Abstract
Kondo lattice materials, where localized magnetic moments couple to itinerant electrons, provide a very rich backdrop for strong electron correlations. They are known to realize many exotic phenomena, with a dramatic example being recent observations of quantum oscillations and metallic thermal conduction in insulators, implying the emergence of enigmatic chargeneutral fermions. Here, we show that thermal conductivity and specific heat measurements in insulating YbIr_{3}Si_{7} reveal emergent neutral excitations, whose properties are sensitively changed by a fielddriven transition between two antiferromagnetic phases. In the lowfield phase, a significant violation of the WiedemannFranz law demonstrates that YbIr_{3}Si_{7} is a charge insulator but a thermal metal. In the highfield phase, thermal conductivity exhibits a sharp drop below 300 mK, indicating a transition from a thermal metal into an insulator/semimetal driven by the magnetic transition. These results suggest that spin degrees of freedom directly couple to the neutral fermions, whose emergent Fermi surface undergoes a fielddriven instability at low temperatures.
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Introduction
Strong electron interactions often lead to the emergence of manybody insulating ground states. Recently, surprising properties have aroused considerable interest in the research of the strongly correlated insulators, SmB_{6} and YbB_{12} with simple cubic crystal structures^{1}. In these Kondo lattice compounds, the band gap opens up at low temperatures due to the hybridization of localized f electrons with conduction electrons^{2}. In particular, quantum oscillations (QOs)^{3,4,5,6,7}, specific heat^{6,8,9}, and thermal conductivity^{6,8,10,11} experiments have posed a significant paradox, revealing gapless excitations in the bulk, in apparent contradiction with the charge gap seen in transport measurements. While the angular dependence of the QO frequencies suggests a threedimensional (3D) bulk Fermi surface in SmB_{6}^{4} and YbB_{12}^{5}, both materials remain robustly insulating to high magnetic fields (in SmB_{6}, a 2D Fermi surface has also been reported^{3,12}). Various theoretical models of the QOs in these insulators have been proposed so far^{13,14,15,16,17,18,19,20,21,22}. Another striking aspect is a nonzero lowtemperature linear specificheat coefficient γ ~ 10 mJ K^{−2} mol^{−1} for SmB_{6}^{6} and ~ 4 mJ K^{−2} mol^{−1} for YbB_{12}^{8,9} in zero field. As the specific heat is measured in the bulk insulating state, these results indicate the existence of gapless and chargeneutral excitations in the bulk consistent with an emergent Fermi surface of neutral fermions.
However, there are distinct differences in the gapless excitations in these correlated insulators. In SmB_{6}, the QOs are observed only in the magnetization (de Haasvan Alphen, dHvA, effect). The dHvA oscillations strongly deviate below 1 K from the LifshitzKosevich theory, which is based on Fermi liquid theory^{4}. In contrast, in YbB_{12}, the QOs are observed not only in the magnetization, but also in the resistivity (Shubnikovde Haas, SdH, effect) and both dHvA and SdH oscillations obey the LifshitzKosevich theory down to 50 mK^{5}. Moreover, in YbB_{12}, a finite residual temperaturelinear (Tlinear) term in the thermal conductivity K_{0} ≡ κ/T(T → 0) is observed, demonstrating the presence of gapless and itinerant neutral fermions^{8}. On the other hand, K_{0} in SmB_{6} has been controversial. While K_{0} of SmB_{6} has been reported to be very small but finite^{6}, the absence of K_{0} has been reported in^{10,11}.
A fascinating question is whether the QOs have any relationship to the neutral fermions. YbB_{12}undergoes an insulator–metal transition at μ_{0}H ~ 50 T, also confirmed by the recent SdH oscillation measurements^{7}. By tracking the Fermi surface area, it has been revealed that the same quasiparticle band gives rise to the SdH oscillations in both insulating and metallic states. By using a twofluid picture, it has been pointed out that neutral quasiparticles coexist with charged fermions^{7}. In addition, it has been shown that K_{0} depends on magnetic fields in YbB_{12}, suggesting that the neutral fermions can couple to magnetic fields^{8}. These results suggest that the neutral fermions may be crucial for explaining the QOs in YbB_{12} and other Kondo lattice insulators. Various theoretical models that invoke novel itinerant lowenergy neutral excitations within the charge gap that can produce QO signals have been proposed, including Majorana Fermi liquids^{17,18,22} and a spin liquid with spinon Fermi surface^{15,16}. However, the nature of the neutral fermions is largely elusive and continues to be hotly debated. As the Kondo hybridization between magnetic moments and conduction electrons is the origin of the charge gap formation in these insulators, it is crucially important to clarify how the neutral fermions couple to the magnetic degrees of freedom. Thus, more systematic investigations on a new class of materials are highly desired to clarify the relationship between QOs, chargeneutral fermions, and magnetic properties.
