The pronounced enhancement of the effective mass is the primary phenomenon associated with strongly correlated electrons. In the presence of local moments, the large effective mass is thought to arise from Kondo coupling, the interaction between itinerant and localized electrons. However, in d electron systems, the origin is not clear because of the competing Hund's rule coupling. Here we experimentally address the microscopic origin for the heaviest d fermion in a vanadium spinel LiV2O4 having geometrical frustration. Utilizing orbital-selective 51V NMR, we elucidate the orbital-dependent local moment that exhibits no long-range magnetic order despite persistent antiferromagnetic correlations. A frustrated spin liquid, Hund-coupled to itinerant electrons, has a crucial role in forming heavy fermions with large residual entropy. Our method is important for the microscopic observation of the orbital-selective localization in a wide range of materials including iron pnictides, cobaltates, manganites and ruthnates.
Electrons in metals behave as quasiparticle, whose mass often becomes extremely heavy when metallic phases are close to the quantum critical boundary for the insulating or magnetic phase1. The microscopic understanding of heavy quasiparticles (HQs) has been a goal of modern many-body quantum statistics. An established route for HQs is antiferromagnetic Kondo coupling between localized f spins and itinerant electrons in rare earth metals. In contrast to the f-electron case, the presence of localized spins is not apparent for d-electron systems. Alternative routes driving d HQ formation have been challenging issues in strongly correlated electron physics.
A representative d HQ material is the vanadium spinel LiV2O4 (refs 2,3), which has a highly frustrated pyrochlore lattice for the B site V3.5+ (3d1.5) ions (Fig. 1a). Anisotropic orbital-dependent intersite interactions give an itinerant orbital and a more localized a1g orbital through a small trigonal distortion of the VO6 octahedron (Fig. 1b)4,5. The HQ was initially explained by off-site Kondo exchange interactions, JK, between localized a1g moments and itinerant electrons (Fig. 1b)4. In the t2g manifold, however, the strong on-site ferromagnetic Hund's exchange interaction, JH, can overcome JK. Many alternative scenarios, such as geometrical frustration via antiferromagnetic interactions JAF,6,7,8, electron correlations9,10 and spin-orbital fluctuations11,12 have been proposed.
Experimentally, the interpretations have been unclear for HQs in LiV2O4. Charge-sensitive probes such as resistivity2, photoemission13 and optical14 measurements showed crossover from a high-temperature incoherent metal to a low-temperature Fermi liquid across the characteristic temperature T *~ 20–30 K. In contrast, spin-sensitive probes including static spin susceptibility2,15 and inelastic neutron scattering16 measurements imply local moments with antiferromagnetic correlations at low temperatures. Furthermore, anomalous temperature T dependences on the specific heat C/T and the Hall coefficient conflict with those expected in a conventional Fermi liquid2. Despite the theoretical view of orbital-selective interactions, no experimental effort has been made to detect the orbital degrees of freedom.
Here, we address the first experimental approach for microscopic observations of the d HQ via orbital-resolved nuclear magnetic resonance (ORNMR) measurements in LiV2O4. The previous NMR experiments using closed-shell Li sites2,15 only measured the net spin susceptibility proportional to the bulk value because the hyperfine interactions at the Li site surrounded by 12 vanadium sites average out the anisotropy. Our ORNMR spectroscopic approach using on-site 51V spins on a high-quality single crystal is sensitive to the orbital-dependent local spin susceptibility, which is beneficial for probing strongly correlated electrons with the orbital degrees of freedom.
ORNMR Knight shift
In LiV2O4, the spin susceptibility χs consists of the a1g and components: χs=χa+χe (hereafter the superscripts a and e denote a1g and , respectively). The NMR frequency shift called the Knight shift, (x, y and z are the principal axes), measures the spin susceptibility via the hyperfine interaction with the hyperfine coupling tensor Ai between the nuclear spin I and the paramagnetic spin polarization for the i-th electron under an external magnetic field. Whereas the isotropic shift due to the core polarization is proportional to χs, the anisotropic part due to the orbital-specific dipole hyperfine interaction17 is expressed by using the principal z component of the coupling constants, and ,
where N is Avogadro's number and μB is the Bohr magneton. In contrast to measures the hyperfine-weighted average of the orbital-dependent spin susceptibility. In the ionic limit, and are given by a quadratic combination of the angular momentum (see Methods) with the reversed sign and the same amplitude, (ref. 17). Hence, we can distinguish which orbital dominates the spin susceptibility from the sign of and obtain the orbital occupation from the amplitude. Namely, should be positive (negative) for χa>χe (χa<χe).
