Spin-orbital States Research Articles

Spin-orbit entanglement driven by the Jahn-Teller effect

Spin-orbit entanglement in 4d and 5d transition metal systems can enhance electronic correlations, leading to nontrivial ground states and the emergence of exotic excitations. There is also an interest to investigate spin-orbit entanglement in 3d compounds, though this is challenging due to their smaller spin-orbit coupling. Here we demonstrate that the Jahn-Teller effect in Mn3+ reduces the energy gap between high- and low- spin-orbital states that lead to enhanced spin-orbit entanglement. Our results show a rare example of synergistic effects of Jahn-Teller and spin-orbit interactions and provide a way to entangle different degrees of freedom in d-metal oxides, which may allow paths to explore the interplay of orbital, lattice and spins in 3d correlated systems.

Combined 3d-4f magnetic interactions to manipulate long-range ferromagnetism in monoclinic double perovskites Gd-substituted [formula omitted][formula omitted]

Oxide double perovskites (DPs) with half-filled 4f Rare-earth (RE) elements along with partially-filled 3d transition-metals (TM) are rich source of exotic spin-orbital properties. Herein, we have investigated explicitly, the structural properties, spin-orbital ground states of the bulk phases and electronic structure of the RE-double perovskites Sm2−xGdxCoMnO6x=0.0,0.5,1.0,1.5 by synthesizing high quality single-phase samples and undertaking high precision experimental measurements of the transport and magnetic properties down to 8 K. In fact, we are able to show that with Gd-substitution, it lowers the value of the magnetoresistance in Sm2CoMnO6 (SCMO) and it is consistent with the observed Curie-Weiss temperature (TCW) which is lower than the observed ordering temperature (TC) upon Gd-substitution of Sm3+. Native presence of magnetic impurities (Co3+ and Mn3+) with low-spin S = 0 configurations in the SCMO vs. the nominal higher spin-state (Co2+ and Mn4+) with S = 3/2 is predicted. It essentially tunes super-exchange mechanism and relative ground states as compared to Gd2CoMnO6 (GCMO). The synergy of crystal structure and electronic structure to the magnetic properties are thus, established with first-principles calculations data, employing self-consistent Hubbard U (Uself) determination for both 3d- and 4f-site’s electronic correlation. The computed quenched spin-moments (∼20–40 %) in the experimental observation at the Sm3+(4f5) and Gd3+(4f7) sites from the ideal value 10.0 μB and 14.0 μB per f.u. is established, respectively. It could be due to magnetic impurities in their ferri-magnetic ground state ordering (FiM3), as also confirmed from present first-principles calculations in the pristine limit. In SCMO x=0.0, the valence band is dominated with the RE(4 f)-orbitals along with the TM(3d)- and O(2p)-orbitals, resulting it as a charge-transfer type semiconductor, while in GCMO x=1.5, Gd(4 f)-orbitals are localized well below 6.5 eV from Fermi level, EF = 0 eV, resulting it as a Mott-insulator-like, mediated via 3d-3d transitions. The long-range ferro-magnetic spin-ordering ground states in SCMO is established through non-collinear spin-orbital ground states determinations. Indeed, the long-range insulating FM ordering in SCMO from combined 3d-4f-sub-lattice’s magnetic interactions, but found to be broken with Gd-substitution (x = 0.5–1.5) aligning all 3d-4f site’s spins into the in-plane. Observed coercive field (HC) and remanence asymmetry (Mr), the nature of exchange-bias transformed from inverse to conventional type, as soon as it contains a small fraction of the Gd-substitution (x = 0.5).

Hybrid spin-orbit exciton-magnon excitations in FePS3

FePS3 is a layered van der Waals (vdW) Ising antiferromagnet that has recently been studied in the context of true 2D magnetism and emerged as an ideal material platform for investigating strong spin-phonon coupling, and non-linear magneto-optical phenomena. In this work, we demonstrate an important unresolved role of spin-orbit coupling (SOC) in the ground state and excitations of this compound. Combining first-principles calculations with linear flavor wave theory (LFWT), we find strong mixing and spectral overlap of different spin-orbital single-ion states. Low-lying excitations form hybrid spin-orbit exciton/magnon modes. Complete parameterization of the low-energy model requires nearly half a million coupling constants. Despite this complexity, such a model can be inexpensively derived using local many-body-based approaches, which yield quantitative agreement with recent experiments. The results highlight the importance of SOC even in first-row transition metals and provide essential insight into the properties of 2D magnets with unquenched orbital moments.