Recently a new insulating Kondo lattice compound YbIr_{3}Si_{7} has been discovered^{23}. YbIr_{3}Si_{7} has a trigonal ScRh_{3}Si_{7}type crystal structure (Fig. 1a). The magnetization and neutron diffraction data show that Yb ions are very close to the trivalent state in the bulk^{23}. In zero field, antiferromagnetic (AFM) order occurs below the Néel temperature T_{N} = 4.0 K. Neutron diffraction measurements report^{23} that, in the AFM state corresponding to the Γ_{1} state, all the Yb^{3+} moments are oriented along the crystallographic c axis ([001]). Each Yb^{3+} moment is aligned antiparallel with its six nearest neighbors in the nearly cubic Yb sublattice and parallel with its coplanar next nearest neighbors. The ordered moment is ~1.5 μ_{B}/Yb^{3+}. We note that in YbIr_{3}Si_{7}, the number of free charge carriers has been suggested to be much fewer than the number of local moments^{23}. It has therefore been proposed^{23} that the system becomes insulating at low temperatures as all the free carriers are consumed in the formation of Kondo singlets. Thus, YbIr_{3}Si_{7} has insulating bulk and longrange magnetic correlations, and is distinct from other simple Kondo insulators, such as SmB_{6} and YbB_{12}. Interestingly, thickness analysis of the electric transport shows that YbIr_{3}Si_{7} harbors conducting surface states whose origin is, however, not topological but rather has to do with the valence change to Yb^{2+} near the sample surface^{23}.
In this paper, we investigate the lowenergy excitations in the AFM insulating state of YbIr_{3}Si_{7} by the lowtemperature specific heat and thermal conductivity measurements. We find that both γ and K_{0} are finite at low fields, demonstrating the presence of mobile and gapless excitations of neutral fermions in the bulk insulating state, i.e., YbIr_{3}Si_{7} is a charge insulator but a thermal metal. The AFM order of this compound can be widely tuned by the external magnetic fields. More precisely, the chargeneutral quasiparticle excitations are either gapless or gapped with an extremely small excitation energy gap, much smaller than the base temperature 90 mK of our thermal conductivity measurements. Most surprisingly, a spinflop transition from AFI to AFII phase at μ_{0}H ≈ 2.5 T gives rise to an opening of a tiny gap or a linearly vanishing density of states (DOS) of neutral fermions, indicating a transition from a thermal metal into an insulator/semimetal. These results suggest that spin degrees of freedom directly couple to the neutral fermions, whose emergent Fermi surface undergoes a transformation in applied field.
Results
Magnetic phases
Resistivity
Figure 1b depicts the Tdependence of the inplane resistivity ρ of YbIr_{3}Si_{7} single crystals (#1 and #2) plotted on a log–log scale. At T ~ 150 K, ρ(T) changes its slope, which is attributed to the onset of Kondo correlations. Below ~150 K, ρ(T) increases rapidly with decreasing T. As shown in the inset of Fig. 1b, ρ(T) increases exponentially as \(\rho (T)\propto \exp ({{{\Delta }}}_{c}/{k}_{{{{{{\mathrm{B}}}}}}}T)\) with the charge gap Δ_{c} ~ 5.9 and ~6.5 meV for sample #1 and #2, respectively. At around T_{N}, ρ(T) is suppressed and increases again with decreasing T down to ~0.3 K. Upon further reducing the temperature, ρ(T) saturates down to the lowest temperature. Figure 1c depicts the low temperature resistivity in magnetic field applied parallel to the c axis (H∥c). The suppression of ρ(T) at T_{N} is reduced in magnetic field and is absent above μ_{0}H = 3 T, consistent with the previously reported data^{23}.
It has been shown that the lowtemperature saturation of ρ(T) arises from the surface state^{23}. In fact, the difference of the saturation values of ρ between crystals #1 (~1 Ωcm) and #2 (~2.1 Ωcm) can be quantitatively explained by the area and thickness of the crystal planes used for the measurements. Similar phenomena have been reported in SmB_{6} and YbB_{12}, in which the metallic conductivity takes place at the surfaces of the crystal, while electronic transport suggests the opening of a finite charge gap in the bulk at low temperatures. These metallic surface in SmB_{6} and YbB_{12} has been attributed to the topological insulating properties at low temperatures^{24}. In fact, the metallic surface states have been resolved by angleresolved photoemission spectroscopy (ARPES)^{25,26}. In particular, spinARPES experiments in SmB_{6} have revealed the spinmomentum locking of the surface quasiparticles as expected from topologically protected Dirac cones^{25}. In YbIr_{3}Si_{7}, on the other hand, the recent photoemission spectroscopy measurements revealed that the surface conduction originates from a change of valence from Yb^{+3} in the bulk to Yb^{+2} on the surface, without invoking topological arguments^{23}.
