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1 Materials and Methods Single crystals of Pr 2 Ir 2 O 7 were grown by a flux method [S1]. Energy dispersive x-ray analysis found no impurity phases, no inhomogeneities and a ratio between Pr and Ir of 1:1.03(3). Single crystal four-circle x-ray diffraction measurements confirmed its single phase with a pyrochlore crystallographic structure [S1]. Transport and magnetization properties were measured at low temperatures down to 0.03 K and under magnetic fields up to 7 T. For temperatures T 0.5 K, the field dependence was measured with a slow fixed rate of 1 Oe/s, while for T 0.7 K, the field was held for 2 min prior to each measurement. For each region in temperature, we used both procedures obtaining basically the same results. For each value in temperature, the field dependence was obtained after cooling the sample at zero field, i.e. the temperature was raised to 2 K and then the sample was cooled down again to a target temperature at zero field. Transport measurements were performed by using a conventional four-probe technique with the current applied along the [110] direction. To obtain the Hall voltage, the superimposed magnetoresistivity signal was subtracted by reversing the field direction. For the field dependence shown in Fig. 3a within the main text, the field was initially swept from zero up to 7 T, and then a hysteresis loop was measured from 7 T to 7 T. In Fig. 3a in the main text, we show the resulting Hall conductivity only for positive field values. The virgin curve between 0 and 7 T is not displayed because of the uncertainty in the magnitude of the superimposed magnetoresistivity signal. The magnetization above and below 2 K was measured respectively, with a commercial SQUID magnetometer and the Faraday method in a dilution refrigerator [S2]. The nonlinear susceptibility was obtained by fitting at each temperature the M(B) curves to the expression M = χ 1 B + χ 3 B 3 /3!. When analyzing our results for T T f, we added a constant term M 0 to describe the effect of the spin freezing. Macroscopically broken time reversal symmetry The macroscopically broken time reversal symmetry means that the time-reversal operation, which inverts the spin and orbital angular momenta and the wavevector, S S, L L, and k k, as well as the fictitious magnetic field b nk b n k, should not be compensated by any other symmetry operations of the crystal, e.g., translation, spatial inversion, reflection, rotation, and their combinations. 1

2 Macroscopically broken time reversal symmetry The macroscopically broken time reversal symmetry means that the time-reversal operation, which inverts the spin and orbital angular momenta and the wavevector, S S, L L, and k k, as well as the fictitious magnetic field b nk b n k, should not be compensated by any other symmetry operations of the crystal, e.g., translation, spatial inversion, reflection, rotation, and their combinations. Hall and longitudinal resistivities Figure S1 shows the temperature dependence of the Hall resistivity ρ H (left axis) and the longitudinal resistivity ρ (right axis) under a magnetic field of B = 0.05 T along the [111] direction. ρ H clearly exhibits a bifurcation between the zero-field cooled (ZFC) and field-cooled (FC) processes below 1.5 K, while ρ does not show any bifurcation. Correspondingly, a bifurcation is visible in σ H but not in σ as shown in Fig. 2a and in the inset of Fig. 2b within the main text because of the small Hall angle Θ H Metamagnetic transition and 2-in, 2-out correlation Figure S2 shows the field 3 dependence of the magnetization along 570 the [100], [110], and [111] directions at 0.1 K. The clear anisotropy observed at high fields ρ is fully consistent with an Isinglike anisotropy for Pr 4f moments ZFC &[S3,S4]. FC As shown in the inset of Fig. S2 and in Fig. 3b within the main text, our measurements at 0.03 and 0.06 K clearly reveal a first-order metamagnetic 2 ρ 560 transition at B c 2.3 T for fields H along the [111] direction. The associated anomaly is observed already at 0.1 K in the M vs. B curve for fields along the [111] direction (Fig. S2). No anomaly is seen for fields applied along the other two crystallographic directions. The fact that the metamagnetic 1 ZFC transition is observed only for fields 550along the [111] direction is a clear evidence for the 2-in, 2-out spin-configuration of Pr 4f moments, and for a FM coupling between the nearest neighbors. In general, four Ising moments on a tetrahedron form two distinct configurations, 0 depending on the sign of the nearest-neighbor FC 540 interaction: an allin, all-out and the 2-in, 2-out (Fig. 1b in the main text) spin-configuration, respectively for antiferromagnetic (AF) and ferromagnetic B = 0.05 T (FM) interactions. Locally, the all-in, all-out state has no net magnetization. Therefore, // [111] to induce a finite magnetization for fields applied along each one of the crystallographic directions, a 1metamagnetic10 transition would have to occur. However, this is not what is observed in our experiment. In contrast, for the 2-in, 2-out spinconfiguration, a metamagnetic transition would T (K) occur only for fields along the [111] direction ρ (µω µωcm) H Figure S1 Temperature dependence of the Hall 2 resistivity ρ H (left axis) and the longitudinal resistivity ρ (right axis) under a magnetic field of B = 0.05 T along the [111] direction. Both the zero-field cooled (ZFC) and field-cooled (FC) results are plotted. ρ (µω µωcm) as precisely seen here and in the spin ice compound Dy 2 Ti 2 O 7 [S5], stabilizing the 3-in, 1-out configuration at high fields. In addition, a FM nearest-neighbor coupling is inferred from the analysis of the RKKY interaction based on a simple Fermi gas model using a carrier concentration 20 %/Pr obtained from Hall effect measurements [S6]. A first-principles calculation predicts a single Fermi surface also with a carrier concentration of 20 %/Pr [S7]. The AF Curie-Weiss temperature observed in our experiments may originate from quantum melting of spin ice [S8,S9], and/or contributions from longer range interaction, for instance, RKKY interaction as indicated by a recent Monte-Carlo simulation [S10] using the experimentally extracted Fermi vector k F. Finally, we note that an analogy with dipolar spin-ice systems is significantly useful. The magnetization M c just below the metamagnetic transition field B c is known to reach M c = g J J z /3 in the kagome-ice state [S5,S11] (Fig. 1c in the main text). Using the Landé factor g J for Pr 3+ and the effective total angular momentum J z obtained from p eff = g J J z = 2.7µ B /Pr [S3], one estimates a value of 0.9 µ B /Pr for M c, which is close to the observed value 2

