In electronic solids with strong spin-orbit interactions (SOIs), the spin and orbital degrees of freedom of an electron are quantum mechanically entangled, which may result in an exotic multipolar order instead of a conventional dipolar order such as a magnetic order. Such a higher-degree order is called "hidden order" because of difficulties in experimental detection. Moreover, the number of candidate compounds is limited, especially rare in d electron systems, in which an interplay between SOIs and Coulomb interactions is expected to cause rich physics. Here, we employ state-of-the-art synchrotron X-ray diffraction techniques on a high-quality single crystal to probe subtle symmetry breaking induced by a multipolar order. We unequivocally demonstrate that the double-perovskite Ba2MgReO6 exhibits successive transitions to quadrupolar and then dipolar orders upon cooling, which is consistent with a theory considering SOIs. Our findings are a significant step towards understanding the intriguing physics of multipoles realized by spin-orbit-entangled 5d electrons.
The Kitaev model, characterized by bond-dependent Ising spin interactions among spin-orbit entangled dipole moments in Mott insulators, offered a new approach to quantum spin liquids. Motivated by another type of bond-dependent interaction among quadrupole moments in $5d^2$ Mott insulators, we provide a microscopic route to uncover the Kitaev multipolar liquid, featuring fractionalized excitations out of non-Kramers doublets carrying multipole moments. The key ingredient is the magnetic field that allows for bond-anisotropic quadrupole-octupole interactions via mixing with the excited triplet …
Motivated by the rapid experimental progress on the spin-orbit-coupled Mott insulators, we propose and study a generic spin model that describes the interaction between the non-Kramers doublets on a triangular lattice and is relevant for triangular lattice rare-earth magnets. We predict that the system supports both pure quadrupolar orders and intertwined multipolar orders in the phase diagram. Besides the multipolar orders, we explore the magnetic excitations to reveal the dynamic properties of the systems. Due to the peculiar properties of the non-Kramers doublets and the selective coupling …
Discovery of novel spin-orbital entangled quantum ground states paves an important avenue for controllable quantum materials via unique couplings to the lattice and other external perturbations. In this work, motivated by recent experiments on cubic heavy fermion materials with multipolar local moments, we theoretically investigate strongly-interacting spin-orbital entangled quantum ground states in multipolar quantum impurity systems. Here itinerant electrons are interacting with the local moments carrying quadrupolar and octupolar moments, in contrast to the conventional Kondo problem with d…
When the ground state of a localized ion is a non-Kramers doublet, such localized ions may carry multipolar moments. For example, Pr$^{3+}$ ions in a cubic environment would possess quadrupolar and octupolar, but no magnetic dipole, moments. When such multipolar moments are placed in a metallic host, unusual interactions between these local moments and conduction electrons arise, in contrast to the familiar magnetic dipole interactions in the classic Kondo problem. In this work, we consider the interaction between a single quadrupolar-octupolar local moment and conduction electrons with $p$-or…
Multipolar magnetism is an emerging field of quantum materials research. The building blocks of multipolar phenomena are magnetic ions with a non-Kramers doublet, where the orbital and spin degrees of freedom are inextricably intertwined, leading to unusual spin-orbital entangled states. The detection of such subtle forms of matter has, however, been difficult due to a limited number of appropriate experimental tools. In this work, motivated by a recent magnetostriction experiment on Pr$_2$Zr$_2$O$_7$, we theoretically investigate how multipolar quantum spin ice, an elusive three dimensional q…
Between Mott and cluster Mott: spin-orbit entangled dimer singlets in Ba$_3$CeRu$_2$O$_9$
University of Cologne（科隆大学）；Stockholm University（斯德哥尔摩大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步辐射中心）；IFW Dresden（德累斯顿联邦物理研究所）；Technische Universität Dresden（德累斯顿工业大学）
Recent experiments have discovered multipolar orders in a variety of $d$-orbital Mott insulators. Motivated by uncovering the exchange interactions which underlie octupolar order proposed in the osmate double perovskites, we study a two-site model using exact diagonalization on a five-orbital Hamiltonian, incorporating spin-orbit coupling (SOC) and interactions, and including both intra-orbital and inter-orbital hopping. Using an exact Schrieffer-Wolff transformation, we then extract an effective pseudospin Hamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolar coupling d…