Phase diagram
Figure 2a displays the Tdependence of the specific heat divided by temperature, C(T)/T of crystal #1 in zero and finite magnetic fields applied for H∥c. Specific heat shows a very sharp peak at T_{N} = 4.0 K in zero field. As indicated by arrows in Fig. 1c, the resistivity shows an anomaly at around T_{N} determined by the specific heat. As depicted in Fig. 2a, C/T increases on approaching T_{N} from above. Similar phenomena have been observed in many antiferromagnets, such as CeRhIn_{5}^{27}. This increase of the specific heat above T_{N} is attributed the entropy release associated with the shortrange AFM order or fluctuations. In YbIr_{3}Si_{7}, the magnetic field suppresses the peak height considerably and shifts T_{N} to lower temperatures. The temperature dependence of C/T changes dramatically at higher fields^{23}. Above μ_{0}H ≈ 3 T, C(T)/T again exhibits a sharp peak, and the peak height increases rapidly, followed by a nearly saturated behavior above μ_{0}H = 5 T. In contrast to lower fields, T_{N} is nearly independent of applied magnetic field. Figure 2b depicts C/T plotted as a function of T^{2} at low temperatures. An upturn of C(T)/T at very low temperature (T ≲ 0.6 K) is attributed to the nuclear Schottky anomaly of the Yb ions.
The specificheat data clearly indicate the presence of two distinct AFM phases, i.e., lowfield AFI and highfield AFII phases. To determine the phase boundary between these two phases below T_{N}, we measured the Hdependence of the magnetization M of crystal #3 taken from the same batch as crystal #1 for H∥c, as shown in Fig. 3a. At around μ_{0}H ≈ 2.5 T, M(H) curves show inflection points at low temperatures. To see this more clearly, we plot the field derivative of the magnetization dM/dH in Fig. 3b. At low temperatures, dM/dH shows a distinct peak as a function of H, which is attributed to the phase transition between the AFI and AFII phases. The peak field of dM/dH is independent of temperature. In addition, no discernible hysteresis is observed between upsweep and downsweep magnetization measurements. Therefore, the AFI and AFII phases are likely separated by a weak firstorder phase transition. Above 2.5 T, the magnetization increases gradually with H without showing saturation. Moreover, as shown by the specific heat, a sharp phase transition is observed even above 2.5 T. In addition, the transition temperature slowly decreases with H. These results support that the AFII is an AFM state, inconsistent with the ferromagnetic state.
Figure 4 displays the H–T phase diagram for H∥c axis determined by the specific heat and magnetization measurements. The Néel temperatures are determined by the peak temperature of C(T)/T. To obtain information on the nature of the phase transition, we performed nuclear magnetic resonance (NMR) measurements for H∥c using another crystal (#4) taken from the same batch as crystal #1 and #3. Figure 5a, b depicts the magneticfield swept ^{29}SiNMR spectrum in the AFI and AFII phases, respectively. There are two crystallographically inequivalent Si sites, Si(1) and Si(2), as illustrated in Fig. 1a. For comparison, the NMR spectrum at 4.2 K above T_{N} are also shown by gray dotted lines. In the AFI phase, the NMR spectrum splits into three peaks. The peaks in the higher and lower magnetic fields indicate that an internal magnetic field at the Si(2) site is parallel to the external magnetic field, which is shown in the inset of Fig. 5a. This spin structure is consistent to that reported by neutron diffraction measurements. The middle peak arises from the Si(1) site at which an internal magnetic field from the Yb magnetic moment is canceled. On the other hand, in the AFII phase, only one peak is observed. This peak slightly shifts to a lower field below T_{N}. This small shift suggests that dominant magnetic moments are oriented perpendicular to the external magnetic field, as shown in the inset of Fig. 5b, although the tilted angle from the ab plane cannot be determined precisely in the present measurements. Thus, the NMR experiment reveals the spinflop transition in which the magnetic moments oriented along the c axis in the AFI phase are rotated to the ab plane in the AFII phase.
Gapless excitations in the insulating state
Specific heat
The specific heat of nonmagnetic and isostructural LuIr_{3}Si_{7} is plotted in Fig. 2a, b to estimate the phonon contribution. The phonon specific heat is negligibly small in the whole temperature range. As shown in Fig. 2b, C(T)/T at low temperatures varies rapidly with T at high fields. As the field is lowered, the Tdependence becomes weaker. Except for the very low Tregime, where C(T)/T shows an upturn due to the nuclear Schottky anomaly of Yb ions, C(T)/T increases with upward curvature with increasing T.