Perfect Zeeman Anisotropy in Rotationally Symmetric Quantum Dots with Strong Spin-Orbit Interaction.

In nanoscale structures with rotational symmetry, such as quantum rings, the orbital motion of electrons combined with a spin-orbit interaction can produce a very strong and anisotropic Zeeman effect. Since symmetry is sensitive to electric fields, ring-like geometries provide an opportunity to manipulate magnetic properties over an exceptionally wide range. In this work, we show that it is possible to form rotationally symmetric confinement potentials inside a semiconductor quantum dot, resulting in electron orbitals with large orbital angular momentum and strong spin-orbit interactions. We find complete suppression of Zeeman spin splitting for magnetic fields applied in the quantum dot plane, similar to the expected behavior of an ideal quantum ring. Spin splitting reappears as orbital interactions are activated with symmetry-breaking electric fields. For two valence electrons, representing a common basis for spin-qubits, we find that modulating the rotational symmetry may offer new prospects for realizing tunable protection and interaction of spin-orbital states.

Quantum criticality enabled by intertwined degrees of freedom

Strange metals appear in a wide range of correlated materials. Electronic localization-delocalization and the expected loss of quasiparticles characterize beyond-Landau metallic quantum critical points and the associated strange metals. Typical settings involve local spins. Systems that contain entwined degrees of freedom offer new platforms to realize unusual forms of quantum criticality. Here, we study the fate of an SU(4) spin-orbital Kondo state in a multipolar Bose-Fermi Kondo model, which provides an effective description of a multipolar Kondo lattice, using a renormalization-group method. We show that at zero temperature, a generic trajectory in the model's parameter space contains two quantum critical points, which are associated with the destruction of Kondo entanglement in the orbital and spin channels, respectively. Our asymptotically exact results reveal an overall phase diagram, provide the theoretical basis to understand puzzling recent experiments of a multipolar heavy fermion metal, and point to a means of designing different forms of quantum criticality and strange metallicity in a variety of strongly correlated systems.

Contrasting electronic states of RuI3 and RuCl3

The spin-orbital entangled states are of great interest as they hold exotic phases and intriguing properties. Here we use first-principles calculations to investigate the electronic and magnetic properties of ${\mathrm{RuI}}_{3}$ and ${\mathrm{RuCl}}_{3}$ in both bulk and monolayer cases. Our results show that ${\mathrm{RuI}}_{3}$ bulk is a paramagnetic metal, which is in agreement with recent experiments. We find that the ${\mathrm{Ru}}^{3+}$ ion of ${\mathrm{RuI}}_{3}$ is in the spin-orbital entangled ${j}_{\mathrm{eff}}=\frac{1}{2}$ state. More interestingly, a metal-insulator transition occurs from ${\mathrm{RuI}}_{3}$ bulk to monolayer, and this is mainly due to the band narrowing with the decreasing lattice dimensionality and to the Ru-I hybridization altered by the I $5p$ spin-orbit coupling. In contrast, ${\mathrm{RuCl}}_{3}$ bulk and monolayer both show Mott-insulating behavior, the ${\mathrm{Ru}}^{3+}$ ion is in the formal $S=\frac{1}{2}$ and $L=1$ state with a large in-plane orbital moment, and this result well explains the experimental large effective magnetic moment of ${\mathrm{RuCl}}_{3}$ and the strong in-plane magnetization. The present work demonstrates the contrasting spin-orbital states and the varying properties of ${\mathrm{RuI}}_{3}$ and ${\mathrm{RuCl}}_{3}$.

Inverse Orbital Torque via Spin-Orbital Intertwined States

Spin torque induced by orbital current has become of great interest, as it allows the use of lighter elements in the development of advanced spin-orbitronic devices. Meanwhile, the inverse orbital torque (IOT) effect, in which an orbital current creates a charge current, has been difficult to detect. This study uses pumped spin-orbital current in Y${}_{3}$Fe${}_{5}$O${}_{12}$/Pt/CuO${}_{x}$ heterostructures to investigate the IOT effect. Mixed spin-orbital states propagate to the Pt/CuO${}_{x}$ interface and create a much stronger transverse charge current than that created without the CuO${}_{x}$ coating. This inverse orbital Rashba-Edelstein effect can be used to advance applications in spintronics.