We present first principles calculations of the magnetic and orbital properties of Ba$_2$NaOsO$_6$ (BNOO), a 5$d^1$ Mott insulator with strong spin orbit coupling (SOC) in its low temperature emergent quantum phases. Our computational method takes into direct consideration recent NMR results that established that BNOO develops a local octahedral distortion preceding the formation of long range magnetic order. We found that the two-sublattice canted ferromagnetic ground state identified in Lu \etal, Nature Comm. {\bf 8}, 14407 (2017) is accompanied by a two-sublattice staggered orbital ordering…
Mott insulators form because of strong electron repulsions, being at the heart of strongly correlated electron physics. Conventionally these are understood as classical "traffic jams" of electrons described by a short-ranged entangled product ground state. Exploiting the holographic duality, which maps the physics of densely entangled matter onto gravitational black hole physics, we show how Mott-insulators can be constructed departing from entangled non-Fermi liquid metallic states, such as the strange metals found in cuprate superconductors. These "entangled Mott insulators" have traits in c…
Using tensor network states to unravel the physics of quantum spin liquids in minimal, yet generic microscopic spin or electronic models remains notoriously challenging. A prominent open question concerns the nature of the insulating ground state of two-dimensional half-filled Hubbard-type models on the triangular lattice in the vicinity of the Mott metal-insulator transition, a regime which can be approximated microscopically by a spin-1/2 Heisenberg model supplemented with additional "ring-exchange" interactions. Using a novel and efficient state preparation technique whereby we initialize f…
We present the application of the SU($N$) ($N>2$) spin-wave theory to spin-orbital Mott insulators whose ground states exhibit magnetic orders. When taking both the spin and orbital degrees of freedom into account rather than projecting onto the Kramers doublet, the lowest spin-orbital locking energy levels, due to the inevitable spin-orbital multipole exchange interactions, the SU($N$) spin-wave theory should take the place of the SU($2$) one. To implement the application, we introduce an efficient general local mean field approach which involves all the local fluctuations into the SU($N$) li…
Study of the combined effects of strong electronic correlations with spin-orbit coupling (SOC) represents a central issue in quantum materials research. Predicting emergent properties represents a huge theoretical problem since the presence of SOC implies that the spin is not a good quantum number. Existing theories propose the emergence of a multitude of exotic quantum phases, distinguishable by either local point symmetry breaking or local spin expectation values, even in materials with simple cubic crystal structure such as Ba$_2$NaOsO$_6$. Experimental tests of such theories by local probe…
In 5$d^2$ Mott insulators with strong spin-orbit coupling, the lowest pseudospin states form a non-Kramers doublet, which carries quadrupolar and octupolar moments. A family of double perovskites where magnetic ions form a face-centered cubic (fcc) lattice was suggested to unveil an octupolar order offering a rare example in $d$-orbital systems. The proposed order requires a ferromagnetic (FM) octupolar interaction, since the antiferromagnetic (AFM) Ising model is highly frustrated on the fcc lattice. A microscopic model was recently derived for various lattices: for an edge-sharing octahedra …
Layered perovskite iridates realize a rare class of Mott insulators that are predicted to be strongly spin-orbit coupled analogues of the parent state of cuprate high-temperature superconductors. Recent discoveries of pseudogap, magnetic multipolar ordered and possible $d$-wave superconducting phases in doped Sr$_2$IrO$_4$ have reinforced this analogy among the single layer variants. However, unlike the bilayer cuprates, no electronic instabilities have been reported in the doped bilayer iridate Sr$_3$Ir$_2$O$_7$. Here we show that Sr$_3$Ir$_2$O$_7$ realizes a weak Mott state with no cuprate a…
