**■ Spontaneous Hall Effect as Evidence of Hidden Time-Reversal Symmetry Breaking in a Frustrated Magnet
**

The anomalous Hall effect (AHE) is a fundamental transport phenomenon where
the electric current generates the transverse voltage drop in the normal
plane to the spontaneous magnetization *M* in ferromagnets. This issue has attracted revived interest because of
its topological and dissipationless character, and its potential application
in spintronics. In particular, it has been shown that the intrinsic mechanism
of the AHE can capture the dominant part in moderately dirty metals. This
intrinsic AHE can be understood in terms of the adiabatic motion of the
Bloch electrons under the electric field *E*, which acquire a quantum geometrical phase, the Berry-phase curvature
*b _{nk}*, in the wavevector (

*k*) space because of the relativistic spin-orbit interaction and the spin magnetization. This

*b*acts as a fictitious magnetic field in the

_{nk}*k*space and bends the orbital motion of electrons as in the case of the Lorentz force due to the real magnetic field

*B*. Thus, it causes the AHE characterized by the finite Hall conductivity s

_{H}at

*B*= 0.

Notably, the condition for observing the AHE at *B* = 0 is the *macroscopically* broken time reversal symmetry (*T*), which ensures a nonzero average of *b _{nk}* over the occupied Bloch states. It does not necessarily require a finite ferromagnetic spin alignment, but a noncoplanar spatial distribution of spins. Unconventional scenarios directly relying on a noncoplanar spin texture with the uniform scalar spin chirality have been addressed for ferromagnets. The macroscopically broken time-reversal symmetry and a resultant nonzero s

_{H}in the absence of a uniform spin magnetization have also been proposed in antiferromagnetic (AF) states and spin-liquid states with the scalar spin chirality. In this exotic example of a chiral spin-liquid, the uniform scalar spin chirality shows the LRO, but the spin magnetic moment does not. However, the AHE at zero field has never been observed to date in the absence of the uniform spin magnetization associated with the ferromagnetism or the spin freezing.

We have recently discovered the macroscopically *T* broken spin-liquid state in a metallic magnet. In particular, we observe
the spontaneous Hall effect in the geometrically frustrated Kondo lattice
Pr_{2}Ir_{2}O_{7} even above its spin freezing temperature *T*_{f} ∼ 0.3 K. A clear hysteresis is observed in the Hall conductivity around zero field below the onset temperature *T*_{H} ∼ 1.5 K, whereas that in the magnetization curve appears only below *T*_{f} within an experimental accuracy. Namely, a large anomalous Hall conductivity
s_{H} is found even at zero field where the magnetization practically vanishes (Fig. 1), in sharp contrast to the conventional AHE in ferromagnets. This indicates an emergence of a hidden order at TH that macroscopically breaks the time reversal symmetry without invoking a LRO of dipolar spins. The phenomenon may be understood in terms of a formation of the uniform spin chirality out of “2-in, 2-out” configurations (inset of Fig. 1) of localized magnetic moments of Pr^{3+} ions in an analogy to spin-ice systems.

[1] Yo Machida, Satoru Nakatsuji, Shigeki Onoda, Takashi Tayama, Toshiro Sakakibara, Nature,Fig. 1: Temperature dependence of the remnant Hall conductivity s_{H}(B=0) (left axis) and remnant magnetizationM(B=0) (right axis) at zero field for the pyrocholore oxide Pr_{2}Ir_{2}O_{7}, obtained after a field sweep down from 7 T in the hysteresis loop measurements. Inset: "2-in, 2-out" configuration of four ＜111＞ Ising spins on a tetrahedron unit of the pyrochlore lattice.

