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Orbital-exchange and fractional quantum number excitations in an f-electron metal, Yb2Pt2Pb

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Science  03 Jun 2016:
Vol. 352, Issue 6290, pp. 1206-1210
DOI: 10.1126/science.aaf0981

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, Yb2Pt2Pb, 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 Yb2Pt2Pb 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 CuEmbedded Image. 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 Embedded Image 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 Yb2Pt2Pb that profoundly challenge this conventional wisdom.

The unusual properties of Yb2Pt2Pb derive in part from its crystal structure (Fig. 1, A and B), where the YbEmbedded Image 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 Embedded Image planes. Equally important is the strong spin-orbit coupling, which combines spin and orbital degrees of freedom into a large, Embedded Image 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 YbEmbedded Image moments, producing a Kramers doublet ground state that is a nearly pure state of the total angular momentum, Embedded Image, Embedded Image. The estimated anisotropy of the Landé Embedded Image factor is in good agreement with that of the measured magnetization, Embedded Image (47), implying strong Ising anisotropy in Yb2Pt2Pb, which confines the individual Yb moments to two orthogonal sublattices in the ab plane.

Fig. 1 Quantum orbital-spin chains in Yb2Pt2Pb.

(A) Crystal structure of Yb2Pt2Pb; red arrows show the lattice axes. (B) The double chain magnetic structure for Embedded Image K without magnetic field (top) and in a 4 T field applied along the (1-10) direction (bottom). The Yb orbitals are depicted as the isosurfaces, at 1 part per million electronic density, of the 4f Embedded Image hydrogenic wave functions for an effective Slater nuclear charge of Embedded ImageYb (29). Blue arrows indicate the ordered magnetic moment directions, which are parallel to the local Ising easy axes, horizontal for the (110) and vertical for the (1-10) sublattice. Crystal axes are shown by black arrows. Thick red arrow shows magnetic-field direction. (C) Orbital overlaps for antiferromagnetic (B = 0) and fully saturated (B = 4 T) state. (D) Illustration of the two-spinon excitation process via spin flip (magnon) creation in Embedded Image antiferromagnetic chain. Such processes correspond to the change of angular momentum, Embedded Image, and are allowed by selection rules that govern interaction with a physical field, such as magnetic field of a neutron, or a photon. (E) For Embedded Image, angular momenta in Yb2Pt2Pb, flipping magnetic moment requires Embedded Image and therefore cannot be induced via single-particle processes. The only processes allowed by the selection rules are those with Embedded Image, such as when two electrons hop, exchanging their orbitals, i.e., pairwise permutations of electrons. Permutation of two nearest-neighbor electrons creates two spinons, while further-neighbor hopping, such as the permutation of the next-nearest-neighbor electrons in opposite-polarity orbitals illustrated in the figure, results in a four-spinon state.

The quantum states of the Embedded Image Ising doublet are the superpositions of its “up” and “down” components, Embedded Image, 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 Embedded Image can only change the total angular momentum quantum number by Embedded Image; 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 Yb2Pt2Pb would display only static, classical Ising behavior, but our data are not consistent with this picture.

Here we report neutron-scattering experiments on Yb2Pt2Pb that reveal a continuum of low-energy quantum excitations that display the distinctive spinon dispersion along the c axis (Fig. 2A), typical of the Embedded Image Heisenberg-Ising XXZ spin Hamiltonian (8), Embedded Image(1)where J is the Heisenberg spin-exchange coupling and Embedded Image is its anisotropy. This observation provides definitive evidence that the Yb moments in Yb2Pt2Pb behave as quantum-mechanical spins-1/2 (9). The spinon spectrum Embedded Image is fully gapped, but the gap is much smaller than the excitation bandwidth, indicating only moderate Ising anisotropy, Embedded Image. The lack of any wave vector Embedded Image 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.

Fig. 2 Fractional spinon excitations in Yb2Pt2Pb.

