The specific heat of Y_{1-x}Pr_{x}Ir_{2}Zn_{20} with *x* = 0.044 has been measured down to 0.08 K in magnetic fields of *B* $\u2264$ 12 T. The 4*f* contribution to the specific heat divided by temperature *C*_{4f}/*T* diverges as −ln*T* in *B* = 0 and 2 T applied along the [100] direction. In *B* = 4 T, however, *C*_{4f}(*T*) exhibits a broad maximum, which shifts to higher temperatures with increasing *B*. This behavior can be explained by the Zeeman effect which splits the ground state doublet through the first excited triplet. The 4*f* contribution to the entropy, *S*_{4f}(*T*), in *B* $\u2265$ 4 T is well reproduced by the crystalline electric field calculation, whereas the calculation fails to reproduce the *S*_{4f}(*T*) data in *B* ≤ 2 T. We propose that the on-site interaction of the active quadrupole in the ground state doublet with two-channel conduction bands gives rise to the –ln*T* variation of *C*_{4f}/*T*.

A class of compounds containing Ce, Pr, Yb, and U ions exhibit non-Fermi liquid (NFL) behaviors in physical properties at low temperatures.^{1} The NFL behaviors have been explained in terms of two concepts. One is a magnetic instability in the vicinity of the quantum critical point, where the quantum fluctuations are divergently enhanced by the competition between the on-site single-channel Kondo effect and the inter-site Ruderman-Kittel-Kasuya-Yosida interaction.^{2} This competition have been explained by the Doniach phase diagram.^{3} The other is a single-site two-channel Kondo effect, where an impurity spin is over-screened by two-channel conduction bands.^{4} When an electric quadrupole of the impurity is regarded as the local spin, the two-channel Kondo effect is translated to the quadrupole Kondo effect.^{5} The theory of the quadrupole Kondo model predicted novel properties, e.g., the entropy of (1/2)*R*ln2 remaining at *T* = 0 and single-site NFL behaviors such as specific heat divided by temperature *C*/*T* $\u221d$ –ln*T*, quadrupole susceptibility *χ*_{Q} $\u221d$ –ln*T*, and electrical resistivity *ρ*/*ρ*_{0} $\u221d$ 1 + *A*$T$, where *ρ*_{0} is the residual resistivity and *A* is the coefficient.^{5} Despite much experimental effort on diluted Pr and U alloys,^{6–8} the single-site quadrupole Kondo effect has not been established yet. It is mostly because a superconducting impurity phase and atomic disorder affect the properties of Heusler-type Pr compounds and the crystalline electric field (CEF) ground state in diluted U systems is not well defined.

The Pr-based compounds Pr*T*_{2}*X*_{20} (*T* = transition metal, *X* = Zn, Al, and Cd) crystallizing in a cubic structure have attracted much attention.^{9} The CEF ground state of the Pr^{3+} ion with 4*f* ^{2} configuration is the non-Kramers Γ_{3} doublet with active quadrupoles. The quadrupole degrees of freedom give rise to a variety of phenomena such as long-range quadrupole order, unconventional superconductivity, and NFL behaviors.^{9} Notably, PrIr_{2}Zn_{20} shows an antiferroquadrupolar (AFQ) order at *T*_{Q} = 0.11 K below which a superconducting transition occurs at *T*_{c} = 0.05 K.^{10} Moreover, the magnetic specific heat divided by temperature, *C*_{m}/*T*, shows –ln*T* dependence and *ρ*(*T*) exhibits an upward curvature in a moderately wide temperature range above *T*_{Q}. It suggests the formation of a quadrupole Kondo lattice.^{11} Recently, we have studied its diluted Pr system Y_{1-x}Pr_{x}Ir_{2}Zn_{20} for *x* ≤ 0.44. The CEF ground state retains the non-Kramers Γ_{3} doublet with active quadrupoles.^{12,13} For *x* ≤ 0.44, *C*_{m}/*T* and the differential electrical resistivity *Δρ* are scaled well by using a characteristic temperature *T*_{0} defined as the temperature where the magnetic entropy reaches 0.75*R*ln2.^{13} Here, *T*_{0} may indicate an energy scale for the NFL behaviors due to the multi-channel Kondo model.^{5,14} Furthermore, for *x* < 0.05, the NFL behaviors of *C*_{m}/*T* $\u221d$ –ln*T* and *Δρ* $\u221d$ 1 + *A*$T$ were observed at *T* < *T*_{0}.^{13} The anomalous temperature dependences are explained by the calculation with the single-site quadrupole Kondo model.