**463**, 210 (2010).

**■ Magnetic control of chiral domains in the chiral spin states of Pr _{2}Ir_{2}O_{7}**

Electronic transport under the influence of a background spin texture opened fundamental themes in physics, such as giant/colossal magnetoresistive effects, and the anomalous Hall-effect (AHE). These magnetotransport phenomena provided the conceptual basis for current research in spintronics and have so far been intensively investigated in ferromagnetic, transition metals, oxides, and semiconductors. The recent discovery of a chiral spin liquid in the metallic frustrated magnet Pr2Ir2O7 through the AHE at zero magnetic field below 1.5 K, provides a unique laboratory for studying novel magnetotransport phenomena in the absence of magnetic dipole long range order [1-3]. In this study, we uncover a strong anisotropy in both the AHE and the magnetoresistance of the chiral spin states observed in cubic single crystals of Pr2Ir2O7 [4]. Hysteresis in the AHE appearing below 1.5 K, is most pronounced when the magnetic field is cycled along the [111] direction, as shown in Fig. 2. This indicates that delocalized orbital currents generating an associated magnetic moment and which are linearly coupled to the chiral order parameter breaking the parity and time-reversal invariance, circulate within the [111] kagome planes. The hysteresis loop in the AHE characterizing the chiral spin state, closes at a metamagnetic transition critical field Bc observed only for fields along the [111] direction. These observations pose the intriguing possibility that Dirac magnetic monopoles and antimonopole, which are deconfined and interact through the magnetic Coulomb law in the classical dipolar spin ice, now have a finite quantum-mechanical average in the zero-field chiral spin-liquid state which is dominated by 2-in, 2-out spin-ice configurations. Above Bc and only for fields applied along the [111] direction, we observe a large positive magnetoresistance and Shubnikov de Haas oscillations. This suggests that densely populated magnetic monopoles and antimonopoles, reconstruct the electronic structure of the Ir conduction electrons in the basal kagome planes.

[1] S. Nakatsuji, Y. Machida, Y. Maeno, T. Tayama, T. Sakakibara, J. van Duijn, L. Balicas, J. N. Millican, R. T. Macaluso, and Julia Y. Chan, Physical Review LettersFig. 2: Hall resistivityr_{xy}=R_{xy}t, wheretis the sample thickness, for a Pr_{2}Ir_{2}O_{7}single crystal, as a function of magnetic fieldHapplied respectively along the (a) [111], (b) [110], and (c) [100] crystallographic directions and forT= 500 mK, respectively. These traces were measured for increasing (blue traces) and decreasing (gray traces) field scans. Notice how the hysteresis depends strongly on the orientation of the magnetic-field. (d) Magnetization as a function ofH, for increasing and decreasing field scans, and respectively forT= 0.06 and 0.5 K. Notice the absence of hysteresis among the traces acquired atT= 0.5 K, but its presence in the Hall response acquired at the same temperature, as shown in (a). (d) A schematic picture to show the fictitious field// [111], generating delocalized orbital currents within the [111] kagome planes, which are linearly coupled to the chiral order parameter breaking the parity and time-reversal invariance.K

**96**, 087204 (2006).

[2] Y. Machida, S. Nakatsuji, Y. Maeno, T. Tayama, T. Sakakibara, S. Onoda, Physical Review Letters

**98**, 057203 (2007).

[3] Yo Machida, Satoru Nakatsuji, Shigeki Onoda, Takashi Tayama, Toshiro Sakakibara, Nature

**463**, 210 (2010).

[4] L. Balicas, S. Nakatsuji, Y. Machida, and S. Onoda, Physical Review Letters

**106**, 217204 (2011).

**■
Quantum criticality in a metallic spin liquid system Pr _{2}Ir_{2}O_{7}
**

At finite temperatures, electronic magnetic moments in magnetic materials are thermally fluctuating
and one may have magnetic phase transitions accompanied by critical thermal fluctuations.
This transition temperature can be lowered by controlling external parameters like pressure and magnetic field.
In temperature near absolute zero, thermal fluctuations will be reduced, allowing quantum fluctuations to take effect.
When the quantum fluctuations are strong enough, one may have a phase transition at absolute zero, namely quantum phase transition.
In the vicinity of the quantum phase transition point, anomalous magnetic and metallic behaviors may be
observed, such as high-*T*_{c} superconductivity in the cuprates and iron pnictides,
and unconventional superconductivity in the heavy fermion systems.
These phenomena are stemmed from the anomalous metallic state around the quantum critical point.

