## Orbitals and charge go their separate ways

In certain materials at very low temperatures, an electron's spin can separate from its charge, zooming through the crystal in the form of a “spinon.” Such materials are usually one-dimensional, and their atoms have spins of 1/2. Wu *et al.* observed related behavior in a three-dimensional metal, Yb_{2}Pt_{2}Pb, where the Yb ions have a large magnetic moment that has its origin in the electrons' orbital motion rather than their spin. Neutron-scattering measurements indicated that these large magnetic moments can flip their direction through an exchange process similar to the one that occurs in spin 1/2 systems. This process results in effective charge-orbital separation.

*Science*, this issue p. 1206

## Abstract

Exotic quantum states and fractionalized magnetic excitations, such as spinons in one-dimensional chains, are generally expected to occur in 3d transition metal systems with spin 1/2. Our neutron-scattering experiments on the 4f-electron metal Yb_{2}Pt_{2}Pb overturn this conventional wisdom. We observe broad magnetic continuum dispersing in only one direction, which indicates that the underlying elementary excitations are spinons carrying fractional spin-1/2. These spinons are the emergent quantum dynamics of the anisotropic, orbital-dominated Yb moments. Owing to their unusual origin, only longitudinal spin fluctuations are measurable, whereas the transverse excitations such as spin waves are virtually invisible to magnetic neutron scattering. The proliferation of these orbital spinons strips the electrons of their orbital identity, resulting in charge-orbital separation.

It is generally believed that fractional quantum excitations such as spinons in one-dimensional (1D) spin chains proliferate and govern magnetism only in systems with small and isotropic atomic magnetic moments, such as spin-1/2 Cu. In contrast, large and anisotropic orbital-dominated moments, such as those produced by strong spin-orbit coupling in the rare earths, are considered to be classical, becoming static as temperature because the conventional Heisenberg-Dirac exchange interaction (*1*, *2*) cannot reverse their directions. Here we present the results of neutron-scattering measurements on the 3D compound Yb_{2}Pt_{2}Pb that profoundly challenge this conventional wisdom.

The unusual properties of Yb_{2}Pt_{2}Pb derive in part from its crystal structure (Fig. 1, A and B), where the Yb ions form ladders along the *c* axis, separated by Pt and Pb; the rungs of the ladders (dashed lines in Fig. 1A) lie on the orthogonal bonds of the Shastry-Sutherland lattice (SSL) (*3*) in the planes. Equally important is the strong spin-orbit coupling, which combines spin and orbital degrees of freedom into a large, Yb moment. The absence of a Kondo effect indicates minimal coupling of Yb to the conduction electrons of this excellent metal (*4*, *5*). A point-charge model (*6*) indicates that the crystal electric field (CEF) lifts the eightfold degeneracy of the Yb moments, producing a Kramers doublet ground state that is a nearly pure state of the total angular momentum, , . The estimated anisotropy of the Landé factor is in good agreement with that of the measured magnetization, (*4*–*7*), implying strong Ising anisotropy in Yb_{2}Pt_{2}Pb, which confines the individual Yb moments to two orthogonal sublattices in the *ab* plane.

The quantum states of the Ising doublet are the superpositions of its “up” and “down” components, , and therefore the doublet can be viewed as an effective quantum spin-1/2. However, familiar interactions like the Zeeman, Heisenberg-Dirac exchange, and dipole interactions that are bilinear in can only change the total angular momentum quantum number by ; they have no matrix elements that would allow transitions between the moment-reversed states of the ground state wave function. Only multiple virtual processes involving excited states could reverse individual Yb moments, but these processes are expected to be very weak because the ground and first excited states are separated by as much as 25 meV, according to specific heat (*4*) and inelastic neutron-scattering measurements (*6*). This would suggest that Yb_{2}Pt_{2}Pb would display only static, classical Ising behavior, but our data are not consistent with this picture.