Figure 6a–h displays C(T)/T vs. T^{2} at low temperatures. Obviously, the extrapolation of C(T)/T above 1 K to T = 0 yields finite intercepts for all fields, indicating the presence of a finite γ. The gapless magnon modes are expected to give rise to C/T ∝ T^{α} with α = 1 and 2 for 2D and 3D systems, respectively. In zero field, C/T increases more steeply than C/T ∝ T^{2} line above 1 K. Therefore, an additional source of specific heat (other than phonons, whose contribution has been subtracted) is required, which we associate with magnons. These magnon excitations are gapped even in zero field. This magnon gap is attributed to a large magnetic anisotropy of Yb^{3+} ions due to the strong LS coupling. As shown in Fig. 6a–h, the specific heat can be fitted by
for all fields. Here, \({C}_{{{{{{{{\rm{mag}}}}}}}}}(T)={\beta }_{M}{T}^{3}\exp ({{{\Delta }}}_{M}/{k}_{{{{{{\mathrm{B}}}}}}}T)\) is the magnon contributions with an excitation gap Δ_{M} and a coefficient β_{M}, and \({C}_{{{{{{\mathrm{Sch}}}}}}}(T)=\frac{{{{\Delta }}}^{2}}{{k}_{{{{{{\mathrm{B}}}}}}}{T}^{2}}\frac{{e}^{{{\Delta }}/{k}_{{{{{{\mathrm{B}}}}}}}T}}{{(1+{e}^{{{\Delta }}/{k}_{{{{{{\mathrm{B}}}}}}}T})}^{2}}\) is the twolevel nuclear Schottky term, where Δ is the corresponding energy splitting. The lowtemperature Schottky contribution is well fitted by Δ/k_{B} ≈ 0.1 K, which is field independent. Field dependence of Δ_{M} is shown in Supplementary Fig. 1. Near the boundary between the AFI and AFII phases, Δ_{M} is strongly suppressed. In the lowfield regime in the AFI phase and in highfield regime in the AFII phase, Δ_{M} is nearly independent of H. In the ordered phase, Δ_{M} is determined by the competition between the Zeeman field and the molecular field due to the magnetic moment around the magnetic ions. When magnetic order is stabilized, the molecular field dominates and Δ_{M} is not seriously influenced by the Zeeman field. Because the magnetic order is suppressed and magnetic fluctuations are enhanced near the phase boundary, Δ_{M} is suppressed, consistent with the observed behavior of Δ_{M}.
In Fig. 7a, the Hdependence of γ obtained by the fitting of Eq. (1) is shown. In the wholefield regime, γ is finite. In the lowfield regime, γ is nearly constant. Remarkably, γ is enhanced above ~ 2 T, and peaks in the vicinity of the phase boundary. Upon entering the AFII phase, γ is first suppressed but then increases gradually with H. To confirm that this Hdependence of γ is not due to a fitting ambiguity, we also plot C(T)/T at 0.7 K, where the Schottky contribution is negligible. The similar Hdependence of C(0.7 K)/T indicates that the enhancement of γ near the phase boundary is an intrinsic property. As the system is insulating, finite γ indicates the presence of a finite DOS of chargeneutral excitations. More precisely, the chargeneutral quasiparticle excitations in the AFI and AFII phases are either gapless or gapped with an extremely small excitation energy gap, much smaller than 0.7 K. To check the reproducibility of the data, we measured the specific heat of crystal #2 grown in the different batch. As shown in Supplementary Fig. 2, C/T of #2 well coincides with that of #1, suggesting that the finite γ is a universal property of this system. We shall discuss the field dependence of γ in more detail below.
The strong suppression of the peak height of C(T)/T and the reduction of T_{N}, and the magnon gap approaching the phase boundary between the AFI and AFII phases suggest a possible influence from a putative fieldinduced AFM quantum critical point (QCP). In fact, the magnitude of the magnetic moment is expected to be strongly reduced with approaching an AFM QCP, which leads to the suppression of the peak height of the specific heat, as reported in CeRhIn_{5}^{27}, and of magnon gap. In YbIr_{3}Si_{7}, however, the putative AFM QCP is avoided by a transition into the AFII phase. Nevertheless, the quantum critical fluctuations emanating from an avoided QCP in the AFII phase would lead to the reduction of the magnetic moment. These results lead us to consider that the enhancement of γ in the AFI phase near the phase boundary is caused by the AFM quantum critical fluctuations. The striking enhancement of γ near the AFM QCP has been reported in several classes of strongly correlated electron systems, including heavy fermions^{27} and iron pnictides^{28}. The present results suggest that fluctuations emanating from an avoided AFM QCP largely modify the DOS of the neutral fermions.
Thermal conductivity
The specific heat involves both localized and itinerant excitations. Therefore, a finite γ does not always indicate the presence of mobile gapless excitations. In fact, amorphous solids and spin glasses exhibit a finite γ, although the excitations in these systems are localized. Moreover, the Schottky anomaly in the specific heat often prevents the analysis of C at very low temperatures. In contrast, the thermal conductivity is determined exclusively by itinerant excitations. In addition, it is free from the Schottky anomaly, enabling us to extend the measurements down to lower temperatures. In particular, a finite intercept K_{0} provides the most direct and compelling evidence for the presence of the itinerant and gapless fermionic excitations, analogous to the excitations near the Fermi surface in pure metals.
Figure 8a–h shows κ/T of crystal #1 plotted as a function of T (main panels) and T^{2} (insets) in zero and finite magnetic fields for H∥c at very low temperatures. In the AFI phase, the Tdependence of κ/T shows a convex downward curvature for κ/T vs. T plot, but a convex upward curvature for κ/T vs. T^{2} plot. As shown in Fig. 8d–h, the behavior of the thermal conductivity in the AFII phase at μ_{0}H ≥ 2.5 T is fundamentally different from that in the AFI phase; the temperature dependence of κ/T shows a concave downward curvature below ~0.3 K. As shown by dashed lines in the insets of Fig. 8d–h, κ/T increases nearly proportional to T^{2} in the hightemperature regime. At very low temperatures, κ/T deviates from the T^{2}dependence.
