Abstract
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 LiV_{2}O_{4} having geometrical frustration. Utilizing orbitalselective ^{51}V NMR, we elucidate the orbitaldependent local moment that exhibits no longrange magnetic order despite persistent antiferromagnetic correlations. A frustrated spin liquid, Hundcoupled 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 orbitalselective localization in a wide range of materials including iron pnictides, cobaltates, manganites and ruthnates.
Introduction
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 phase^{1}. The microscopic understanding of heavy quasiparticles (HQs) has been a goal of modern manybody 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 felectron case, the presence of localized spins is not apparent for delectron 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 LiV_{2}O_{4} (refs 2,3), which has a highly frustrated pyrochlore lattice for the B site V^{3.5+} (3d^{1.5}) ions (Fig. 1a). Anisotropic orbitaldependent intersite interactions give an itinerant orbital and a more localized a_{1g} orbital through a small trigonal distortion of the VO_{6} octahedron (Fig. 1b)^{4,5}. The HQ was initially explained by offsite Kondo exchange interactions, J_{K}, between localized a_{1g} moments and itinerant electrons (Fig. 1b)^{4}. In the t_{2g} manifold, however, the strong onsite ferromagnetic Hund's exchange interaction, J_{H}, can overcome J_{K}. Many alternative scenarios, such as geometrical frustration via antiferromagnetic interactions J_{AF},^{6,7,8}, electron correlations^{9,10} and spinorbital fluctuations^{11,12} have been proposed.
Experimentally, the interpretations have been unclear for HQs in LiV_{2}O_{4}. Chargesensitive probes such as resistivity^{2}, photoemission^{13} and optical^{14} measurements showed crossover from a hightemperature incoherent metal to a lowtemperature Fermi liquid across the characteristic temperature T *~ 20–30 K. In contrast, spinsensitive probes including static spin susceptibility^{2,15} and inelastic neutron scattering^{16} 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 liquid^{2}. Despite the theoretical view of orbitalselective 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 orbitalresolved nuclear magnetic resonance (ORNMR) measurements in LiV_{2}O_{4}. The previous NMR experiments using closedshell Li sites^{2,15} 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 onsite ^{51}V spins on a highquality single crystal is sensitive to the orbitaldependent local spin susceptibility, which is beneficial for probing strongly correlated electrons with the orbital degrees of freedom.
Results
ORNMR Knight shift
In LiV_{2}O_{4}, the spin susceptibility χ^{s} consists of the a_{1g} and components: χ^{s}=χ^{a}+χ^{e} (hereafter the superscripts a and e denote a_{1g} 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 A_{i} between the nuclear spin I and the paramagnetic spin polarization for the ith electron under an external magnetic field. Whereas the isotropic shift due to the core polarization is proportional to χ^{s}, the anisotropic part due to the orbitalspecific dipole hyperfine interaction^{17} is expressed by using the principal z component of the coupling constants, and ,
where N is Avogadro's number and μ_{B} is the Bohr magneton. In contrast to measures the hyperfineweighted average of the orbitaldependent 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 χ^{a}>χ^{e} (χ^{a}<χ^{e}).
The ^{51}V Knight shift tensors of LiV_{2}O_{4} are determined from the angle dependence of K for 2–300 K, as shown in Fig. 2. The ^{51}V NMR spectra were detectable only at specific angles, where the nuclear quadrupole interaction almost vanishes, because the nuclear spinspin 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 VO_{6} crystal field (Fig. 1b). At 300 and 2 K, the relationship shows the a_{1g}dominant spin susceptibility (χ^{a}>χ^{e}). The result is consistent with the localized nature of the a_{1g} orbital, as theoretically suggested^{4,5,6,10}.
Temperature dependence of orbital occupations
To address HQ formation, we measured the thermal variations of the ^{51}V Knight shifts and in comparison with the ^{7}Li 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–Weisslike increase at high temperatures, followed by a broad maximum at approximately 20 K. The results agree with the spin susceptibility for highquality crystals, free from a Curietail increase at low temperatures^{2,3}. The onsite ^{51}V Knight shift probes the spin susceptibility with greater sensitivity than the offsite ^{7}Li one and shows a smooth decrease below T^{*}. Below 5 K, K^{s} becomes nearly T independent, as observed in the Fermi liquid.