Here we report neutron-scattering experiments on Yb_{2}Pt_{2}Pb that reveal a continuum of low-energy quantum excitations that display the distinctive spinon dispersion along the *c* axis (Fig. 2A), typical of the Heisenberg-Ising XXZ spin Hamiltonian (*8*),
(1)where *J* is the Heisenberg spin-exchange coupling and is its anisotropy. This observation provides definitive evidence that the Yb moments in Yb_{2}Pt_{2}Pb behave as quantum-mechanical spins-1/2 (*9*). The spinon spectrum is fully gapped, but the gap is much smaller than the excitation bandwidth, indicating only moderate Ising anisotropy, . The lack of any wave vector dispersion for this gap (Fig. 2B), or for the scattering intensity in the *ab* plane (Fig. 2C), indicates that the dispersing excitations are confined to the ladder rails, which form an array of weakly coupled spin-1/2 chains.

The overall wave vector dependence of the energy-integrated intensity (Fig. 2, C and D) reveals that the excitations in each of the two orthogonal sublattices of Yb moments in Yb_{2}Pt_{2}Pb are longitudinally polarized. This is clearly demonstrated in Fig. 2D, where the dependence on in the scattering plane is very accurately described by the projections of Yb moments on the wave vector, consistent with the polarization factor in the neutron-scattering cross-section, which is only sensitive to magnetic fluctuations perpendicular to . The longitudinal character of magnetic excitations in Yb_{2}Pt_{2}Pb is a direct consequence of the strong orbital anisotropy imposed by the crystal field and the resulting strongly anisotropic Landé *g-*factor. Even if the effective spin Hamiltonian that describes the low-energy dynamics in Yb_{2}Pt_{2}Pb has modes involving transverse spin fluctuations, such as spin waves, they virtually do not couple to physical fields at our disposal and are de facto invisible in experiments. In particular, the measured longitudinal spectrum, which is typical of a spin-1/2 XXZ chain (Fig. 2), indicates the presence of transverse spinon excitations (*8*, *10*–*12*), but these are not seen in experiments. That the XY-part of the effective spin Hamiltonian Eq. 1 is unobservable results from the well-understood effect of quantum selection rules. The direct consequence for our measurements is that we do not observe a (transverse) magnon, which is expected (*13*) when a magnetic field *B* = 4 T applied along (1-10) crystal direction saturates Yb moments that are parallel to the field (*4*–*7*), bringing this sublattice to the ferromagnetic (FM) state (Fig. 1B). Instead, FM chains do not contribute to magnetic scattering, and this allows us to use the 4 T data as a background that can isolate their contribution at *B* = 0 (Fig. 2D).

To establish the hierarchy of energy scales in the effective XXZ Hamiltonian, we fit the energy cuts at different values of to a phenomenological half-Lorentzian line shape (*14*), which accounts both for the sharp continuum onset and its broad, asymmetric extent to higher energies (Fig. 3, A and B). We can thus very accurately determine the lower boundary, , of the spinon continuum (points in Fig. 2A), which we fit to the exact Bethe-Ansatz expression for the XXZ Hamiltonian (Eq. 1) (*8*, *10*, *11*),

Here is the gap and the bandwidth of the spinon dispersion, , both of which are functions of the *J* and parameters of the Hamiltonian Eq. 1 (*6*). The fit yields values meV and meV for the spinon dispersion parameters, which correspond to , and *J* meV in the effective spin-1/2 XXZ Hamiltonian, and the excitation gap at , meV. Despite the strong anisotropy of the individual Yb moments, their inferred coupling in the spin chain is surprisingly close to the isotropic Heisenberg limit , as evidenced by the smallness of the excitation gap compared to their observed bandwidth meV (Fig. 2A).

Computations carried out on the XXZ Hamiltonian Eq. 1 closely reproduce key aspects of our experimental results. The mixed Heisenberg-Ising character of Yb_{2}Pt_{2}Pb is evident in the broad peak at in the structure factor found by integrating the experimental and computed spectra in energy (Fig. 3C). is intermediate between the near divergence expected for isotropic interactions () and the leading-order Ising expression (*15*) , where is Yb magnetic moment, being the effective spin-1/2 *g*-factor for the local Ising direction. Crystal electric field calculations for the Yb ground state doublet in Yb_{2}Pt_{2}Pb indicate and (*6*), so that magnetic neutron-scattering intensity, which is proportional to , is at least a factor of 100 weaker for the transverse, XY-polarized fluctuations, in agreement with what we observe.