In the present magnetic insulating system, the thermal conductivity can be written as a sum of the phonon, magnon, and nonphononic quasiparticle contributions, κ = κ_{ph} + κ_{mag} + κ_{qp}. We first discuss κ_{mag}. Because the magnon gap is Δ_{M}/k_{B} ~ 2 K except for the phase boundary regime, the magnon contribution is expected to become exponentially small in the temperature range shown in Fig. 8a–h. We note that regardless of whether the magnons are gapped or gapless, κ_{mag}/T(T → 0) = 0. We next discuss the contribution of κ_{ph}. We point out that magnon–phonon scatterings do not play an important role in the phonon thermal conductivity because of the following reasons. If the magnon–phonon coupling are strong, the magnetic field would open up a gap in magnon spectrum, leading to the suppression of the magnon–phonon scattering, thus resulting in the enhancement of κ_{ph} with H. However, as shown in Supplementary Fig. 3, κ/T above 90 mK decreases monotonically with magnetic field up to 12 T. Moreover, κ(H) changes little near the phase boundary between AFI and AFII, although the specific heat is distinctly enhanced by the magnetic fluctuations (Fig. 7a). As the phonon DOS does not change at the phase boundary, the observed field dependence indicates that phonons are little affected by magnons and nonphononic quasiparticles.
At low temperatures, κ_{ph} is given by \({\kappa }_{{{{{{\mathrm{ph}}}}}}}=\frac{1}{3}{\beta }_{{{{{{\mathrm{ph}}}}}}}\langle {v}_{s}\rangle {\ell }_{{{{{{\mathrm{ph}}}}}}}{T}^{3}\), where β_{ph} is the phonon specificheat coefficient obtained by the Debye phonon specific heat C_{ph} = β_{ph}T^{3}, 〈v_{s}〉 is the acoustic phonon velocity, and ℓ_{ph} is the effective mean free path of acoustic phonons. When ℓ_{ph} becomes comparable to the crystal size at very low temperatures (boundary limit), ℓ_{ph} is approximately limited by the effective diameter of the crystal \({d}_{{{{{{\mathrm{eff}}}}}}}=\frac{2}{\pi }\left(\int\nolimits_{0}^{\alpha }\frac{t}{\cos \theta }{{{{{\mathrm{d}}}}}}\theta +\int\nolimits_{\alpha }^{\pi /2}\frac{w}{\sin \theta }{{{{{\mathrm{d}}}}}}\theta \right)\), where w and t are the width and thickness of the crystal, respectively, and \(\alpha =\arctan (w/t)\). Using β_{ph} = 0.45 mJ mol^{−1}K^{4} for LuIr_{3}Si_{7}, we estimate 〈v_{s}〉 ≈ 3000 m s^{−1}. We then find that ℓ_{ph} reaches the crystal size below 0.2 K. Using d_{eff} = 0.17 mm, we estimate the phonon thermal conductivity in the boundary limit \({\kappa }_{{{{{{\mathrm{ph}}}}}}}^{b}/T\approx 4.9\) mW K^{−2} m^{−1} at the lowest temperature 0.08 K, which is smaller than the observed κ/T for all fields. As \({\kappa }_{ph}^{b}\) gives the upper limit of the phonon thermal conductivity, this indicates that the thermal conductivity is dominated by κ_{qp} in the lowtemperature regime. Thus, the temperature and field dependencies of the thermal conductivity are mainly determined by the nonphononic quasiparticle contributions.