The vicinity of a Mott insulating phase has constantly been a fertile ground for finding exotic quantum states, most notably the high Tc cuprates and colossal magnetoresistance manganites. The layered transition metal dichalcogenide 1T-TaS2 represents another intriguing example, in which the Mott insulator phase is intimately entangled with a series of complex charge-density-wave (CDW) orders. More interestingly, it has been recently found that 1T-TaS2 undergoes a Mott-insulator-to-superconductor transition induced by high pressure, charge doping, or isovalent substitution. The nature of the M…
We point out the universal presence of the excitonic superexchange in spin-orbit-coupled Mott insulators. It is observed that, the restriction to the lowest spin-orbit-entangled "$J$" states may sometimes be insufficient to characterize the microscopic physics, and the virtual excitonic processes via the upper "$J$" states provide an important correction to the superexchange. We illustrate this excitonic superexchange from a two-dimensional $5d$ iridate Sr$_2$IrO$_4$ and explain its physical consequences such as the orbital-like coupling to the external magnetic flux and the nonlinear magnetic…
Quantum magnets with spin $J=2$, which arise in spin-orbit coupled Mott insulators, can potentially display multipolar orders. We carry out an exact diagonalization study of a simple octahedral crystal field Hamiltonian for two electrons, incorporating spin-orbit coupling (SOC) and interactions, finding that either explicitly including the $e_g$ orbitals, or going beyond the rotationally invariant Coulomb interaction within the $t_{2g}$ sector, causes a degeneracy breaking of the $J\!=\!2$ level degeneracy. This can lead to a low-lying non-Kramers doublet carrying quadrupolar and octupolar mom…
We investigate the electronic structure and magnetic properties of a $J_\text{eff} = 1/2$ iridate Ca$_4$IrO$_6$ and the implications of doping electrons and holes using ab initio density functional theory. Our calculations considering spin-orbit interaction reveal that although the Mott-insulating parent compound transforms into a conductor upon doping, antiferromagnetism sustains in the doped system, albeit with a grossly noncollinear arrangement of the spins. We find a strong spin-orbit interaction and magneto-crystalline anisotropy, causing frustration in the system, possibly leading to the…
We study the orbitally frustrated singlet-triplet models that emerge in the context of spin-orbit coupled Mott insulators with $t_{2g}^4$ electronic configuration. In these compounds, low-energy magnetic degrees of freedom can be cast in terms of three-flavor "triplon" operators describing the transitions between spin-orbit entangled $J=0$ ionic ground state and excited $J=1$ levels. In contrast to a conventional, flavor-isotropic O(3) singlet-triplet models, spin-orbit entangled triplon interactions are flavor-and-bond selective and thus highly frustrated. In a honeycomb lattice, we find clos…
Complex oxides with $4d$ and $5d$ transition-metal ions recently emerged as a new paradigm in correlated electron physics, due to the interplay between spin-orbit coupling and electron interactions. For $4d$ and $5d$ ions, the spin-orbit coupling, $ζ$, can be as large as 0.2-0.4 eV, which is comparable with and often exceeds other relevant parameters such as Hund's coupling $J_{\rm H}$, noncubic crystal field splitting $Δ$, and the electron hopping amplitude $t$. This gives rise to a variety of spin-orbit-entangled degrees of freedom and, crucially, non-trivial interactions between them that d…
Dynamical correlations of Jeff = 1/2 isospins in the paramagnetic state of spin-orbital Mott insu- lator Sr2IrO4 was revealed by resonant magnetic x-ray diffuse scattering. We found two-dimensional antiferromagnetic fluctuation with a large in-plane correlation length exceeding 100 lattice spacings at even 20 K above the mangnetic ordering temperature. In marked contrast to the naive expecta- tion of strong magnetic anisotropy associated with an enhanced spin-orbit coupling, we discovered isotropic isospin correlation that is well described by the two-dimensional S = 1/2 quantum Heisen- berg m…
The isospin-phonon coupling in Sr2IrO4 is investigated by temperature dependent phonon Raman scattering. Anomalous behavior in the frequency of all studied optical phonons is observed below the magnetic transition temperature TN ~240 K. The strongest effect is detected for the A1g mode at 272 cm-1 associated with the modulation of the Ir-O-Ir bond angle. Additionally, the A1g mode at 560 cm-1 shows a Fano asymmetric lineshape sensitive to TN, supporting the existence of low-energy (~70 meV) electronic excitations that are renormalized by the magnetic order. These results reveal new aspects of …