Large-bias spectroscopy of Yu-Shiba-Rusinov states in a double quantum dot

We have performed tunnel transport spectroscopy on a quantum dot (QD) molecule proximitized by a superconducting contact. In such a system, the scattering between QD spins and Bogoliubov quasiparticles leads to the formation of Yu-Shiba-Rusinov (YSR) states within the superconducting gap. In this work, we investigate interactions appearing when one- and two-electron spin states in a double-QD energetically align with the superconducting gap edge. We find that the inter-dot spin-triplet state interacts considerably stronger with the superconductor than the corresponding singlet, pointing to stronger screening. By forming a ring molecule with a significant orbital contribution to the effective g-factor, we observe interactions of all four spin-orbital one-electron states with the superconductor under a weak magnetic field.

Long-range spin-orbital order in the spin-orbital SU(2)×SU(2)×U(1) model

By using the tensor-network state algorithm, we study a spin-orbital model with SU(2)$\times$SU(2)$\times$U(1) symmetry on the triangular lattice. This model was proposed to describe some triangular $d^1$ materials and was argued to host a spin-orbital liquid ground state. In our work the trial wavefunction of its ground state is approximated by an infinite projected entangled simplex state and optimized by the imaginary-time evolution. Contrary to the previous conjecture, we find that the two SU(2) symmetries are broken, resulting in a stripe spin-orbital order with the same magnitude $m=0.085(10)$. This value is about half of that in the spin-1/2 triangular Heisenberg antiferromagnet. Our result demonstrates that although the long-sought spin-orbital liquid is absent in this model the spin-orbital order is significantly reduced due to the enhanced quantum fluctuation. This suggests that high-symmetry spin-orbital models are promising in searching for exotic states of matter in condensed-matter physics.

Spin-orbital Yu-Shiba-Rusinov states in single Kondo molecular magnet

Studies of single-spin objects are essential for designing emergent quantum states. We investigate a molecular magnet Tb2Pc3 interacting with a superconducting Pb(111) substrate, which hosts unprecedented Yu-Shiba-Rusinov (YSR) subgap states, dubbed spin-orbital YSR states. Upon adsorption of the molecule on Pb, the degeneracy of its lowest unoccupied molecular orbitals (LUMO) is lifted, and the lower LUMO forms a radical spin via charge transfer. This leads to Kondo screening and subgap states. Intriguingly, the YSR states display two pairs of resonances with clearly distinct behavior. The energy of the inner pair exhibits prominent inter and intra molecular variation, and it strongly depends on the tip height. The outer pair, however, shifts only slightly. As is unveiled through theoretical calculations, the two pairs of YSR states originate from the ligand spin and charge-fluctuating higher LUMO, coexisting in a single molecule, but only weakly coupled presumably due to different spatial distribution. Our work paves the way for understanding complex many-body excitations and constructing molecule-based topological superconductivity.

Bond ordering and molecular spin-orbital fluctuations in the cluster Mott insulator GaTa4Se8

For materials where spin-orbit coupling is competitive with electronic correlations, the spatially anisotropic spin-orbital wave functions can stabilize degenerate states that lead to many and diverse quantum phases of matter. Here we find evidence for a dynamical spin-orbital state preceding a ${T}^{*}$ = 50 K order-disorder spin-orbital ordering transition in the $j=3/2$ lacunar spinel ${\mathrm{GaTa}}_{4}{\mathrm{Se}}_{8}$. Above ${T}^{*}, {\mathrm{GaTa}}_{4}{\mathrm{Se}}_{8}$ has an average cubic crystal structure, but total scattering measurements indicate local noncubic distortions of ${\mathrm{Ta}}_{4}$ tetrahedral clusters for all measured temperatures $2<T<300$ K. Inelastic neutron-scattering measurements reveal the dynamic nature of these local distortions through symmetry forbidden optical phonon modes that modulate $j=3/2$ molecular orbital occupation as well as intercluster Ta-Se bonds. Spin-orbital ordering at ${T}^{*}$ cannot be attributed to a classic Jahn-Teller mechanism and, based on our findings, we propose that intercluster interactions acting on the scale of ${T}^{*}$ act to break global symmetry. The resulting staggered intercluster dimerization pattern doubles the unit cell, reflecting a spin-orbital valence bond ground state.

Spin-Orbital States and Strong Antiferromagnetism of Layered Eu2SrFe2O6 and Sr3Fe2O4Cl2.