Fig. 3: Schematic phase diagram of temperatureTvs. external parameters in strongly correlated electron systems.

Intensive studies on quantum critical phenomena have been done on the groups of compounds so called heavy fermion systems.
It is well known that the tuning of the external parameters by applying pressure or magnetic field suppresses the magnetic order
down to absolute zero, leading to the emergence of quantum criticality along with the evolution of exotic phase such as
anomalous superconductors. Here we have revealed the existence of quantum criticality in a metallic spin liquid system
Pr_{2}Ir_{2}O_{7}, which may provide a key insight on the mechanism of
large spontaneous anomalous Hall effect observed in its spin liquid state [1-4].
In this material, Pr ions form the pyrochlore structure, where the vertices of the corner sharing network of tetrahedra are
occupied by Pr ions' Ising type spins, and the ferromagnetic interaction between them causes geometrically frustrated spin ice state.
As a result, the ground state is considered to be spin liquid, having no dipolar magnetic order. In addition,
the spontaneous Hall effect appears in this spin liquid phase, suggesting the emergence of a chiral spin liquid state which has finite spin chirality.

In this study, we performed the precise magnetocaloric effect measurements on Pr_{2}Ir_{2}O_{7} [5].
The magnetic Grüneisen ratio, which measures the change of temperature with magnetic field under adiabatic conditions,
generally is known to diverge at the quantum critical point, and thus this physical quantity is very sensitive to the existence of quantum criticality.

The results show the divergence of magnetic Grüneisen ratio, indicating the existence of quantum criticality.
In addition, the critical scaling which examines the position of a quantum critical point, suggests that
Pr_{2}Ir_{2}O_{7} has a quantum critical point at zero magnetic field. Therefore,
this system is located at a zero-field quantum critical point without tuning of any external parameter.

Fig. 4: Temperature dependence of the magnetic Grüneisen ratioΓ_{H}of Pr_{2}Ir_{2}O_{7}. The divergent behaviour ofΓ_{H}appears with decreasing of a magnetic field.

Fig. 5: Critical scaling of the magnetic Grüneisen ratioΓ_{H}for Pr_{2}Ir_{2}O_{7}. This analysis evidences a zero-field quantum critical point in the material.

In summary, we have found the zero-field quantum criticality in Pr_{2}Ir_{2}O_{7} as indicated by
the divergent Grüneisen ratio and zero-field quantum critical point. All these results suggest that
the chiral spin liquid phase accompanied with the spontaneous Hall effect emerges
under the influence of the quantum criticality led by geometrical frustration.
This supports the manifestation of novel type of "quantum critical spin liquid" states.
Our study highlights the spin ice as the parent state of the chiral spin liquid state,
inducing the spontaneous Hall effects and many other intriguing quantum magnetic phenomena
as represented by coherent propagation of monopoles. Our findings of quantum criticality
which emerges in a spin liquid state of a highly frustrated metal require further studies,
both experimental and theoretical, on the group of frustrated metals.

**96**, 087204 (2006).

[2] Y. Machida, S. Nakatsuji, Y. Maeno, T. Tayama, T. Sakakibara, and S. Onoda, Physical Review Letters