The -integrated scattering in Fig. 3C yields a fluctuating moment /Yb at 0.1 K, about half as large as the ordered moment determined in previous work (*16*). The energy integral of the local autocorrelation function , which is obtained by integrating the measured intensity in , yields a similar result, /Yb (Fig. 3D), with the difference indicating a systematic error resulting from different data binning. The sum rule for the effective spin-1/2 dictates that the integral intensity in each polarization channel is (). Therefore, the sum gives a total Yb moment, . Combining the inelastic spectrum and the elastic order parameter measurements in Yb_{2}Pt_{2}Pb (*6*), we find that is between 3.8 and 4.4 [] for temperatures from 0.1 to 100 K (Fig. 4A), fully consistent with the predictions of the point charge model. The spinons provide virtually all of the magnetic dynamics in Yb_{2}Pt_{2}Pb, and they are completely captured by our experiments. This result immediately rules out a naïve explanation that the observed longitudinal magnetic response could originate from the two-magnon continuum, as in conventional magnets, because in that case the continuum would comprise only a small part of the dynamical spectral weight (*17*, *18*). Moreover, stable magnons do not exist in an antiferromagnetic spin-1/2 chain, where the elementary excitations are spinons, and the system’s one-dimensionality is clearly established by the measured dispersion (Fig. 2, A to C). Finally, the static spin susceptibility (*T*) computed for spin-1/2 XXZ chain with closely reproduces direct measurements of (Fig. 4B).

Further comparison with the exact result (*19*) for the XXZ model (Eq. 1), however, indicates that the fluctuations measured in Yb_{2}Pt_{2}Pb at 0.1 K are stronger than the predicted spinon contribution to the dynamical spin structure factor, which for is only of the ordered spin contribution . Figure 3D makes it clear that the calculated underestimates the contribution of the high-energy states in Yb_{2}Pt_{2}Pb. Direct comparison of the detailed energy dependencies of the measured (Fig. 2A) and computed (broadened by the instrumental resolution of 0.1 meV) (Fig. 4C) spectra of longitudinal excitations reveals that there is considerable spectral weight present in the experimental data above the upper boundary of the two-spinon continuum, , that is absent in the computed spectrum (*10*, *11*). A somewhat better agreement can be obtained by fitting the measured intensity to the calculated longitudinal structure factor and adjusting and *J* as fit parameters instead of adopting the values determined from the lower boundary of the continuum. This results in and *J* = 0.205 meV (Fig. 4D), shifting the two-spinon spectral weight to higher energy and also providing better agreement with the measured susceptibility (Fig. 4B) and (Fig. 3C). However, this improvement is achieved at the cost of the excellent experimental and theoretical agreement for the lower spinon boundary, which, in fact, is determined very precisely from the line fits (Fig. 3, A and B). This dilemma is resolved by noting that the observed high-energy magnetic spectral weight in Yb_{2}Pt_{2}Pb is consistent with a substantial contribution of four-spinon states, whose upper boundaries (*12*) are shown by the broken lines in Fig. 2A. This result is quite unexpected, given that two-spinon excitations account for all but a few percent of the total spectral weight (*12*, *13*, *20*) in the nearest-neighbor Heisenberg-Ising chain.

We now show that these seemingly perplexing experimental results can be understood in terms of the interplay of 4f-electron exchange, strong spin-orbit coupling, and a crystal field that lifts the large orbital degeneracy of the multiplet. The intersite electron hopping in the f-electron Hamiltonian for Yb_{2}Pt_{2}Pb, which we adopt in the form of a 1D Hubbard model (*6*), leads to an electronic interaction (*21*) whose physical nature is not a Heisenberg-Dirac spin exchange (*1*, *2*), but rather an orbital exchange (Fig. 1), a realization that has been appreciated in the physics of Kondo effect (*22*, *23*) and more recently in certain cold-atom systems (*24*).

The orbital-exchange interaction in Yb_{2}Pt_{2}Pb is a natural generalization of the Heisenberg-Dirac spin exchange between the two electrons, and has the same physical origin in the electronic Coulomb repulsion (*1*, *2*). The magnetism in Yb_{2}Pt_{2}Pb is tied to the wave function of a single 4f hole with orbital momentum , having sixfold symmetry around the quantization axis, given by the magnetic structure as perpendicular to the rails of Yb ladders in Yb_{2}Pt_{2}Pb crystal. The energy cost for hopping between sites, which in Yb_{2}Pt_{2}Pb is synonymous with orbital exchange, is reduced when neighboring Yb ions are in alternating states of , because in that case, the exchange of electrons between the two sites required for hopping involves the overlap of two identical orbital lobes along the ladder rails (Fig. 1, B and C). The sixfold symmetry of the f-orbital breaks the rail-rung equivalence and ensures that this energy advantage is not accrued for hopping in a transverse direction, decoupling the ladder rails. Combined with the weak interactions between orthogonal ladders mandated by the SSL geometry (*4*), this leads to the spin-chain nature of the emergent effective Hamiltonian.

