Designing Kitaev Spin Liquids in Metal-Organic Frameworks

On-Surface Self-Assembly Kinetic Study of Cu-Hexaazatriphenylene 2D Conjugated Metal-Organic Frameworks on Coinage Metals and MoS2 Substrates

Supramolecular coordination self-assembly on solid surfaces provides an effective route to form two-dimensional (2D) metal-organic frameworks (MOFs). In such processes, surface-adsorbate interaction plays a key role in determining the MOFs' structural and chemical properties. Here, we conduct a systematic study of Cu-HAT (HAT = 1,4,5,8,9,12-hexaazatriphenylene) 2D conjugated MOFs (c-MOFs) self-assembled on Cu(111), Au(111), Ag(111), and MoS2 substrates. Using scanning tunneling microscopy and density functional theory calculations, we found that the as-formed Cu3HAT2 c-MOFs on the four substrates exhibit distinctive structural features including lattice constant and molecular conformation. The structural variations can be attributed to the differentiated substrate effects on the 2D c-MOFs, including adsorption energy, lattice commensurability, and surface reactivity. Specifically, the framework grown on MoS2 is nearly identical to its free-standing counterpart. This suggests that the 2D van der Waals (vdW) materials are good candidate substrates for building intrinsic 2D MOFs, which hold promise for next-generation electronic devices.

Two-Dimensional Cobalt(II) Benzoquinone Frameworks for Putative Kitaev Quantum Spin Liquid Candidates

The realization and discovery of quantum spin liquid (QSL) candidate materials are crucial for exploring exotic quantum phenomena and applications associated with QSLs. Most existing metal-organic two-dimensional (2D) quantum spin liquid candidates have structures with spins arranged on the triangular or kagome lattices, whereas honeycomb-structured metal-organic compounds with QSL characteristics are rare. Here, we report the use of 2,5-dihydroxy-1,4-benzoquinone (X2dhbq, X = Cl, Br, H) as the linkers to construct cobalt(II) honeycomb lattices (NEt4)2[Co2(X2dhbq)3] as promising Kitaev-type QSL candidate materials. The high-spin d7 Co2+ has pseudospin-1/2 ground-state doublets, and benzoquinone-based linkers not only provide two separate superexchange pathways that create bond-dependent frustrated interactions but also allow for chemical tunability to mediate magnetic coupling. Our magnetization data show antiferromagnetic interactions between neighboring metal centers with Weiss constants from -5.1 to -8.5 K depending on the X functional group in X2dhbq linkers (X = Cl, Br, H). No magnetic transition or spin freezing could be observed down to 2 K. Low-temperature susceptibility (down to 0.3 K) and specific heat (down to 0.055 K) of (NEt4)2[Co2(H2dhbq)3] were further analyzed. Heat capacity measurements confirmed no long-range order down to 0.055 K, evidenced by the broad peak instead of the λ-like anomaly. Our results indicate that these 2D cobalt benzoquinone frameworks are promising Kitaev QSL candidates with chemical tunability through ligands that can vary the magnetic coupling and frustration.

Local gate control of Mott metal-insulator transition in a 2D metal-organic framework

Electron-electron interactions in materials lead to exotic many-body quantum phenomena, including Mott metal-insulator transitions (MITs), magnetism, quantum spin liquids, and superconductivity. These phases depend on electronic band occupation and can be controlled via the chemical potential. Flat bands in two-dimensional (2D) and layered materials with a kagome lattice enhance electronic correlations. Although theoretically predicted, correlated-electron Mott insulating phases in monolayer 2D metal-organic frameworks (MOFs) with a kagome structure have not yet been realised experimentally. Here, we synthesise a 2D kagome MOF on a 2D insulator. Scanning tunnelling microscopy (STM) and spectroscopy reveal a MOF electronic energy gap of ∼200 meV, consistent with dynamical mean-field theory predictions of a Mott insulator. Combining template-induced (via work function variations of the substrate) and STM probe-induced gating, we locally tune the electron population of the MOF kagome bands and induce Mott MITs. These findings enable technologies based on electrostatic control of many-body quantum phases in 2D MOFs.