The insulating iron compounds Eu2SrFe2O6 and Sr3Fe2O4Cl2 have high-temperature antiferromagnetic (AF) order despite their different layered structures. Here, we carry out density functional calculations and Monte Carlo simulations to study their electronic structures and magnetic properties aided with analyses of the crystal field, magnetic anisotropy, and superexchange. We find that both compounds are Mott insulators and in the high-spin (HS) Fe2+ state (S = 2) accompanied by the weakened crystal field. Although they have different local coordination and crystal fields, the Fe2+ ions have the same level sequence and ground-state configuration (3z2-r2)2(xz, yz)2(xy)1(x2-y2)1. Then, the multiorbital superexchange produces strong AF couplings, and the (3z2-r2)/(xz, yz) mixing via the spin-orbit coupling (SOC) yields a small in-plane orbital moment and anisotropy. Indeed, by tracing a set of different spin-orbital states, our density functional calculations confirm the strong AF couplings and the easy planar magnetization for both compounds. Moreover, using the derived magnetic parameters, our Monte Carlo simulations give the Néel temperature TN = 420 K (372 K) for the former (the latter), which well reproduce the experimental results. Therefore, the present study provides a unified picture for Eu2SrFe2O6 and Sr3Fe2O4Cl2 concerning their electronic and magnetic properties.

Interplay of the Jahn-Teller effect and spin-orbit coupling: The case of trigonal vibrations

We study an interplay between the orbital degeneracy and spin-orbit coupling giving rise to spin-orbital entangled states. As a specific example, we analyze the interaction of electrons occupying triply degenerate single-ion $t_{2g}$ levels with trigonal vibronic modes (the $t\otimes T$ problem). A more general problem of the electron-lattice interaction involving both tetragonal and trigonal vibrations is also considered. It is shown that the result of such interaction crucially depends on the occupation of $t_{2g}$ levels leading to either the suppression or enhancement of the Jahn-Teller effect by the spin-orbit coupling.

Antiferromagnetism and magnetic frustration in the metalorganic compounds MCl2 -4SC( NH2)2, M = (Mn,Fe)

We report magnetic and specific heat measurements at low temperatures ($>100$ mK) and high magnetic fields (up to 9 T) for the metalorganic compounds ${\mathrm{MnCl}}_{2}\text{\ensuremath{-}}4\mathrm{SC}{({\mathrm{NH}}_{2})}_{2}$ (DTM) and ${\mathrm{FeCl}}_{2}\text{\ensuremath{-}}4\mathrm{SC}{({\mathrm{NH}}_{2})}_{2}$ (DTF). Fits to the experimental data from the Curie-Weiss law and mean-field theory indicate the antiferromagnetic nature of both compounds. For temperatures down to $T=0.56$ K the compound DTM exhibits two successive transitions, which may be associated with different magnetic orderings. In this system the external magnetic field is a tuning parameter that allows us to access such magnetic phases separated by distinct quantum critical points. No indication of magnetic ordering is found in DTF down to 100 mK, making this compound a candidate for a strongly frustrated material with frustration parameter $f=|{\ensuremath{\theta}}_{\mathrm{CW}}|/{T}_{N}>100$. Finally, we discuss the absence of Bose-Einstein condensation of magnons and the possibility of a spin-orbital liquid state in DTF.

Unveiling a critical stripy state in the triangular-lattice SU(4) spin-orbital model

The simplest spin-orbital model can host a nematic spin-orbital liquid state on the triangular lattice. We provide clear evidence that the ground state of the SU(4) Kugel-Khomskii model on the triangular lattice can be well described by a “single” Gutzwiller projected wave function with an emergent parton Fermi surface, despite it exhibits strong finite-size effect in quasi-one-dimensional cylinders. The finite-size effect can be resolved by the fact that the parton Fermi surface consists of open orbits in the reciprocal space. Thereby, a stripy liquid state is expected in the two-dimensional limit, which preserves the SU(4) symmetry while breaks the translational symmetry by doubling the unit cell along one of the lattice vector directions. It is indicative that these stripes are critical and the central charge is c=3, in agreement with the SU(4)1 Wess-Zumino-Witten conformal field theory. All these results are consistent with the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem.

SU(4)-symmetric quantum spin-orbital liquids on various lattices

An emergent $\mathrm{SU}(4)$ symmetry discovered in the microscopic model for $d^1$ honeycomb materials [M.~G.~Yamada, M.~Oshikawa, and G.~Jackeli, Phys. Rev. Lett. \textbf{121}, 097201 (2018).] has enabled us to tailor exotic $\mathrm{SU}(4)$ models in real materials. In the honeycomb structure, the emergent $\mathrm{SU}(4)$ Heisenberg model would potentially have a quantum spin-orbital liquid ground state due to the \textit{multicomponent frustration}, and we can expect similar spin-orbital liquids also in three-dimensinal versions of the honeycomb lattice. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and entangled together due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic \textit{orbitalon} excitations at low temperature.