(A) The dispersion of the spectrum of magnetic excitations along the Embedded Image direction in reciprocal space of Yb2Pt2Pb at T = 0.1 K, obtained by averaging the scattered neutron intensity over the first Brillouin zone in Embedded Image, along the perpendicular, Embedded Image direction. Circles: onset of the excitation continuum determined by fitting the constant-Embedded Image data. Solid white lines: lower and upper boundaries of the two-spinon continua. Dashed lines: upper boundaries of the four-spinon continua (6). (B) The dispersion of the scattered neutron intensity along the Embedded Image direction for Embedded Image. (C) The partial static structure factor, Embedded Image, obtained by integrating the scattered intensity from 0.15 to 1.5 meV. Embedded Image depends on the relative orientation of the scattering vector Embedded Image and the direction of magnetic moment fluctuations (30). (D) At 4 T, Embedded Image (red squares) follows the projection of the (110) sublattice moments on the scattering wave vector (polarization factor Embedded Image, red line), indicating that only magnetic fluctuations along the (110) moments contribute to magnetic scattering. Both polarizations are present when Embedded Image T, consistent with equal longitudinal spinon spectral weight in both (110) and (1-10) sublattices. By subtracting the 4 T data from the 0 T data (black dots), we can isolate the scattering from fluctuations along the field direction, (1-10), which are always perpendicular to the scattering vector Embedded Image and whose polarization factor is constant (black line). The agreement between the polarization factors and our data confirms that only fluctuations of the Yb moments that lie along the respective (110) or (1-10) direction are seen in our experiment and that there is no measurable transverse component of magnetic moment or excitations. Error bars in all figures represent 1 SD.

The overall wave vector dependence of the energy-integrated intensity Embedded Image (Fig. 2, C and D) reveals that the excitations in each of the two orthogonal sublattices of Yb moments in Yb2Pt2Pb are longitudinally polarized. This is clearly demonstrated in Fig. 2D, where the Embedded Image dependence on Embedded Image in the Embedded Image 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 Embedded Image. The longitudinal character of magnetic excitations in Yb2Pt2Pb 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 Yb2Pt2Pb 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, 1012), 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 (47), 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 Embedded Image XXZ Hamiltonian, we fit the energy cuts at different values of Embedded Image 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, Embedded Image, 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),

Fig. 3 Spinon line shapes and the onset of the continuum in Yb2Pt2Pb.

The spectrum of the dynamical structure factor of magnetization fluctuations, Embedded Image, in Yb2Pt2Pb for (A) Embedded Image (Embedded Image) and (B) Embedded Image (Embedded Image), both integrated within 1 Brillouin zone in Embedded Image . The red lines show fits to the “half-Lorentzian” line shape (blue) convoluted with the Gaussian of 0.1 meV full width at half-maximum representing the resolution of the DCS spectrometer (light gray), (14). (C) The energy-integrated scattering function Embedded Image obtained by summing the normalized data over the first Brillouin zone in Embedded Image compares favorably with that calculated for the effective Embedded Image Heisenberg-Ising Hamiltonian with Embedded Image and Embedded Image meV and with the effective g-factor Embedded Image (blue line), and with Embedded Image, Embedded Image meV, and Embedded Image (green line). A fit to the leading Ising-limit (Embedded Image1) expression, Embedded Image, (red line) is less satisfactory, emphasizing that effective spin-1/2 Hamiltonian in Yb2Pt2Pb is not extremely Ising-like. This is consistent with the observation that the gap in the spinon spectrum, Embedded Image meV (B), is markedly smaller than the bandwidth, Embedded Image meV, Fig. 2A. (D) The energy dependence of the Embedded Image-integrated intensity, which represents the local dynamical structure factor, Embedded Image, of magnetization fluctuations in Yb2Pt2Pb. The energy-integral of the Embedded Image inelastic intensity (black dashed line, right scale) gives the square of the total fluctuating magnetic moment of Embedded Image per Yb. Computational results for Embedded Image and its energy integral are compared for Embedded Image (red solid and dashed lines) and Embedded Image (blue solid and dashed lines).

Embedded Image(2)

Here Embedded Image is the gap and Embedded Image the bandwidth of the spinon dispersion, Embedded Image, both of which are functions of the J and Embedded Image parameters of the Hamiltonian Eq. 1 (6). The fit yields values Embedded Image meV and Embedded Image meV for the spinon dispersion parameters, which correspond to Embedded Image, and J Embedded Image meV in the effective spin-1/2 XXZ Hamiltonian, and the excitation gap at Embedded Image Embedded Image, Embedded Image 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 Embedded Image, as evidenced by the smallness of the excitation gap Embedded Image compared to their observed bandwidth Embedded Image 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 Yb2Pt2Pb is evident in the broad peak at Embedded Image in the structure factor Embedded Image found by integrating the experimental and computed spectra in energy (Fig. 3C). Embedded Image is intermediate between the near divergence expected for isotropic interactions (Embedded Image) and the leading-order Ising expression (15) Embedded Image, where Embedded Image is Yb magnetic moment, Embedded Image being the effective spin-1/2 g-factor for the local Ising direction. Crystal electric field calculations for the Yb ground state doublet in Yb2Pt2Pb indicate Embedded Image and Embedded Image (6), so that magnetic neutron-scattering intensity, which is proportional to Embedded Image, is at least a factor of 100 weaker for the transverse, XY-polarized fluctuations, in agreement with what we observe.