The 51V Knight shift tensors of LiV2O4 are determined from the angle dependence of K for 2–300 K, as shown in Fig. 2. The 51V NMR spectra were detectable only at specific angles, where the nuclear quadrupole interaction almost vanishes, because the nuclear spin-spin relaxation times at other angles are too fast. The obtained K traces three cosine curves of equation (2) in the Methods, which satisfies the cubic Fd3 m lattice. The principal z axis of K at ±54° for the two V sites indicates the 3d orbital symmetry governed by the trigonal VO6 crystal field (Fig. 1b). At 300 and 2 K, the relationship shows the a1g-dominant spin susceptibility (χa>χe). The result is consistent with the localized nature of the a1g orbital, as theoretically suggested4,5,6,10.
Temperature dependence of orbital occupations
To address HQ formation, we measured the thermal variations of the 51V Knight shifts and in comparison with the 7Li Knight shift and the bulk spin susceptibility χ (Fig. 3a). Good linearity was observed between these Knight shifts and χ (Supplementary Fig. S1). All of data show a Curie–Weiss-like increase at high temperatures, followed by a broad maximum at approximately 20 K. The results agree with the spin susceptibility for high-quality crystals, free from a Curie-tail increase at low temperatures2,3. The on-site 51V Knight shift probes the spin susceptibility with greater sensitivity than the off-site 7Li one and shows a smooth decrease below T*. Below 5 K, Ks becomes nearly T independent, as observed in the Fermi liquid.
When a1g local moments becomes Fermi liquid with HQ via a1g- hybridization or intersite Kondo coupling below T*, χa is expected to decrease significantly, whereas χe is less sensitive. It could lead a decrease of from equation (1). To inspect this property, we plot against T in Fig. 3b. We find no appreciable change in for 2–300 K. This lack of change signifies that the localized character of the a1g orbital persists to the Fermi liquid state across T*. Although of the f-electron system has not been reported, the non-linear relationship in the K–χ plot may be a manifestation of Kondo coupling in the Ce- and U-based compounds20,21,22.
reflects the 3d orbital polarization when χa and χe scale to electron occupations. From we can evaluate the mixing ratio of the a1g and orbitals in LiV2O4. The singly occupied a1g orbital has , as observed in the insulating pyrochlore material Lu2V2O7 (ref. 18). In contrast, vanishes for equivalent mixing of a1g and , as observed in a less correlated metal V2O3 (Fig. 3b). The observed intermediate in LiV2O4 manifests a significant contribution to the spin susceptibility in throughout the temperature range. Namely, the spin must be polarized via Hund's rule coupling to the localized a1g spins under the magnetic field, although the itinerancy of is much better than that of a1g. The occupation ratio is evaluated as (see Methods), corresponding to the electron numbers of and for 3d1.5. The half-filling a1g occupation is distinct from that expected in the tight-binding calculation without electron correlations, where (ref. 12). However, it is compatible with the strongly localized a1g picture owing to the strong renormalization into the Mott insulating state5,9,10 and provides microscopic evidence for orbital-dependent localization, which is robust across T*.
Dynamical spin susceptibility of the orbital-selective spin liquid
Another interesting issue is the dynamical part probed by the nuclear spin-lattice relaxation rate . is generally given by (ref. 19), where is the wave vector q component of the hyperfine coupling constant normal to the quantization axis, and is the transverse dynamical spin susceptibility at the NMR frequency ω. In a cubic lattice, and are isotropic for the 51V and 7Li sites. (T1T)−1 measured for 51V and 7Li (Fig. 4a) follows the linear relationship , where the linear coefficient C=1.0×103 is close to the square ratio of the hyperfine coupling, and C0=64 s−1 K arises from the T-invariant orbital component. The scaling relation indicates that the Li sites probe spin fluctuations via the net transferred hyperfine interaction and allows us to evaluate unobservable at low temperatures from .