**98**, 057203 (2007).

[3] Y. Machida, S. Nakatsuji, S. Onoda, T. Tayama and T. Sakakibara, Nature

**463**, 210 (2010).

[4]L. Balicas, S. Nakatsuji, Y. Machida, and S. Onoda, Physical Review Letters

**106**, 217204 (2011).

[5] Y. Tokiwa, J. J. Ishikawa, S. Nakatsuji, and P. Gegenwart, Nature Materials

**13**, 356-359 (2014).

**■
Strongly Correlated Zero-gap Semiconductor Pr _{2}Ir_{2}O _{7}
**

In the field of the solid state physics, materials exhibiting novel physical properties are vigorously explored. Zero-gap semiconductors are one fascinating group of materials where topological functionalities lead to high carrier mobility and the quantum Hall effect. It is known that electrons behave as if they are massless in materials such as graphene because of the linear band dispersion near the point where the valence and the conduction bands come in contact with each other. For graphene, new phenomena were discovered one after another and it became the subject of the Nobel Prize in Physics in 2010. So far, the physics of zero-gap semiconductors have only been studied in materials where the interaction between electrons is weak.

An example of a zero-gap structure is a Luttinger semimetal with quadratic band touching whose band dispersion is parabolic near the band touching point as illustrated in Fig. 6. It was predicted more than 40 years ago that materials in a Luttinger semimetal state would show novel electronic states because of the strong electronic correlations that are unobtainable in conventional metals. However, in materials known so far, such as α?Sn and HgTe, it has been difficult to identify experimentally the effects of electronic correlations because the effective mass of electrons is small and hence the electronic correlations are weak.

To clarify the effect of the strong electronic correlations, we focused on
Pr_{2}Ir_{2}O_{7} [1]. It is already known that Pr _{2}Ir_{2}O_{7} is a Luttinger semimetal with
quadratic band touching and that the effective mass of electrons is about 6
times larger than the mass of the free electron in vacuum [2]. We therefore
carried out a terahertz spectroscopy study on the Pr_{2}Ir _{2}O_{7} thin films and observed a very large dielectric constant of about 180
at a temperature of 5 K as shown in Fig. 7 [3]. This value is several tens
of times larger than that of zero-gap semiconductors (e.g. α-Sn and HgTe)
known so far. Additionally, in a Luttinger semimetal state, the dielectric
constant is a measure of the scale of electronic correlations. By using
this fact, when the magnitude of the electronic correlations is estimated
from the dielectric constant, the scale of electronic correlations is about
2 orders of magnitude larger than the kinetic energy.

We have thus demonstrated that electronic correlations are indeed very strong in a Luttinger semimetal with quadratic band touching. In the future, it is expected that further understanding of the role of electronic correlations in determining the physical properties of zero-gap semiconductors will lead to the creation of novel metallic states and new functional materials.

References

[1] T. Ohtsuki, Z. Tian, A. Endo, M. Halim, S. Katsumoto, Y. Kohama, K. Kindo, S. Nakatsuji, and M. Lippmaa, arXiv:1711.07813 (2017).

[2] T. Kondo, M. Nakayama, R. Chen, J. J. Ishikawa, E.-G. Moon, T.
Yamamoto, Y. Ota, W. Malaeb, H. Kanai, Y. Nakashima, Y. Ishida, R. Yoshida,
H. Yamamoto, M. Matsunami, S. Kimura, N. Inami, K. Ono, H. Kumigashira, S.
Nakatsuji, L. Balents, and S. Shin, Nat. Commun. **6**, 10042
(2015).

[3] B. Cheng, T. Ohtsuki, D. Chaudhuri, S. Nakatsuji, M. Lippmaa, and N. P.
Armitage, Nat. Commun. **8**, 2097 (2017).

Authors

T. Ohtsuki, B. Cheng^{a}, D. Chaudhuri^{a}, S. Nakatsuji,
M. Lippmaa, and N. P. Armitage^{a}

^{a}
The Johns Hopkins University

Figure Caption

Fig. 6. Band structure of a Luttinger semimetal, which is a zero-gap semiconductor. The valence band, which is filled with electrons (blue spheres) and the empty conduction band both have a three-dimensional parabolic shape, and are in contact with each other at a single point close to the Fermi level.

Fig. 7. Temperature dependence of the dielectric constant. The low-temperature value is several tens of times larger than that of known zero-gap semiconductors (e.g. α?Sn and HgTe). The dielectric constant becomes larger when the Fermi level approaches the band touching point.