When a_{1g} local moments becomes Fermi liquid with HQ via a_{1g} hybridization or intersite Kondo coupling below T^{*}, χ^{a} is expected to decrease significantly, whereas χ^{e} 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 a_{1g} orbital persists to the Fermi liquid state across T^{*}. Although of the felectron system has not been reported, the nonlinear relationship in the K–χ plot may be a manifestation of Kondo coupling in the Ce and Ubased compounds^{20,21,22}.
reflects the 3d orbital polarization when χ^{a} and χ^{e} scale to electron occupations. From we can evaluate the mixing ratio of the a_{1g} and orbitals in LiV_{2}O_{4}. The singly occupied a_{1g} orbital has , as observed in the insulating pyrochlore material Lu_{2}V_{2}O_{7} (ref. 18). In contrast, vanishes for equivalent mixing of a_{1g} and , as observed in a less correlated metal V_{2}O_{3} (Fig. 3b). The observed intermediate in LiV_{2}O_{4} 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 a_{1g} spins under the magnetic field, although the itinerancy of is much better than that of a_{1g}. The occupation ratio is evaluated as (see Methods), corresponding to the electron numbers of and for 3d^{1.5}. The halffilling a_{1g} occupation is distinct from that expected in the tightbinding calculation without electron correlations, where (ref. 12). However, it is compatible with the strongly localized a_{1g} picture owing to the strong renormalization into the Mott insulating state^{5,9,10} and provides microscopic evidence for orbitaldependent localization, which is robust across T^{*}.
Dynamical spin susceptibility of the orbitalselective spin liquid
Another interesting issue is the dynamical part probed by the nuclear spinlattice 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 ^{51}V and ^{7}Li sites. (T_{1}T)^{−1} measured for ^{51}V and ^{7}Li (Fig. 4a) follows the linear relationship , where the linear coefficient C=1.0×10^{3} is close to the square ratio of the hyperfine coupling, and C_{0}=64 s^{−1} K arises from the Tinvariant 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 χ^{s} at low temperatures likely comes from the shortrange antiferromagnetic correlation with the exchange interaction J_{AF}~150 K. Nevertheless, no longrange magnetic ordering occurs down to 1.5 K, the energy scale of ~J_{AF}/100. It suggests that the frustrated a_{1g} spins form in a quantum liquid at low temperatures with lowlying excitations.
Discussion
Our results provide significant insights into the formation of 3d HQ. As observed in the Tindependent , the a_{1g} 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 entropy^{6,7,8}. The itinerant electrons interact with the underlying spin liquid via the Hund's rule coupling. Thus, the 3d HQ behaviour in LiV_{2}O_{4} 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 Kondolike coherence peak appears on the boundary of the orbitalselective Mott transition for the a_{1g} part^{10}. Even in such a case, χ^{a} may vary across T^{*}, while χ^{e} is invariant. Our results suggest that, if the Kondolike 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 systems^{23}, where localized and itinerant characters coexist. Furthermore, the chirality degrees of freedom might provide appreciable entropy in the pyrochlore lattice^{24} and contribute to the anomalous Hall effect^{3}.
The ORNMR technique offers new holographic experiments that could give microscopic insights into strongly correlated electrons. Various orbitalresolved tools, such as Xray absorption and photoemission spectroscopy, have been recently developed. The Knight shift measurement has a unique advantage in detecting the orbitaldependent local spin susceptibility via the magnetic hyperfine interactions between d spins and onsite nuclear spins. The method has not been achieved in rareearth heavy fermion compounds because of the difficulty in detecting NMR signals for the onsite nuclear spins^{20}. Additional technical improvements in the NMR measurements may reveal the hidden orbitalselective Mott transition in transition metal oxides, such as ruthenates, pnictides and manganites.
Methods
NMR measurements
The ORNMR experiments were performed on a single crystal of LiV_{2}O_{4} synthesized by the selfflux method^{25}. The crystal with the octahedral shape was placed on a twoaxis goniometer and rotated under a fixed magnetic field H_{0}=9.402 and 8.490 T. The NMR spectra were obtained from spinecho signals after two π/2 pulses separated by a time τ. The ^{51}V NMR measurements were made only for powder samples above 50 K (ref. 15) likely due to the fast spinecho decay time T_{2} 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 T_{2}.