J_{eff}=1/2 Hyperoctagon Lattice in Cobalt Oxalate Metal-Organic Framework

We report the magnetic properties of a cobalt oxalate metal-organic framework featuring the hyperoctagon lattice. Our thermodynamic measurements reveal the J_{eff}=1/2 state of the high-spin Co^{2+} (3d^{7}) ion and the two successive magnetic transitions at zero field with two-stage entropy release. ^{13}C-NMR measurements reveal the absence of an internal magnetic field in the intermediate temperature phase. Multiple field-induced phases are observed before full saturation at around 40 T. We argue the unique cobalt oxalate network gives rise to the Kitaev interaction and/or a bond frustration effect, providing an unconventional platform for frustrated magnetism on the hyperoctagon lattice.

Intertwining of Localized (d) and Delocalized (π) Spins in Magnetically Frustrated Two-Dimensional Metal-Organic Frameworks

Two-dimensional metal-organic frameworks (2D MOFs) are emerging as a new class of multifunctional materials for diversified applications, although magnetic properties have not been widely explored. The metal ions and organic ligands in some of the 2D MOFs are arranged in the well-known Kagome lattice, leading to geometric spin frustration. Hence, such systems could be the potential candidates to exhibit an exotic quantum spin liquid (QSL) state, as was observed in Cu3(HHTP)2 (HHTP = hexahydroxytriphenylene), with no magnetic transition down to 38 mK. Hereto, we have investigated the spin intertwining in a bimetallic 2D MOF system, M3(HHTP)2 (M = Cu/Zn), arising from the localized (d-electron) and delocalized (π-electron) S = 1/2 spins from the Cu(II) ions and the HHTP radicals, respectively. The origin of the spin frustration (down to 5K) was critically examined by varying the metal composition in bimetallic systems, CuxZn3-x(HHTP)2 (x = 1, 1.5, 2), containing both S = 1/2 and S = 0 spins. Additionally, to gain a deeper understanding, we studied the spin interaction in the pristine Zn3(HHTP)2 system containing only S = 0 Zn(II) ions. In view of the quantitative estimate of the localized and delocalized spins, the d-π spin correlation appears essential in understanding the unusual magnetic and/or other physical properties of such hybrid organic-inorganic 2D crystalline solids.

Robust Spin Liquidity in 2D Metal-Organic Framework Cu3 (HHTP)2 with S=1 /2 Kagome Lattice

On one hand electron or hole doping of quantum spin liquid (QSL) may unlock high-temperature superconductivity and on the other hand it can disrupt the spin liquidity, giving rise to a magnetically ordered ground state. Recently, a 2D MOF, Cu3 (HHTP)2 (HHTP - 2,3,6,7,10,11-hexahydroxytriphenylene), containing Cu(II) S=1 / 2 ${{ 1/2 }}$ frustrated spins in the Kagome lattice is emerging as a promising QSL candidate. Herein, we present an elegant in situ redox-chemistry strategy of anchoring Cu3 (HHTP)2 crystallites onto diamagnetic reduced graphene oxide (rGO) sheets, resulting in the formation of electron-doped Cu3 (HHTP)2 -rGO composite which exhibited a characteristic semiconducting behavior (5 K to 300 K) with high electrical conductivity of 70 S ⋅ m-1 and a carrier density of ~1.1×1018  cm-3 at 300 K. Remarkably, no magnetic transition in the Cu3 (HHTP)2 -rGO composite was observed down to 1.5 K endorsing the robust spin liquidity of the 2D MOF Cu3 (HHTP)2 . Specific heat capacity measurements led to the estimation of the residual entropy values of 28 % and 34 % of the theoretically expected value for the pristine Cu3 (HHTP)2 and Cu3 (HHTP)2 -rGO composite, establishing the presence of strong quantum fluctuations down to 1.5 K (two times smaller than the value of the exchange interaction J).