The Embedded Image-integrated scattering in Fig. 3C yields a fluctuating moment Embedded Image/Yb at 0.1 K, about half as large as the ordered moment Embedded Image determined in previous work (16). The energy integral of the local autocorrelation function Embedded Image, which is obtained by integrating the measured intensity in Embedded Image, yields a similar result, Embedded Image/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 Embedded Image (Embedded Image). Therefore, the sum Embedded Image gives a total Yb moment, Embedded Image. Combining the inelastic spectrum and the elastic order parameter measurements in Yb2Pt2Pb (6), we find Embedded Image that is between 3.8 and 4.4 Embedded Image [Embedded Image] 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 Yb2Pt2Pb, 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 Embedded Image(T) computed for spin-1/2 XXZ chain with Embedded Image closely reproduces direct measurements of Embedded Image (Fig. 4B).

Fig. 4 Spinons in Yb2Pt2Pb: theory and experiment.

(A) Temperature dependencies of the ordered Yb moment from neutron-diffraction measurements (black circles), the fluctuating moment from the energy and wave-vector integrated normalized Embedded Image (red points), and the total (blue points). (B) The temperature dependence of the static, uniform magnetic susceptibility Embedded Image for Yb2Pt2Pb (black circles), measured with a magnetic properties measurement system (Embedded Image K) and a Hall sensor magnetometer (0.2 KEmbedded Image K), shows good agreement with Embedded Image calculated for the XXZ chain, for Embedded Image and Embedded Image meV (red line, Embedded Image). Agreement is less good for Embedded Image and J Embedded Image meV (blue line, Embedded Image). (C and D) The longitudinal structure factor, Embedded Image, of the XXZ spin-1/2 chain Eq. 1 calculated using the algebraic Bethe ansatz (11, 12, 20) (C) for Embedded Image and Embedded Image meV and (D) for Embedded Image and J Embedded Image meV. The experimentally determined lower boundary Embedded Image is shown (circles) along with the calculated lower and upper two-spinon (solid lines) and the upper four-spinon (broken lines) boundaries, as described in the text. The calculation is normalized to the total experimental intensity by using the sum rule for a single component of the dynamical structure factor, which holds for spin-1/2, Embedded Image.

Further comparison with the exact result (19) for the XXZ model (Eq. 1), however, indicates that the fluctuations measured in Yb2Pt2Pb at 0.1 K are stronger than the predicted spinon contribution to the dynamical spin structure factor, which for Embedded Image is only Embedded Image of the ordered spin contribution Embedded Image. Figure 3D makes it clear that the calculated Embedded Image underestimates the contribution of the high-energy states in Yb2Pt2Pb. 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, Embedded Image, 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 Embedded Image and adjusting Embedded Image and J as fit parameters instead of adopting the values determined from the lower boundary of the continuum. This results in Embedded Image 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 Embedded Image (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 Yb2Pt2Pb 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 Embedded Image multiplet. The intersite electron hopping in the f-electron Hamiltonian for Yb2Pt2Pb, 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 Yb2Pt2Pb 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 Yb2Pt2Pb is tied to the wave function of a single 4f hole with orbital momentum Embedded Image, having sixfold symmetry around the Embedded Image quantization axis, given by the magnetic structure as perpendicular to the rails of Yb ladders in Yb2Pt2Pb crystal. The energy cost for hopping between sites, which in Yb2Pt2Pb is synonymous with orbital exchange, is reduced when neighboring Yb ions are in alternating states of Embedded Image, 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, Embedded Image, which in the cases where only electronic spins are at play, reduces to the usual Heisenberg spin exchange, Embedded Image. For the case of a Embedded Image-manifold, which in the absence of crystal fields is highly degenerate, it has the form of a permutation operator acting on a Embedded Image-dimensional space of two neighboring Yb ions. The permutation operator interchanges states Embedded Image and Embedded Image with equal weights, thus including the process Embedded Image to Embedded Image 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 Embedded Image has the form of the antiferromagnetic Embedded Image XXZ Hamiltonian, it retains the birthmark of its unusual origin in exchange processes that are distinct from those having the conventional Heisenberg Embedded Image form.

The effective spin-1/2 physics emerges in Yb2Pt2Pb 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, Embedded Image 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 Yb2Pt2Pb 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 Yb2Pt2Pb, 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 Yb2Pt2Pb, 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 (2628).

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 (3153)

References and Notes

  1. See supplementary materials on Science Online for details.
  2. More information about the instrument configuration is on the DCS website: www.ncnr.nist.gov/instruments/dcs/.
  3. More information about the instrument configuration is on the CNCS website: http://neutrons.ornl.gov/cncs/.
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.
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