In the present case, is governed by paramagnetic fluctuations of local moments at high temperatures. Above 150 K, the scaling behaviour between and is indeed observed = constant in Fig. 4b). Below 150 K, a progressive increase indicates antiferromagnetic correlation, consistent with the growth of χ(q,ω) at a finite q (=0.64 Å−1) in the inelastic neutron scattering measurements below 80 K (ref. 16). Therefore, the suppression of χs at low temperatures likely comes from the short-range antiferromagnetic correlation with the exchange interaction JAF~150 K. Nevertheless, no long-range magnetic ordering occurs down to 1.5 K, the energy scale of ~JAF/100. It suggests that the frustrated a1g spins form in a quantum liquid at low temperatures with low-lying excitations.
Our results provide significant insights into the formation of 3d HQ. As observed in the T-independent , the a1g spins likely remain incoherent, even entering into a coherent 'Fermi liquid' state, and couple ferromagnetically to spins. No indication of Kondo coupling was observed down to low temperatures despite the large antiferromagnetic fluctuations. The remaining local moments can be highly frustrated and carry large residual entropy6,7,8. The itinerant electrons interact with the underlying spin liquid via the Hund's rule coupling. Thus, the 3d HQ behaviour in LiV2O4 could be mapped on the frustrated ferromagnetic Kondo lattice.
In the absence of antiferromagnetic Kondo coupling, the HQ formation has not been established theoretically. In a Hubbard model calculation, the Kondo-like coherence peak appears on the boundary of the orbital-selective Mott transition for the a1g part10. Even in such a case, χa may vary across T*, while χe is invariant. Our results suggest that, if the Kondo-like peak appears, a large fraction of the incoherent spins still remains and carries entropy. Such fractionalization of the nearly localized electron might be common to strongly correlated electron systems23, where localized and itinerant characters coexist. Furthermore, the chirality degrees of freedom might provide appreciable entropy in the pyrochlore lattice24 and contribute to the anomalous Hall effect3.
The ORNMR technique offers new holographic experiments that could give microscopic insights into strongly correlated electrons. Various orbital-resolved tools, such as X-ray absorption and photoemission spectroscopy, have been recently developed. The Knight shift measurement has a unique advantage in detecting the orbital-dependent local spin susceptibility via the magnetic hyperfine interactions between d spins and on-site nuclear spins. The method has not been achieved in rare-earth heavy fermion compounds because of the difficulty in detecting NMR signals for the on-site nuclear spins20. Additional technical improvements in the NMR measurements may reveal the hidden orbital-selective Mott transition in transition metal oxides, such as ruthenates, pnictides and manganites.
The ORNMR experiments were performed on a single crystal of LiV2O4 synthesized by the self-flux method25. The crystal with the octahedral shape was placed on a two-axis goniometer and rotated under a fixed magnetic field H0=9.402 and 8.490 T. The NMR spectra were obtained from spin-echo signals after two π/2 pulses separated by a time τ. The 51V NMR measurements were made only for powder samples above 50 K (ref. 15) likely due to the fast spin-echo decay time T2 at low temperatures. To overcome this problem, we used a short τ=3–10 μs and a magnetic field precisely (<0.1°) aligned to the crystal axis equivalent to the magic angle of the nuclear quadrupole interaction. Otherwise, the NMR signals were depressed owing to the fast T2.