In zero field, the linear extrapolation of κ/T to T = 0 has almost a zero intercept as seen in κ/T vs. T plot (main panel of Fig. 8a). On the other hand, the extrapolation to T = 0 has finite intercepts for both κ/T vs. T and κ/T vs. T^{2} plots for μ_{0}H = 1 and 2 T (Fig. 8b, c). This indicates that the quasiparticle thermal conductivity contains a finite residual Tlinear term, K_{0} = κ_{qp}/T(T → 0). We note that in the AFI phase, similar magnitude of K_{0}, including vanishingly small K_{0} in zero field, is observed in crystal #2. Moreover, in crystal #2 with similar effective diameter as #1, the magnitude of κ/T at finite temperature is close to that of #1, suggesting similar mean free paths of phonon and quasiparticle in the AFI phase (Supplementary Fig. S4). These results demonstrate the presence of mobile and gapless fermionic excitations in the AFI phase. We stress that the observed finite K_{0} does not originate from charged quasiparticles, in contrast to conventional metals. Evidence for this is provided by the spectacular violation of the WiedemannFranz (WF) law, which connects the electronic thermal conductivity κ^{e} to the electrical resistivity ρ. In metals at low temperatures, the ratio L = κ^{e}ρ/T ≤ L_{0} is satisfied, where \({L}_{0}=({\pi }^{2}/3){({k}_{{{{{{\mathrm{B}}}}}}}/e)}^{2}=2.44\times 1{0}^{8}{{{{{{{\rm{W}}}}}}}}{{\Omega }}\,{{{{{{{{\rm{K}}}}}}}}}^{2}\) is the Lorenz number. The values of K_{0}ρ_{0}, where ρ_{0} is the residual resistivity, are found to be ~2.6 × 10^{3}L_{0} and ~3.5 × 10^{3}L_{0} at μ_{0}H = 1 and 2 T, respectively. Here we used K_{0} = 6.4 and 8.6 mW K^{−2} m^{−1} at μ_{0}H = 1 and 2 T, respectively, and ρ_{0} = 0.99 Ωcm for both fields. It is highly unlikely that the surface metallic region significantly violates the WF law. In fact, it is well known that the WF law holds in the 2D metals, even in the quantum Hall regime. We also note that the WF expectation of L_{0}/ρ_{0} from the metallic surface is less than 2.5 × 10^{−3} mW K^{−2} m^{−1}, which is by far smaller than the experimental resolution. These results lead us to conclude that the neutral fermions in the insulating bulk state are responsible for the observed finite K_{0}. This suggests that, as the bulk resistivity diverges as T → 0, the Lorenz number for the heatcarrying quasiparticles also diverges. We also stress that the finite K_{0} cannot be explained by the magnon excitations, as mentioned above. Thus, the thermal conductivity and specificheat data under magnetic fields in the AFI phase of YbIr_{3}Si_{7} provide evidence for the presence of highly mobile and gapless neutral fermion excitations, which has been similarly reported in YbB_{12}.
We note parenthetically that finite values of both γ and K_{0} in the insulating states have been reported in quantumspinliquid candidates with 2D triangular lattices, including the organic compounds, EtMe_{3}Sb[Pd(dmit)_{2}]_{2}^{29} and κH_{3}(CatEDTTTF)_{2}^{30}, and inorganic compounds, 1TTaS_{2}^{31} and Na_{2}BaCo(PO_{4})_{2}^{32}. In EtMe_{3}Sb[Pd(dmit)_{2}]_{2}, although the presence or absence of finite K_{0} has been controversial among different research groups^{33,34}, it has been shown very recently that differences between data sets are likely to be due to the cooling rate^{35}. In 1TTaS_{2}, as finite K_{0} readily disappears by the introduction of disorder/impurity, the magnitude of K_{0} appears to depend strongly on the sample quality^{31}. These results suggest that highquality single crystals are required to observe the finite K_{0} in quantum spinliquid systems. In the above compounds, finite γ and K_{0} have been discussed in terms of electrically neutral spinons forming the Fermi surface.
In the AFII phase of YbIr_{3}Si_{7}, the magnitude of κ/T is strongly reduced above 200 mK compared to that in the AFI phase. As γ in the AFII phase is close to that of the AFI phase except at the phase boundary, quasiparticle DOS is not largely different between two phases. Therefore, the suppression of κ/T above 200 mK in the AFII phase suggests that scattering time of neutral quasiparticles strongly depends on the magnetic structure. Moreover, a remarkable deviations from the T^{2}dependence and suppression of κ/T at very low temperatures are clearly observed. In the temperature regime where κ/T is suppressed, κ/T depends on T as κ/T ~ T^{q} with q < 1, which cannot be explained by a phonon contribution. Thus, the suppression of κ/T indicates an opening of a tiny gap in the spectrum of the itinerant quasiparticle excitations. As this gap formation occurs below ~0.3 K, the estimate of the gap is two orders of magnitude smaller than the Kondo gap (~60 K). We point out that there are two possible explanations for this behavior at very low temperatures well below ~0.3 K. One is a fully gapped thermal insulating state and the other is thermal semimetallic or nodal metallic state with a linearly vanishing DOS, as indicated by red shaded regime in Fig. 4. To clarify which scenario is realized, future measurements at lower temperature are required.
In the AFII phase, T^{2}dependent κ/T in the hightemperature regime followed by a sharp drop at low temperatures is reproduced in a different crystal (#2, Supplementary Fig. 4). On the other hand, the magnitude of κ/T of crystal #2 is largely enhanced compared to that of crystal #1. Moreover, the temperature at which the quasiparticle gap is formed in crystal #2 is nearly two times larger than that in crystal #1. As the γvalue is very close in both crystals in the AFII phase, this difference is attributed to the mean free path of the neutral fermions.