The leading-order Coulomb contribution for the low-energy manifold of electronic states (*6*, *25*) is given by the two-electron permutation operator, , which in the cases where only electronic spins are at play, reduces to the usual Heisenberg spin exchange, . For the case of a -manifold, which in the absence of crystal fields is highly degenerate, it has the form of a permutation operator acting on a -dimensional space of two neighboring Yb ions. The permutation operator interchanges states and with equal weights, thus including the process to where both moments simultaneously reverse, which cannot be achieved through conventional Heisenberg-Dirac spin exchange (Fig. 1, D and E). The crystal field lifts the degeneracy of the Yb moments, and although the effective interaction that emerges after the projection on the manifold of the lowest Kramers doublets has the form of the antiferromagnetic XXZ Hamiltonian, it retains the birthmark of its unusual origin in exchange processes that are distinct from those having the conventional Heisenberg form.

The effective spin-1/2 physics emerges in Yb_{2}Pt_{2}Pb from the combination of high-energy (Coulomb, spin-orbit, hopping) interactions. The spin-orbit coupling virtually quenches the electronic spin degree of freedom, forcing its alignment with the large orbital moment, and in this way the effective spin-1/2 XXZ model effectively describes the quantum dynamics of the electronic orbital degree of freedom. This is directly evidenced in our experiments by the large, magnetic moment carried by spinons. The orbital exchange sets the scale for these emergent quantum dynamics, which we find by comparing the measured spinon dispersion with computed spectra (Fig. 4).

Because the orbital angular momentum dominates the total Yb moment, magnetic order in Yb_{2}Pt_{2}Pb is synonymous with orbital order, and the configuration depicted in Fig. 1, D and E, is a natural way to understand how permutation of two neighboring electrons generates two spinons in the antiferromagnetic background. This is a process that entails charge-orbital separation, because the electron count per site is unchanged by correlated hopping, but the phases of the orbital wave function on both sites are reversed. Further-neighbor orbital exchange leads to states with four spinons (Fig. 1E). Hence, long-range hopping, either by virtue of the in-chain itinerancy of the 4f electrons or via coupling to the conduction electrons in metallic Yb_{2}Pt_{2}Pb, provides a natural mechanism for the spectral weight of the excitations that we observe above the two-spinon but within the four-spinon continuum boundaries.

Our results provide a specific mechanism for charge-orbital separation in Yb_{2}Pt_{2}Pb, where the proliferation of spinons implies that electrons lose their orbital-phase identity. When united with the previous demonstrations of spin-charge and spin-orbital separation, this finding completes the triad of electron fractionalization phenomena in one dimension (*26*–*28*).

**Correction (21 June 2016):** In the Acknowledgments, the NSF award for work at Stony Brook was incorrectly listed as NSF-DMR-131008; the correct number is NSF-DMR-1310008.

## Supplementary Materials

www.sciencemag.org/content/352/6290/1206/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S10

Tables S1 to S3

## References and Notes

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- More information about the instrument configuration is on the CNCS website: http://neutrons.ornl.gov/cncs/.
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**Acknowledgments:**Work at Brookhaven National Laboratory (I.A.Z., A.M.T., M.S.K.) was supported by the Office of Basic Energy Sciences (BES), Division of Materials Sciences and Engineering, U.S. Department of Energy (DOE), under contract DE-SC00112704. Work at Stony Brook (L.S.W., W.J.G., M.C.A.) was supported by NSF-DMR-1310008. L.S.W. was also supported by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory (ORNL). This research at ORNL’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. DOE. Work at NIST Center for Neutron Research (NCNR) is supported in part by the NSF under Agreement no. DMR-1508249. J.-S.C. and M.B. acknowledge support from the Netherlands Organization for Scientific Research (NWO) and the Foundation for Fundamental Research on Matter (FOM) of the Netherlands.