In the present work, we report the specific heat of Y_{1-x}Pr_{x}Ir_{2}Zn_{20} for *x* = 0.044 in magnetic fields of *B* $\u2264$ 12 T. To obtain distinct NFL signals from the Pr ions, we chose a sample with *x* = 0.044, in which *C*_{m}/*T* shows the –ln*T* behavior down to 0.08 K at *B* = 0.^{13} Since the Pr ions are well isolated in this system, neither magnetic nor quadrupole order was observed. Magnetic fields were applied along the [100] direction whereby the Zeeman effect is expected to split the Γ_{3} doublet much wider than those in *B* || [110] and [111].

A single crystal was grown from a starting composition of *x* = 0.05 in Y_{1-x}Pr_{x}Ir_{2}Zn_{20} by the procedure reported in the previous papers.^{12,13} The Pr composition of the crystal could not be reliably determined by the wavelength dispersive electron-probe microanalysis with 1% resolution. Therefore, we determined the value of *x* as 0.044 by comparing the measured magnetization value at *B* = 1 T and *T* = 1*.*8 K with that calculated by using the CEF parameters for PrIr_{2}Zn_{20}.^{15} The measurements of *C* were done by a thermal relaxation method using a Quantum Design physical property measurement system (PPMS) from 0.4 to 300 K in *B* $\u2264$ 12 T. For 0.08 $\u2264$*T* $\u2264$ 0.5 K, *C* was measured in *B* $\u2264$ 6 T by using a laboratory-built system installed in a commercial Cambridge mFridge mF-ADR/100s adiabatic demagnetization refrigerator.

Figure 1 shows the temperature dependences of *C* for Y_{1-x}Pr_{x}Ir_{2}Zn_{20} with *x* = 0.044 in *B* ≤ 12 T applied along the [100] direction. In *B* = 0, *C*(*T*) is increased on cooling below 1 K and saturated to a constant value below 0.3 K. The data for *T* < 0.3 K is largely enhanced when *B* is increased to 6 T, which can be explained by considering nuclear contribution as described below. The Hamiltonian of a ^{141}Pr nucleus with the spin *I* = 5/2 in a magnetic field ** B** is represented by

where *A*_{hf} = 0.052 K is a hyperfine coupling of a Pr ion,^{16} ** J** is the total angular momentum of 4

*f*electrons,

*g*

_{N}= 1.72 the nuclear

*g*factor, and

*μ*

_{N}the nuclear magneton. By taking the isothermal magnetization

**along**

*M***into consideration, the term of $J\u22c5B$ is equivalent to $(M/gJ\mu B)\u22c5B$, where**

*B**g*

_{J}= 4/5 is the Lande’s

*g*factor for

*J*= 4 and $\mu B$ the Bohr magneton. Here, we used the values of

*M*(

*B*) measured for PrIr

_{2}Zn

_{20}(Ref. 11) as shown with the opened circles in the inset of Fig. 1. The nuclear specific heat

*C*

_{N}(

*T*) calculated for

*B*= 2, 4, and 6 T are shown in Fig. 1 with the colored solid curves. By increasing

*B*,

*C*

_{N}is enhanced and the calculated value reproduces the upturn of the experimental data for

*T*< 0.3 K in

*B*= 6 T. In

*B*= 4 T,

*C*(

*T*) shows a broad maximum at around 0.4 K, which shifts to higher temperatures with increasing

*B*and becomes a shoulder at 3 K in

*B*= 12 T. The evolution of the Schottky specific heat results from the splitting of the CEF ground state doublet by the Zeeman effect via the excited triplet as described later.

To estimate the 4*f-*electron contribution, *C*_{4f}, the calculated *C*_{N} and the phonon contribution *C*_{ph} (black solid curve in Fig. 1) were subtracted from the measured *C* data. Here, *C*_{ph} was calculated by the equation, *C*_{ph} = *xC*_{La}+(1 − *x*)*C*_{Y}, where C_{La} and C_{Y} are the *C* data for nonmagnetic compounds LaIr_{2}Zn_{20} and YIr_{2}Zn_{20}, respectively. The temperature variation of *C*_{4f} is plotted in the inset of Fig. 2. The data of *C*_{4f} in *B* = 4 T exhibit a maximum at around 0.4 K, which shifts to high temperatures as *B* is increased. The magnitude of the maximum does not change in *B* up to 12 T, which is a characteristic of the Schottky specific heat. This indicates that the ground state doublet is split by the magnetic fields for *B* ≥ 4 T. Thereby, the split state loses the quadrupole degrees of freedom.

The main panel of Fig. 2 shows the temperature dependences of *C*_{4f}/*T*. The –ln*T* dependence appears at *T* < 0.5 K in *B* = 0 and 2 T, which agrees with the prediction of the single-site quadrupole Kondo model.^{5} For *B* $\u2265$ 4 T, on the other hand, *C*_{4f}/*T* data deviate from the –ln*T* variation at low temperatures and exhibit a maximum which shifts to higher temperatures.