3 Metamagnetic transition and 2-in, 2-out correlation Figure S2 shows the field dependence of the magnetization along the [100], [110], and [111] directions at 0.1 K. The clear anisotropy observed at high fields is fully consistent with an Isinglike anisotropy for Pr 4f moments [S3,S4]. As shown in the inset of Fig. S2 and in Fig. 3b within the main text, our measurements at 0.03 and 0.06 K clearly reveal a first-order metamagnetic transition at B c 2.3 T for fields along the [111] direction. The associated anomaly is observed already at 0.1 K in the M vs. B curve for fields along the [111] direction (Fig. S2). No anomaly is seen for fields applied along the other two crystallographic directions. The fact that the metamagnetic transition is observed only for fields along the [111] direction is a clear evidence for the 2-in, 2-out spin-configuration of Pr 4f moments, and for a FM coupling between the nearest neighbors. In general, four Ising moments on a tetrahedron form two distinct configurations, depending on the sign of the nearest-neighbor interaction: an allin, all-out and the 2-in, 2-out (Fig. 1b in the main text) spin-configuration, respectively for antiferromagnetic (AF) and ferromagnetic (FM) interactions. Locally, the all-in, all-out state has no net magnetization. Therefore, to induce a finite magnetization for fields applied along each one of the crystallographic directions, a metamagnetic transition would have to occur. However, this is not what is observed in our experiment. In contrast, for the 2-in, 2-out spinconfiguration, a metamagnetic transition would occur only for fields along the [111] direction T = 0.1 K [100] 1 [111] M (µ B /Pr) 0.5 M (µ B /Pr) B (T) B (T) B // [111] 0.03 K [110] Figure S2 Field dependence of the magnetization M(B) for fields along the [100], [110], and [111] directions at 0.1 K. Inset: Hysteresis in the magnetization M(B) at the metamagnetic transition for fields along the [111] direction at 0.03 K. Theoretical calculation For the tight-binding calculation, we took into account four different angles of rotation of a IrO 6 octahedron and the associated triply degenerate 5d t 2g orbitals in the local coordinate frames. No significant effect was found from the small splitting of the Ir 5d levels due to the trigonal crystal-field of the pyrochlore structure. The orbital-dependent dd electron transfer between 3