6-GHz lattice response in a quantum spin-orbital liquid probed by time-resolved resonant x-ray scattering

Long-sought quantum spin liquids usually emerge in geometrically frustrated magnets. Less commonly, unfrustrated magnets can harbor a variety of quantum spin liquids if charge and/or orbital degrees of freedom are involved. Jahn-Teller distortion of ${\mathrm{Cu}}^{2+}{\mathrm{O}}_{6}$ octahedra is absent in hexagonal ${\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$ suggesting a Cu $3d$ spin-orbital liquid state. Here, by means of time-resolved resonant x-ray scattering, we show that hexagonal ${\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$ exhibits a charge-orbital dynamics which is absent in the orthorhombic phase of ${\mathrm{Ba}}_{3}{\mathrm{CuSb}}_{2}{\mathrm{O}}_{9}$ with Jahn-Teller distortion and Cu $3d$ orbital order. The time scale of charge-orbital dynamics manifests in the coherent phonons at 6 GHz. The present work reveals a role of electron-lattice entanglement in the quantum spin-orbital liquid state.

Spin-orbital liquid in Ba3CuSb2O9 stabilized by oxygen holes

Both the Jahn-Teller distortion of Cu$^{2+}$O$_6$ octahedra and magnetic ordering are absent in hexagonal Ba$_3$CuSb$_2$O$_9$ suggesting a Cu 3$d$ spin-orbital liquid state. Here, by means of resonant x-ray scattering and absorption experiment, we show that oxygen 2$p$ holes play crucial role in stabilizing this spin-orbital liquid state. These oxygen holes appear due to the "reaction" Sb$^{5+}$$\rightarrow$Sb$^{3+}$ $+$ two oxygen holes, with these holes being able to attach to Cu ions. The hexagonal phase with oxygen 2$p$ holes exhibits also a novel charge-orbital dynamics which is absent in the orthorhombic phase of Ba$_3$CuSb$_2$O$_9$ with Jahn-Teller distortion and Cu 3$d$ orbital order. The present work opens up a new avenue towards spin-charge-orbital entangled liquid state in transition-metal oxides with small or negative charge transfer energy.

Competing Energy Scales in Topological Superconducting Heterostructures.

Artificially engineered topological superconductivity has emerged as a viable route to create Majorana modes. In this context, proximity-induced superconductivity in materials with a sizable spin–orbit coupling has been intensively investigated in recent years. Although there is convincing evidence that superconductivity may indeed be induced, it has been difficult to elucidate its topological nature. Here, we engineer an artificial topological superconductor by progressively introducing superconductivity (Nb), strong spin–orbital coupling (Pt), and topological states (Bi2Te3). Through spectroscopic imaging of superconducting vortices within the bare s-wave superconducting Nb and within proximitized Pt and Bi2Te3 layers, we detect the emergence of a zero-bias peak that is directly linked to the presence of topological surface states. Our results are rationalized in terms of competing energy trends which are found to impose an upper limit to the size of the minigap separating Majorana and trivial modes, its size being ultimately linked to fundamental materials properties.

Polarization interferometric prism: A versatile tool for generation of vector fields, measurement of topological charges, and implementation of a spin–orbit controlled-Not gate

Optical vortex and vector field are two important types of structured optical fields. Due to their wide applications and unique features in many scientific realms, the generation, manipulation, and measurement of such fields have attracted significant interest and become very important topics. However, most ways to generate vector fields have a trade-off among flexibility, efficiency, stability, and simplicity. Meanwhile, an easy and direct way to measure the topological charges, especially for a high order optical vortex, is still a challenge. Here we design and manufacture a prism: a polarization interferometric prism (PIP) as a single-element interferometer, which can conveniently convert an optical vortex to vector fields with high efficiency and be utilized to precisely measure the topological charge (both absolute value and sign) of an arbitrary optical vortex, even with a high order. Experimentally, we generate a variety of vector fields with global fidelity ranging from 0.963 to 0.993 and measure the topological charge of an optical vortex by counting the number of petals uniformly distributed over a ring on the output intensity patterns. As a versatile tool to generate, manipulate, and detect the spin-orbital state of single photons, PIP can also work in the single-photon regime for quantum information processing. In the experiment, the PIP is utilized as a spin–orbit controlled-Not gate on the generated 28 two-qubit states, achieving the state fidelities ranging from 0.966 to 0.995 and demonstrating the feasibility of the PIP for single photons.