By integrating the *C*_{4f}/*T* data, we estimated the 4*f*-electron contribution to the entropy, *S*_{4f}, which is plotted as function of temperature in Fig. 3. Here, the measured *S*_{4f} is vertically offset by 1.40 J*/*K Pr-mol for *B* = 0 and 0.48 J*/*K Pr-mol for *B* = 2 T, respectively, so that they agree with the data of *B* = 4 T in the temperature range of *T* > 4 K. The *S*_{4f} curve moves to high temperatures with increasing *B*, which is the consequence of the shift of the maximum temperature of *C*_{4f}(*T*) as shown in the inset of Fig. 2. To estimate *S*_{4f}(*T*) in *B*, the CEF effect and the Zeeman effect are considered. The CEF Hamiltonian is represented with the CEF parameters *W* and *X*, and the Stevens operators $Onm$ as follows,

The colored solid curves in Fig. 3 are calculated by considering the CEF and Zeeman effect. Since the ionic radius of the Y^{3+} ion is smaller than that of the Pr^{3+} ion, we modified the CEF parameters for PrIr_{2}Zn_{20}, which were determined by the inelastic neutron scattering experiments,^{15} to be more relevant for the diluted Y(Pr)Ir_{2}Zn_{20} system. Here, we increased the absolute value of *W* to adjust the energy scale of the CEF. The entropy was calculated with the modified CEF parameters of *W* = −1.50 K and *X* = 0.537 as shown with the solid curves. For *B*$\u2265$ 4 T, the experimental data moderately agree with the solid curve. This suggests that the entropy of the Γ_{3} doublet at *B* = 0 is released at higher temperatures when the magnetic fields are increased above 4 T. For *B* = 2 T, however, the calculated curve is larger than the experimental data in the temperature range of *T* < 0.9 K. Moreover, in *B* = 0, the *S*_{4f} data are about 0.25*R*ln2 at the lowest temperature of 0.08 K, although the calculation remains at a constant value of *R*ln2 for *T* < 2 K. This discrepancy indicates that the magnetic entropy must be released by another mechanism except the CEF and Zeeman effects. In fact, *C*_{4f}/*T* for *B* = 0 and 2 T follows –ln*T* in the temperature range where the value of *S*_{4f} is less than *R*ln2. It suggests that the residual entropy and the –ln*T* dependence result from the active quadrupoles in the doublet. Furthermore, the anomalous behaviors agree with the prediction of the single-site quadrupole Kondo model.^{5} Therefore, the single-site quadrupole Kondo effect may arise from the on-site hybridization between the local quadrupoles and the two-channel conduction bands. To confirm that the quadrupole Kondo effect causes the NFL behaviors of *C*_{4f} and the residual entropy in *S*_{4f}, measurements of specific heat and electrical resistivity in magnetic fields are currently under investigation using different Pr concentrations.

In conclusion, we have reported the specific heat of the Pr diluted system Y_{1-x}Pr_{x}Ir_{2}Zn_{20} with *x* = 0.044 in magnetic fields of *B* ≤ 12 T applied along the [100] direction. We obtained the 4*f*-contribution to the specific heat *C*_{4f} and estimated the entropy *S*_{4f}. In *B* = 0 and 2 T, *C*_{4f}/*T* shows the –ln*T* dependence in the temperature range 0.08 < *T* < 0.5 K. In *B* ≥ 4 T, however, *C*_{4f} shows a broad maximum which shifts to higher temperatures with increasing *B*. The *S*_{4f} (*T*) data in *B* $\u2265$ 4 T can be reproduced by considering the splitting of the Γ_{3} doublet due to the Zeeman effect via the first excited Γ_{4} triplet. In magnetic fields below 2 T, however, the calculated entropy is much larger than that evaluated from the experimental data. Therefore, the –ln*T* behaviors of *C*_{4f}/*T* in *B* = 0 and 2 T may arise from the single-site quadrupole Kondo effect, where the active quadrupoles of the Γ_{3} doublet hybridize with the two-channel conduction bands.

We are grateful to S. Hoshino, J. Otsuki, H. Kusunose, A. Tsuruta, K. Miyake, R. J. Yamada, G. B. Park, K.-I. Imura, and A. Tanaka for valuable discussions. The specific heat measurements with the PPMS and the mFridge mF-ADR 100/s refrigerator were performed at N-BARD, Hiroshima University. This work was financially supported by Grants-in-Aid from MEXT/JSPS of Japan, Nos., JP15KK0169, JP15H05886, JP16H01076 (J-Physics), and JP18H01182.