4 T (K) Figure S1 Temperature dependence of the Hall resistivity ρ H (left axis) and the longitudinal resistivity ρ (right axis) under a magnetic field of B = 0.05 T along the [111] direction. Both the zero-field cooled (ZFC) and field-cooled (FC) results are plotted. as precisely seen here and in the spin ice compound Dy 2 Ti 2 O 7 [S5], stabilizing the 3-in, 1-out configuration at high fields. In addition, a FM nearest-neighbor coupling is inferred from the analysis of the RKKY interaction based on a simple Fermi gas model using a carrier concentration 20 %/Pr obtained from Hall effect measurements [S6]. A first-principles calculation predicts a single Fermi surface also with a carrier concentration of 20 %/Pr [S7]. The AF Curie-Weiss temperature observed in our experiments may originate from quantum melting of spin ice [S8,S9], and/or contributions from longer range interaction, for instance, RKKY interaction as indicated by a recent Monte-Carlo simulation [S10] using the experimentally extracted Fermi vector k F. Finally, we note that an analogy with dipolar spin-ice systems is significantly useful. The magnetization M c just below the metamagnetic transition field B c is known to reach M c = g J J z /3 in the kagome-ice state [S5,S11] (Fig. 1c in the main text). Using the Landé factor g J for Pr 3+ and the effective total angular momentum J z obtained from p eff = g J J z = 2.7µ B /Pr [S3], one estimates a value of 0.9 µ B /Pr for M c, which is close to the observed value M c 0.8 µ B /Pr. From B c one can also estimate the effective nearest-neighbor FM coupling Jff eff for spin-ice systems as Jff eff (1/3)g J J z µ B B c /k B. With the observed B c =2.3 T, one obtains Jff eff 1.4 K for Pr 2 Ir 2 O 7, which is close to the temperature ( 2 K) where a peak is seen Figure in the S2 magnetic Field dependence specific heat of the C m magnetization as in spin-icem(b) systems for[s11] fields(fig. along 2cthe in the [100], main [110], text). and [111] directions at 0.1 K. Inset: Hysteresis in the magnetization M(B) at the metamagnetic transition for fields along the [111] direction at 0.03 K. Theoretical calculation 3 For the tight-binding calculation, we took into account four different angles of rotation of a IrO 6 octahedron and the associated triply degenerate 5d t 2g orbitals in the local coordinate frames. No significant effect was found from the small splitting of the Ir 5d levels due to the trigonal crystal-field of the pyrochlore structure. The orbital-dependent dd electron transfer between the nearest-neighbor Ir sites was estimated from the Slater-Koster table [S12]. The amplitude was chosen so that the total bandwidth becomes of the order of 3 ev as obtained by the firstprinciples band calculation [S7], which also uncovered a single electron-like Fermi surface with a carrier concentration comparable to the experimental estimate of 0.2 per Ir. The relativistic spin-orbit interaction for the 5d electrons is large, and it has finite matrix elements within the t 2g manifold. We took the spin-orbit coupling strength of λ =0.2 ev, which was also estimated from band structure calculations. The effective AF Kondo coupling J fd to the Pr 4f moments was estimated to be 4 mev. The calculations have been performed with wavevector meshes for the zero-field-magnetic configuration shown in Fig. 1d in the main text. An energy broadening of 0.05 ev has been introduced for practical calculations, which is comparable to the relaxation rate obtained from the observed longitudinal conductivity σ Ω 1 cm 1. The results are shown in Fig. S3 (left axis) as a function of the number of 5d electrons per Ir site. For the expected Ir 4+ configuration with 5d 5, it gives 0.2 Ω 1 cm 1 for the zero-field spin configuration shown in Fig. 1d in the main text. 4 4

5 5 4 High magnetic field σ H (Ω -1 cm -1 ) Zero magnetic field Number of d electrons/ir Figure S3 Theoretical calculation of the Hall conductivity σ H as a function of the number of the 5d electrons per Ir site for the magnetic configurations under zero magnetic field shown in Figure 1d of the main text (left axis). Also shown is the Hall conductivity calculated for the 3-in, 1-out configuration at high magnetic field (right axis). -10 The uniform orbital moment at zero magnetic field can be calculated from L = m h 2 dk (2π) 3 n n T H (µ ε n (k))a nn (k) a n n(k)(2µ ε n (k) ε n (k)) (1) with the electron mass m [S13,S14]. Note that only the interband matrix elements with the Fermi level being located in between the two band energies contribute to L. Therefore, large parts giving rise to the intrinsic AHE are cancelled out by each other. In particular, for the configuration shown in Fig. 1d in the main text, the orbital magnetization along the [111] field direction are obtained as µ B /PrIrO

6 Supplementary References [S1] Millican, J. N. et al. Mater. Res. Bull. 42, 928 (2007). [S2] Sakakibara, T., Mitamura, H., Tayama, T. & Amitsuka, H. Jpn. J. Appl. Phys. 33, 5067 (1994). [S3] Nakatsuji, S. et al. Phys. Rev. Lett. 96, (2006). [S4] Machida, Y. et al. J. Phys. Chem. Solids 66, 1435 (2005). [S5] Sakakibara, T., Tayama, T., Hiroi, Z., Matsuhira, K. & Takagi, S. Phys. Rev. Lett. 90, (2003). [S6] Machida, Y. et al. Phys. Rev. Lett. 98, (2007). [S7] Onoda, S., Ishii, F., & Ozaki, T. paper presented at the 63th Autumn Meeting of the Japanese Physical Society, Morioka, Japan, 22 September [S8] Molavian, H. R., Gingras, M. J. & Canals, B. Phys. Rev. Lett. 98, (2007). [S9] Onoda, S. & Tanaka, Y. ArXiv: [S10] Ikeda, A. & Kawamura, H. J. Phys. Soc. Jpn. 77, (2008). [S11] Bramwell, S. T. & Gingras, M. J. P. Science 294, 1495 (2001). [S12] Harrison, W. A. Electronic Structure and the Properties of Solids (Dover, New York, 1989). [S13] Xiao, D., Shi, J. & Niu, Q. Phys. Rev. Lett. 95, (2005). [S14] Ceresoli, D., Thonhauser, T., Vanderbilt, D. & Resta, R. Phys. Rev. B 74, (2006). 6