The angular dependence of the 51V Knight shift K(θ) with the local trigonal symmetry is fitted into the general formula26
for the V1 site and
for the V2 or V3 site related by a mirror symmetry, where
Magnetic hyperfine interactions
Magnetic hyperfine interactions in 3d systems are generally given by17
where li, si and I denote operators of orbital and spin of the i-th electron and the nuclear spin, respectively, the coefficient using the Bohr magneton μB, the nuclear gyromagnetic ratio γn, the Plank's constant and a radial expectation value r−3. The first term represents the orbital contribution that quenches in crystals but partly revives via the Van–Vleck process under the magnetic field. The second term arises from a Fermi contact interaction due to the core polarization of inner s spins, giving the isotropic hyperfine coupling constant (κ~0.5 for vanadates18, where κ is a dimensionless parameter). The third term denotes anisotropic dipole interactions determined by 3d orbital occupations, where the principal components are expressed as by using the equivalent operator of 3d angular momentum,
with the spin-orbit coupling parameter λ and the second-order matrix elements between the ground and excited states, . In the LS-coupling, the sum of the terms for several electrons can be replaced by
where L and S are total orbital and spin, respectively, and .
Analysis of the 51V Knight shift
The experimental observable is the Knight shift tensor K=(Kx, Ky, Kz) defined as the resonance frequency shift due to the hyperfine interaction of equation (4). The spin component Ks is obtained by subtracting the small orbital component K0 including the chemical shift and the Van–Vleck shift from the K–χ plot in Supplementary Figure S1. For a paramagnetic system, si in equation (4) is replaced by the effective electron spin polarization proportional to spin susceptibility χs. In a multi-orbital system, χs is composed of orbital-dependent spin susceptibilities. Then Ks is expressed by the sum of hyperfine fields from 3d spins. Whereas the isotropic part of Ks, , is given by
the anisotropic part, , is expressed as the arithmetic average of the orbital-dependent spin susceptibility weighted by the principal z component of the hyperfine coupling tensor, as shown inequation (1). In LiV2O4, with a local trigonal distortion, 1.5 electrons are filled in two orbitals, a1g and , whose qαβ are equivalent to those of and , respectively, taking the principal axes along the trigonal axis: qxx=qyy=1, qzz=−2 for (a1g)1, while the values are numerically the same but reversed in sign for . qαα vanishes when the two orbitals are equally occupied. Using the relation is expressed as
The good linearity in K–χ plots indicates , where f is the fraction of χa in χs. Then can be further reduced to
for the negligible as expected from the small K0. To experimentally obtain the effective orbital polarization , we take a ratio of Equations (7) and (9)) and cancel out the numerical constants and χs. Namely,
Here for a fully a1g polarized case (f=1), close to the experimentally obtained in LuV2O7 (ref. 18). From the experimental result in LiV2O4, we obtained f~0.8, corresponding to the occupation ratio and hence n~1 for a1g and n~0.25 for .
Electric hyperfine interactions
The electrostatic hyperfine interaction can be a direct probe for 3d orbital order. In the presence of the anisotropic electric field gradient around the nuclear spin, the 51V NMR spectrum is split into seven lines for I=7/2. The NMR spectrum becomes sharpest at , identifying the magic angle where the nuclear quadrupole splitting frequency δv vanishes. Then δv should have a maximum at θ0=54.7° satisfying δv~(3cos2θ0−1)=0, which is exactly equivalent to the local trigonal symmetry. We observed a quadrupole splitting frequency δvx=90 kHz in the 51V NMR spectra at H0||, by using a very short pulse interval time τ=3 μs. From the lattice symmetry, we can obtain . We confirmed that δvx was independent of temperature down to 2 K and hence the orbital occupation was invariant across T*.
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We thankful for the technical assistance by S. Inoue, and valuable discussion with N. Kawakami, Y. Motome, S. Watanabe, T. Tsunetsugu and S. Sachdev. This work was financially supported by the Grant-in-Aid for Scientific Research on the Priority Area, Novel State of Matter Induced by Frustration, (No. 22014006) from the MEXT and the Grants-in-Aid for Scientific Research (No. 22684018 and 24340080) from the JSPS.
The authors declare no competing financial interests.
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Shimizu, Y., Takeda, H., Tanaka, M. et al. An orbital-selective spin liquid in a frustrated heavy fermion spinel LiV2O4. Nat Commun 3, 981 (2012). https://doi.org/10.1038/ncomms1979
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