The angular dependence of the ^{51}V Knight shift K(θ) with the local trigonal symmetry is fitted into the general formula^{26}
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 by^{17}
where l_{i}, s_{i} and I denote operators of orbital and spin of the ith electron and the nuclear spin, respectively, the coefficient using the Bohr magneton μ_{B}, the nuclear gyromagnetic ratio γ_{n}, 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 vanadates^{18}, 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 spinorbit coupling parameter λ and the secondorder matrix elements between the ground and excited states, . In the LScoupling, 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 ^{51}V Knight shift
The experimental observable is the Knight shift tensor K=(K_{x}, K_{y}, K_{z}) defined as the resonance frequency shift due to the hyperfine interaction of equation (4). The spin component K^{s} is obtained by subtracting the small orbital component K_{0} including the chemical shift and the Van–Vleck shift from the K–χ plot in Supplementary Figure S1. For a paramagnetic system, s_{i} in equation (4) is replaced by the effective electron spin polarization proportional to spin susceptibility χ^{s}. In a multiorbital system, χ^{s} is composed of orbitaldependent spin susceptibilities. Then K^{s} is expressed by the sum of hyperfine fields from 3d spins. Whereas the isotropic part of K^{s}, , is given by
the anisotropic part, , is expressed as the arithmetic average of the orbitaldependent spin susceptibility weighted by the principal z component of the hyperfine coupling tensor, as shown inequation (1). In LiV_{2}O_{4}, with a local trigonal distortion, 1.5 electrons are filled in two orbitals, a_{1g} and , whose q_{αβ} are equivalent to those of and , respectively, taking the principal axes along the trigonal axis: q_{xx}=q_{yy}=1, q_{zz}=−2 for (a_{1g})^{1}, while the values are numerically the same but reversed in sign for . q_{αα} 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 χ^{a} in χ^{s}. Then can be further reduced to
for the negligible as expected from the small K_{0}. To experimentally obtain the effective orbital polarization , we take a ratio of Equations (7) and (9)) and cancel out the numerical constants and χ^{s}. Namely,
Here for a fully a_{1g} polarized case (f=1), close to the experimentally obtained in LuV_{2}O_{7} (ref. 18). From the experimental result in LiV_{2}O_{4}, we obtained f~0.8, corresponding to the occupation ratio and hence n~1 for a_{1g} 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 ^{51}V NMR spectrum is split into seven lines for I=7/2. The NMR spectrum becomes sharpest at [001], identifying the magic angle where the nuclear quadrupole splitting frequency δv vanishes. Then δv should have a maximum at θ_{0}=54.7° satisfying δv~(3cos^{2}θ_{0}−1)=0, which is exactly equivalent to the local trigonal symmetry. We observed a quadrupole splitting frequency δv_{x}=90 kHz in the ^{51}V NMR spectra at H_{0}[110], by using a very short pulse interval time τ=3 μs. From the lattice symmetry, we can obtain . We confirmed that δv_{x} was independent of temperature down to 2 K and hence the orbital occupation was invariant across T^{*}.
Additional information
How to cite this article: Shimizu Y. et al. An orbitalselective spin liquid in a frustrated heavy fermion spinel LiV_{2}O_{4}. Nat. Commun. 3:981 doi: 10.1038/ncomms1979 (2012).
References
 1
Gegenwart, P., Si, Q. & Steglich, F. Quantum criticality in heavyfermion metals. Nat. Phys. 4, 186–197 (2008).
 2
Kondo, S. et al. LiV2O4: a heavy fermion transition metal oxide. Phys. Rev. Lett. 78, 3729–3733 (1997).
 3
Urano, C. et al. LiV2O4 spinel as a heavymass Fermi liquid: anomalous transport and role of geometrical frustration. Phys. Rev. Lett. 85, 1052–1055 (2000).
 4
Anisimov, A. I. et al. Electronic structure of the heavy fermion metal LiV2O4 . Phys. Rev. Lett. 83, 364–367 (1999).
 5
Nekrasov, I. A. et al. Orbital state and magnetic properties of LiV2O4 . Phys. Rev. B 67, 085111 (2003).
 6
Lacroix, C. Heavy fermion behavior of itinerant frustrated systems: βMn, Y(Sc)Mn2 and LiV2O4 . Can. J. Phys. 79, 1469–1473 (2001).
 7
Burdin, S., Grempel, D. R. & Georges, A. Heavyfermion and spinliquid behaviour in a Kondo lattice with magnetic frustration. Phys. Rev. B 66, 045111 (2002).
 8
Hopkinson, J. & Coleman, P. LiV2O4: frustration induced heavy fermion metal. Phys. Rev. Lett. 89, 267201 (2002).
 9
Kusunose, H., Yotsuhashi, S. & Miyake, K. Formaiton of a heavy quasipartcle state in the twoband Hubbard model. Phys. Rev. B 62, 4403 (2002).