$B$-site ordered 4$d^1$ and 5$d^1$ double perovskites have a number of potential novel ground states including multipolar order, quantum spin liquids and valence bond glass states. These arise from the complex interactions of spin-orbital entangled $J_{eff}$ = 3/2 pseudospins on the geometrically frustrated fcc lattice. The 4$d^1$ Mo$^{5+}$ perovskite Ba$_2$YMoO$_6$ has been suggested to have a valence bond glass ground state. Here we report on the low temperature properties of powder samples of isostructural Ba$_2$LuMoO$_6$: the only other known cubic 4$d^1$ perovskite with one magnetic catio…
Pairwise entanglement, calculated separately for charge and spin degrees of freedom, is proposed as a ground-state signature of the Mott transition in correlated nanoscopic systems. Utilizing the exact diagonalization - ab initio method (EDABI), for chains containing $N\leqslant{}16$ hydrogenic-like atoms (at the half filling), we find that the vanishing of the nearest-neighbor charge concurrence indicates the crossover from a partly-localized quantum liquid to the Mott insulator. Spin concurrence remains nonzero at the insulating phase, showing that the decopling of spin and charge degrees of…
Transition metal oxides exhibit various competing phases and exotic phenomena depending on how their reaction to the rich degeneracy of the $d$-orbital. Large spin-orbit coupling (SOC) reduces this degeneracy in a unique way by providing a spin-orbital-entangled ground state for 4$d$ and 5$d$ transition metal compounds. In particular, the spin-orbital-entangled Kramers doublet, known as the $J_{\mathbf{eff}}$=1/2 pseudospin, appears in layered iridates and $α$-RuCl$_3$, manifesting a relativistic Mott insulating phase. Such entanglement, however, seems barely attainable in 3$d$ transition meta…
The pyrochlore iridate In$_2$Ir$_2$O$_7$ is a strong $J_{\mathrm{eff}} = 1/2$ Mott insulator with frustrated magnetism. Despite the large trigonal crystal field, a small admixture of $J_{\mathrm{eff}} = 3/2$ component in the $J_{\mathrm{eff}} = 1/2$ bands and a small splitting of $J_{\mathrm{eff}} = 3/2$ bands are observed as compared with other pyrochlore iridates A$_2$Ir$_2$O$_7$ (A: trivalent cation). We argue that the reduced inter-site hopping between the $J_{\mathrm{eff}} = 1/2$ and the $J_{\mathrm{eff}} = 3/2$ manifold plays a predominant role in the distinct behavior of In$_2$Ir$_2$O$_…
Candidate materials for Kitaev spin liquid phase have been intensively studied recently because of their potential applications in fault-tolerant quantum computing. Although most of the studies on Kitaev spin liquid have been done in 4$d$ and 5$d$ based transition metal compounds, recently there has been a growing research interest in Co-based quasi-two-dimensional honeycomb magnets, such as BaCo$_2$(AsO$_4$)$_2$ because of formation of spin-orbit-entangled $J_{\rm eff}$ = 1/2 pseudospin moments at Co$^{2+}$ sites and potential realizations of Kitaev-like magnetism therein. Here, we obtain hig…
Mott Criticality as the Confinement Transition of a Pseudogap-Mott Metal
Indian Institute of Science Education and Research Kolkata（印度科学教育研究所加尔各答分校）；The Institute of Mathematical Sciences, HBNI（数学科学研究所，HBNI）；National Institute of Science, Education and Research, HBNI（国家科学、教育和研究学院，HBNI）；Jawaharlal Nehru Center for Advanced Scientific Research（贾瓦哈拉尔·尼赫鲁高级科学研究中心）；Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
Motivated by experimental and theoretical interest in realizing multipolar orders in $d$-orbital materials, we discuss the quantum magnetism of $J\!=\!2$ ions which can be realized in spin-orbit coupled oxides with $5d^2$ transition metal ions. Based on the crystal field environment, we argue for a splitting of the $J\!=\!2$ multiplet, leading to a low lying non-Kramers doublet which hosts quadrupolar and octupolar moments. We discuss a microscopic mechanism whereby the combined perturbative effects of orbital repulsion and antiferromagnetic Heisenberg spin interactions leads to ferro-octupola…
The recent discovery of skyrmions in Cu$_2$OSeO$_3$ has established a new platform to create and manipulate skyrmionic spin textures. We use high-field electron spin resonance (ESR) spectroscopy combining a terahertz free electron laser and pulsed magnetic fields up to 64 T to probe and quantify its microscopic spin-spin interactions. Besides providing direct access to the long-wavelength Goldstone mode, this technique probes also the high-energy part of the excitation spectrum which is inaccessible by standard low-frequency ESR. Fitting the behavior of the observed modes in magnetic field to …
Non-magnetic ground state in A$_2$WCl$_6$ (A = Cs, Rb, K): A face-centered cubic system of spin-orbit-entangled $J$ = 2 states