We note that the specific heat measurements cannot resolve this gap formation due to the Schottky anomaly. The quasiparticle thermal conductivity κ_{qp} is written as κ_{qp}/T = K_{0} + f(T). We find that the Tdependent part of κ/T, f(T) + κ_{ph}/T, can be fitted by a powerlaw dependence on T, as depicted in the insets of Supplementary Fig. 5a–h. The detailed Tdependence of κ_{qp} is difficult to determine due to the presence of a small but finite κ_{ph}/T. The filled red circles in Fig. 7b represent K_{0} obtained by extrapolating κ/T to T = 0 using the powerlaw fits in the AFI phase. After the initial rapid increase, K_{0} increases slowly. In the AFII phase, the quasiparticle contribution before the gap formation is obtained by the extrapolation from the hightemperature regime to T = 0. The filled red squares in Fig. 7b show this quasiparticle contribution. For comparison, we also plot κ/T at 90 mK. Interestingly, κ/T obtained by hightemperature extrapolation appears to lie on top of the extrapolation from the AFI phase. This suggests that K_{0} steadily increases with magnetic field, but is strongly affected by the spinflop transition. This field dependence indicates that the itinerant neutral fermions couple to the magnetic field and are strongly influenced by the magnetic ordering.
Discussion
The combined results of the specific heat and thermal conductivity provide pivotal information on the neutral fermions observed in insulating materials. As shown by Fig. 7a, b, γ and K_{0} exhibit very different Hdependence. In particular, at zero field, while γ is finite, no sizable K_{0} is observed. We note that the result at zero field bears a resemblance to that of SmB_{6}. On the other hand, finite γ and K_{0} values in YbB_{12} are similar to those of YbIr_{3}Si_{7} in a finite field in the AFI phase, although there is no signature of the phase transition at low field, as shown in Fig. 3a, b. In YbB_{12}, the γ value is nearly sample independent, while K_{0} values are strongly sample dependent, which is attributed to the amount of the impurities/defects determining the mean free path of the quasiparticles. In contrast, in the present study, we find strong field dependence of K_{0} in a single sample, which cannot be due to the change in the impurity scattering. The rapid enhancement of K_{0} = \(\frac{1}{3}\)γvℓ, where v and ℓ are velocity and mean free path of the neutral fermions, respectively, at low H is attributed either to the increase of γ or to the increase of ℓ. As γ is nearly constant in the AFI phase except in the vicinity of the phase boundary, the enhancement of K_{0} is attributed to the enhancement of ℓ of the quasiparticles. On the other hand, the absence of an enhancement of K_{0} near the avoided QCP, despite the enhancement of γ, may be because K_{0} is proportional to γτ, where τ is the scattering time. To explain why K_{0} is not seriously affected by the enhancement of γ near the QCP, it is required that τ is inversely proportional to the DOS of the neutral fermions, τ ∝ 1/γ. Such a mechanism is, for example, observed in dwave superconducting materials, which show a universal residual thermal conductivity^{36}.
The nature and behavior of the novel chargeneutral fermions are not well understood; there are very few experimental results that can be used as tests of the various theoretical models, which include 3D Majorana fermions, composite magnetoexcitons, and spinons in fractionalized Fermi liquids. In this sense, our observations that the itinerant neutral fermions are very sensitive to the magnetic ordering can put significant restrictions on the various theories. The tiny gap formation (or a linearly vanishing DOS) in the AFII phase indicates a transition from a thermal metal into an insulator (or a thermal semimetal), while the material remains an electrical insulator. This result demonstrates that the Fermi surface of the chargeneutral fermions becomes unstable towards gap formation at low temperatures, which is driven by the magnetic transition of the insulator. Therefore it is natural to consider that the neutral fermions are composed of strongly magnetically coupled c and f electrons through the Kondo effect. In this situation, neutral fermion excitations will be affected by AFM order and fluctuations. As revealed by the thermal conductivity measurements in two crystals, the scattering time of the chargeneutral fermions is largely different only in the AFII phase. It is an open question why such a difference is present only in the thermally insulating/semiconducting phase but is absent in the thermally metallic phase. The clarification of the f(T) term in κ_{qp} would be key for understanding this remarkable difference between the AFI and AFII phases and is also an important future issue to understand the coupling between the neutral fermions and spin degrees of freedom.
In summary, we performed specific heat, thermal conductivity, and NMR measurements of bulk insulating YbIr_{3}Si_{7} at low temperatures. In the lowfield AFI phase, we find finite γ and K_{0}, demonstrating the emergence of itinerant gapless excitations even in the magnetic ground state of a Kondo insulator. A spectacular violation of the WF law directly indicates that YbIr_{3}Si_{7} is a charge insulator but a thermal metal. More precisely, the chargeneutral quasiparticle excitations are either gapless or gapped with an extremely small excitation energy gap, much smaller than the base temperature 90 mK of our thermal conductivity measurements. A spinflop transition at μ_{0}H ~ 2.5 T is revealed by NMR measurements. With approaching the spinflop transition, γ is largely enhanced. Remarkably, inside the highfield AFII phase, κ/T exhibits a sharp drop at very low temperatures, indicating the opening of a tiny gap much smaller than the Kondo gap or a linear vanishing DOS of the neutral excitations. This demonstrates a fieldinduced transition from a thermal metal into an insulator/semimetal driven by the spinflop transition. The present results demonstrate that the neutral fermions are directly coupled to the spin degrees of freedom, which have never been considered in existing theories. Our experimental observations impose a strong constraint on the theories of chargeneutral fermions. Thus, YbIr_{3}Si_{7} provides an intriguing platform for studying the neutral fermions in strongly correlated insulators.