 10
Arita, R., Held, K., Lukoyanov, A. V. & Anisomov, V. I. Doped Mott insulator as the origin of heavy fermion behaviour in LiV2O4 . Phys. Rev. Lett. 98, 166402 (2007).
 11
Yamashita, Y. & Ueda, K. Spinorbital fluctuations and a large mass enhancement in LiV2O4 . Phys. Rev. B 67, 195107 (2003).
 12
Hattori, K. & Tsunetsugu, H. Effective Hamiltonian of a threeorbital Hubbard model on the pyrochlore lattice: application to LiV2O4 . Phys. Rev. B 79, 035115 (2009).
 13
Shimoyamada, A. et al. Heavyfermionlike state in a transition metal oxide LiV2O4 single crystal: indication of Kondo resonance in the photoemission spectrum. Phys. Rev. Lett. 96, 026403 (2006).
 14
Jönsson, P. E. et al. Correlationdriven heavyfermion formation in LiV2O4 . Phys. Rev. Lett. 99, 167402 (2007).
 15
Mahajan, A. V. et al. ^{7}Li and ^{51}V NMR study of the heavyfermion compound LiV2O4 . Phys. Rev. B 57, 8890–8899 (1998).
 16
Lee, S. H. et al. Spin fluctuations in a magnetically frustrated metal LiV2O4 . Phys. Rev. Lett. 86, 5554–5557 (2001).
 17
Abragam, A. & Bleaney, B. Electron Paramagnetic Resonance of Transition Ions, (Oxford University Press, London, 1970).
 18
Kiyama, T. et al. Direct observation of the orbital state in Lu2V2O7: a ^{51}V NMR study. Phys. Rev. B 73, 184422 (2006).
 19
Moriya, T. The effect of the electronelectron on the nuclear spinlattice relaxation rate in metals. J. Phys. Soc. Jpn. 18, 516–520 (1963).
 20
Curro, N. J., Young, B. L., Schmalian, J. & Pines, D. Scaling in the emergent behavior of heavyelectron materials. Phys. Rev. B 70, 235117 (2004).
 21
Sakai, H. B. et al. ^{59}Co NMR shift anomalies and spin dynamics in the normal state of superconducting CeCoIn5: verification of twodimensional antiferromagnetic spin fluctuations. Phys. Rev. B 82, 020501R (2010).
 22
Kambe, S. et al. Onecomponent description of magnetic excitations in the heavyfermion compound CeIrIn5 . Phys. Rev. B 81, 140405R (2010).
 23
Sachdev, S. Holographic metals and the fractionalized Fermi liquid. Phys. Rev. Lett. 105, 151602 (2010).
 24
Udagawa, M. & Motome, Y. Chiralitydriven mass enhancement in the kagome Hubbard model. Phys. Rev. Lett. 104, 106409 (2010).
 25
Matsushita, Y., Ueda, H. & Ueda, Y. Flux crystal growth and thermal stabilities of LiV2O4 . Nat. Mater. 4, 845–850 (2005).
 26
Volkoff, G. M., Petch, H. E. & Smellie, D. W. Nuclear electric quadrupole interaction in single crystals. Can. J. Phys. 30, 270–289 (1952).
Acknowledgements
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 GrantinAid for Scientific Research on the Priority Area, Novel State of Matter Induced by Frustration, (No. 22014006) from the MEXT and the GrantsinAid for Scientific Research (No. 22684018 and 24340080) from the JSPS.
Author information
Affiliations
Contributions
The research reported in the manuscript was carried out by H. Takeda, M.T. and Y.S. under the supervision of M.I. The sample was synthesized by S.N. and H. Takagi. The manuscript was written by Y.S.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figure S1 (PDF 49 kb)
Rights and permissions
About this article
Cite this article
Shimizu, Y., Takeda, H., Tanaka, M. et al. An orbitalselective spin liquid in a frustrated heavy fermion spinel LiV_{2}O_{4}. Nat Commun 3, 981 (2012). https://doi.org/10.1038/ncomms1979
Received:
Accepted:
Published:
Further reading

Orbital molecules in vanadium oxide spinels
Physical Review B (2020)

High pressure crystal structures of orthovanadates and their properties
Journal of Applied Physics (2020)

Occupation switching of d orbitals in vanadium dioxide probed via hyperfine interactions
Physical Review B (2020)

Giant, unconventional anomalous Hall effect in the metallic frustrated magnet candidate, KV3Sb5
Science Advances (2020)

Degenerate antiferromagnetic states in spinel oxide LiV2O4
Chinese Physics B (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.