermal conductivity at each field. Thus, κ -κe represents the thermal conductivity of phonon part, since there is no magnetic thermal conductivity in Pr2Ir2O7, as illustrated in the main text. F(x) is the scaling function. It is also held for sample B (see Supplementary   Fig. 4b). Note that the x variable T/H 4/3 in the function F(x) is exactly the same as the scaling law found in the magnetic Grüneisen ratio. Observation of the scaling law in the thermal conductivity is appealing since it is rather rare that the heat transport data scale as a function of a single parameter. It gives a new viewpoint towards such quantum magnets that we hope will stimulate the theoretical study.

Supplementary Note 3: Reproducibility of heat transport results
We performed transport measurements on another Pr2Ir2O7 single crystal (Sample B), and obtained similar results to Sample A. Sample B was cut and polished into a rectangular shape with length l = 0.72 mm, width w = 0.26 mm and thickness t = 0.20 mm. The electric and thermal conductivity were measured by standard four-wire method.
Supplementary Fig. 3a shows the temperature dependence of the longitudinal resistivity of Sample B at zero field. The Kondo effect is also evidenced by the minimum at 45 K (see the inset of Supplementary Fig. 3a), confirming the good quality of our sample. The plots of ρ(T) at 0, 3, and 5 T are presented in Supplementary Fig. 3b. By extrapolating to the zero-temperature limit, we can get the residual resistivity ρ0 = 757, 743, and 755 µΩ cm for µ0H = 0, 3, and 5 T, respectively.

Supplementary Note 4: High-temperature thermal conductivities
The high-temperature thermal conductivities of Sample B are shown in Supplementary Fig. 5a. The magnetic fields are applied along the [111] direction. A broad peak is observed at around 10 K at each field. The /T data at 0H = 5 T still overlap with those at 7 T below 9 K, indicating that besides the electron thermal conductivity, the  above 5 T is purely contributed by phonons without magnetic scatterings until thermal fluctuations of spins dominate over polarization effects by magnetic fields at even higher temperatures (T > 9 K). Unexpectedly, contrary to the ultralow-temperature results, the zero-field data are the largest among the data at other fields and the  data are suppressed more strongly with increasing the fields. In other words, negative MTCs are observed at temperatures above 5 K, which is opposite to the giant positive MTCs at sub-Kelvin region. Therefore, there must exist a crossover from the positive MTCs at low temperatures to the negative MTCs at high temperatures between 1 K and 5 K, as shown in the yellow part of Supplementary Fig. 5b.
This crossover region is in good correspondence with the energy scale of renormalized interactions between Pr moments, which is w = 1.7 K. When T < w, the susceptibility and Hall resistivity start to exhibit logarithmically diverging behaviors and the anomalous Hall effect emerges (Ref. 6-8 in the main text). The magnetic specific heat also exhibits a broad peak at this temperature (Ref. 6 in the main text). These all indicate a spin-liquid state below this temperature. This particular energy scale comes from the partial screening of Pr moments due to the Kondo effect, which renormalizes the AFM interaction from the RKKY interaction energy scale of about 20 K to w = 1.7 K. As a result, w is a critical temperature dividing two separate states: when T < w, the underscreened moments form a correlated spin liquid, where "2-in, 2-out" configurations are the ground states; when T > w, the Kondo effect starts to lead to the screening of the 4f moments (Ref. 6,8 in the main text). Therefore, the crossover at this temperature region may be due to the different physical states above and below w. In this context, the origin of the negative MTC at high temperatures must be different from the positive MTC at low temperatures, and should be related to the Kondo screening of 4f moments. Note that the negative MTC itself is rather unexpected and interesting, and calls for a careful research and analysis in the future.