Methods
Crystal growth and resistivity
Single crystals of YbIr_{3}Si_{7} have been grown using the laser pedestal technique. The crystals #1, #3, and #4 were taken from the same batch and crystal #2 was taken from the different batch. The crystals were cut from the asgrown ingot and polished into a rectangular shape, with the longest direction corresponding to the a axis and the shortest direction to the c axis. Back scattering Xray Laue diffraction was used to orient the crystals. The dimension of the samples used for transport and heat capacity measurements are 2.7 × 1.6 × 0.10 mm^{3} (#1) and 2.3 × 0.66 × 0.3 mm^{3} (#2). Resistivity were measured with a.c. technique in a standard fourcontact configuration. The same contacts were used for the thermal conductivity measurements.
Specific heat
The specific heat of YbIr_{3}Si_{7} single crystals was measured by a longrelaxation calorimetry using a bare chip Cernox sensor, which is used as a thermometer and a heater. The specific heat of the addenda including the grease was measured before the sample was mounted. The specific heat of the sample is obtained by subtracting the addenda from the total specific heat measured with the sample.
Nuclear magnetic resonance
^{29}Sinuclear magnetic resonance (NMR) measurements were performed in applied magnetic field parallel to the c axis. A conventional spinecho technique was used.
Thermal conductivity
The thermal conductivity were measured by the steadystate method, applying the current q along the a axis. Magnetic field was applied along the c axis. The thermal gradients ∇T were detected by RuO_{2} thermometers, and κ were obtained as κ = q/∇T. The RuO_{2} thermometers were calibrated by a commercial RuO_{2} thermometer (LakeShore) with zerofield calibrations. The field dependence of the RuO_{2} thermometers was calibrated by using a Coulomb blockade thermometer, which is insensitive to the magnetic fields. The validity of the calibration of the RuO_{2} thermometers was carefully checked by confirming the WF law of a thin gold wire in magnetic fields in the same setup.
Discussions of thermal decoupling and thermal leakage
(1) Decoupling: In metals, thermal decoupling of phonons and electrons caused by the poor contact leads to the anomalous rapid suppression in the thermal conductivity at low temperatures, as reported in cuprate superconductors^{37}. To ensure good thermal contacts, we sputtered gold on fresh surface of the crystal and then attached the contacts with silver epoxy. At room temperature and at ~100 K, the contact resistance is much less than 1 Ω. While the downturn of the thermal conductivity with decreasing T is observed only in the AFII phase above 2.5 T, it is absent in the AFI phase at lower fields. The fact that the finite residual temperaturelinear term in the thermal conductivity is observed down to the lowest temperature in the AFI phase provides evidence that quasiparticles are well coupled to phonons, i.e., the absence of thermal decoupling. (2) Thermal leakage: We measured the thermal conductivity of thin stainless wire, whose thermal resistance is about two orders of magnitude larger than the present crystal, in the same setup. We confirmed the WiedemannFranz law in the stainless wire, demonstrating negligibly small thermal leakage.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
A.H.N. and E.M. acknowledge fruitful discussions with Chris Hooley. Y.M. acknowledges discussion with H. Kontani, Lu Li, and J. Singleton. L.Q., and E.M. acknowledge support from the U.S. Department of Energy for Grant No. DESC0019503. A.H.N. was supported by the National Science Foundation grant No. DMR1917511 and the Robert A. Welch Foundation grant C1818. This work is supported by GrantsinAid for Scientific Research (KAKENHI) (Nos. JP15H02106, JP18H01177, JP18H01178, JP18H01180, JP18H05227, JP19H00649, JP20H02600, JP18K03511, and JP20H05159) and on Innovative Areas “Quantum Liquid Crystals” (No. JP19H05824) from Japan Society for the Promotion of Science (JSPS), and JST CREST (JPMJCR19T5).
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L.Q. and E.M. grew the highquality singlecrystalline samples. Y.S., T.T., Y.K., and S. Kasahara performed resistivity, magnetization, specific heat, and thermal conductivity measurements. T.K., S. Kitagawa, and K.I. performed nuclear magnetic resonance measurements. Y.S., T.T., Y.K., and S. Kasahara analyzed the data. Y.S., S.S., Y.K., R.P., T.S., A.H.N., E.M., and Y.M. discussed and interpreted the results. Y.S., Y.K., R.P., T.S., A.H.N., and Y.M. prepared the manuscript, with input from all other coauthors.
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Sato, Y., Suetsugu, S., Tominaga, T. et al. Chargeneutral fermions and magnetic fielddriven instability in insulating YbIr_{3}Si_{7}. Nat Commun 13, 394 (2022). https://doi.org/10.1038/s41467021275419
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DOI: https://doi.org/10.1038/s41467021275419
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