Max Planck Institute for Solid State Research（马克斯·普朗克固相研究所）；Institute for Functional Matter and Quantum Technologies, University of Stuttgart（斯图加特大学功能物质与量子技术研究所）；National Institute for Materials Science（国立材料研究所）；Synchrotron Radiation Research Center, National Institutes for Quantum Science and Technology（量子科学技术研究机构同步辐射研究中心）；Graduate School of Engineering and Science, Shibaura Institute of Technology（芝浦工业大学理工学研究科）；Nuclear Structure Research Group, RIKEN Nishina Center for Accelerator-Based Science（RIKEN仁科加速器科学中心核结构研究组）；ISIS Neutron and Muon Source, STFC Rutherford Appleton Laboratory（STFC卢瑟福阿普尔顿实验室ISIS中子和μ子源）；Department of Physics, University of Tokyo（东京大学物理学系）
Motivated by the recent activities on the Ni-based diamond lattice antiferromagnet NiRh$_2$O$_4$, we theoretically explore on a general ground the unique spin and orbital physics for the Ni$^{2+}$ ions with a $3d^8$ electron configuration in the tetrahedral crystal field environment and on a diamond lattice Mott insulator. The superexchange interaction between the local moments usually favors magnetic orders. Due to the particular electron configuration of the Ni$^{2+}$ ion with a partially filled upper $t_{2g}$ level and a fully filled lower $e_g$ level, the atomic spin-orbit coupling becomes…
We performed resonant x-ray diffraction experiments at the $L$ absorption edges for the post-perovskite-type compound CaIrO$_{3}$ with $(t_{2g})^5$ electronic configuration. By observing the magnetic signals, we could clearly see that the magnetic structure was a striped order with an antiferromagnetic moment along the c-axis and that the wavefunction of a $t_{2g}$ hole is strongly spin-orbit entangled, the $J_{\rm eff} =1/2$ state. The observed spin arrangement is consistent with theoretical work predicting a unique superexchange interaction in the $J_{\rm eff} =1/2$ state and points to the u…
Properties of the doped spin 3/2 Mott insulator near half filling
本文对自旋为3/2的哈伯德模型在半填充附近具有大的反Hunds规则耦合的情形，给出了一个精确的广义Bogoliubov变换.由于我们的变换是么正的，因此我们可以在全希尔伯特空间中采用标准的近似平均场理论方法来分析掺杂的莫特绝缘体，这与基于截断希尔伯特空间补充硬核约束的传统方法相反。在完全半填充的基态是绝缘（莫特）单态，根据我们的分析，非费米液体序参量$ Δ$通常与扩展的s波超导性，将出现自洽尽快有限密度$ n $孔被引入。非费米液体行为是将Mott单重态映射到Fock真空的幺正变换的非线性性质的结果，该幺正变换在有效拉格朗日量中引入了诸如$Δn$的反常项。我们的分析使用的方法，很容易推广到多带哈伯德模型，并可以提供一种机制，即非费米液体序参量的密度成正比的莫特绝缘体与局部纠缠基态。对于更复杂的系统，这样的序参量可以自然地与各种其他序参量共存。
In the search for quantum spin liquids, candidate materials for the Kitaev model and its extensions have been intensively explored during the past decade, as the models realize the exact quantum spin liquids in the ground state. Thus far, insulating magnets in the low-spin $d^5$ electron configuration under the strong spin-orbit coupling have been studied for realizing the Kitaev-type bond-dependent anisotropic interactions between the spin-orbital entangled Kramers doublets. To extend the candidates, here we investigate the systems in a high-spin $d^7$ electron configuration, whose ground sta…
Spin-orbit entangled magnetic dipoles, often referred to as pseudospins, provide a new avenue to explore novel magnetism inconceivable in the weak spin-orbit coupling limit, but the nature of their low-energy interactions remains to be understood. We present a comprehensive study of the static magnetism and low-energy pseudospin dynamics in the archetypal spin-orbit Mott insulator Sr2IrO4. We find that in order to understand even basic magnetization measurements, a formerly overlooked in-plane anisotropy is fundamental. In addition to magnetometry, we use neutron diffraction, inelastic neutron…
The concept of the entanglement between spin and orbital degrees of freedom plays a crucial role in understanding various phases and exotic ground states in a broad class of materials, including orbitally ordered materials and spin liquids. We investigate how the spin-orbital entanglement in a Mott insulator depends on the value of the spin-orbit coupling of the relativistic origin. To this end, we numerically diagonalize a 1D spin-orbital model with the 'Kugel-Khomskii' exchange interactions between spins and orbitals on different sites supplemented by the on-site spin-orbit coupling. In the …
We establish the double perovskite Ba$_2$CeIrO$_6$ as a nearly ideal model system for j=1/2 moments, with resonant inelastic x-ray scattering indicating a deviation of less than 1% from the ideally cubic j=1/2 state. The local j=1/2 moments form an fcc lattice and are found to order antiferromagnetically at $T_N$=14K, more than an order of magnitude below the Curie-Weiss temperature. Model calculations show that the geometric frustration of the fcc Heisenberg antiferromagnet is further enhanced by a next-nearest neighbor exchange, indicated by ab initio theory. Magnetic order is driven by a bo…