An exact chiral amorphous spin liquid

Topological insulator phases of non-interacting particles have been generalized from periodic crystals to amorphous lattices, which raises the question whether topologically ordered quantum many-body phases may similarly exist in amorphous systems? Here we construct a soluble chiral amorphous quantum spin liquid by extending the Kitaev honeycomb model to random lattices with fixed coordination number three. The model retains its exact solubility but the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian quantum spin liquid phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.

A quantum spin liquid candidate isolated in a two-dimensional CoIIRhIII bimetallic oxalate network

A quantum spin liquid (QSL) is an elusive state of matter characterized by the absence of long-range magnetic order, even at zero temperature, and by the presence of exotic quasiparticle excitations. In spite of their relevance for quantum communication, topological quantum computation and the understanding of strongly correlated systems, like high-temperature superconductors, the unequivocal experimental identification of materials behaving as QSLs remains challenging. Here, we present a novel 2D heterometallic oxalate complex formed by high-spin Co(ii) ions alternating with diamagnetic Rh(iii) in a honeycomb lattice. This complex meets the key requirements to become a QSL: a spin ½ ground state for Co(ii), determined by spin-orbit coupling and crystal field, a magnetically-frustrated triangular lattice due to the presence of antiferromagnetic correlations, strongly suppressed direct exchange interactions and the presence of equivalent interfering superexchange paths between Co centres. A combination of electronic paramagnetic resonance, specific heat and ac magnetic susceptibility measurements in a wide range of frequencies and temperatures shows the presence of strong antiferromagnetic correlations concomitant with no signs of magnetic ordering down to 15 mK. These results show that bimetallic oxalates are appealing QSL candidates as well as versatile systems to chemically fine tune key aspects of a QSL, like magnetic frustration and superexchange path geometries.

Low-Dimensional Metal-Organic Magnets as a Route toward the S = 2 Haldane Phase

Metal-organic magnets (MOMs), modular magnetic materials where metal atoms are connected by organic linkers, are promising candidates for next-generation quantum technologies. MOMs readily form low-dimensional structures and so are ideal systems to realize physical examples of key quantum models, including the Haldane phase, where a topological excitation gap occurs in integer-spin antiferromagnetic (AFM) chains. Thus, far the Haldane phase has only been identified for S = 1, with S ≥ 2 still unrealized because the larger spin imposes more stringent requirements on the magnetic interactions. Here, we report the structure and magnetic properties of CrCl₂(pym) (pym = pyrimidine), a new quasi-1D S = 2 AFM MOM. We show, using X-ray and neutron diffraction, bulk property measurements, density-functional theory calculations, and inelastic neutron spectroscopy (INS), that CrCl₂(pym) consists of AFM CrCl₂ spin chains (J₁ = -1.13(4) meV) which are weakly ferromagnetically coupled through bridging pym (J₂ = 0.10(2) meV), with easy-axis anisotropy (D = -0.15(3) meV). We find that, although small compared to J₁, these additional interactions are sufficient to prevent observation of the Haldane phase in this material. Nevertheless, the proximity to the Haldane phase together with the modularity of MOMs suggests that layered Cr(II) MOMs are a promising family to search for the elusive S = 2 Haldane phase.

Enhanced Curie Temperature of Two-Dimensional Cr(II) Aromatic Heterocyclic Metal-Organic Framework Magnets via Strengthened Orbital Hybridization

Two-dimensional (2D) metal-organic frameworks (MOFs) with room-temperature magnetism are highly desirable but challenging due to the weak superexchange interaction between metal atoms. For this purpose, strengthening the hybridization between metal ion and organic linkage presents an experiment-feasible chemical solution to enhance the Curie temperature. Here, we report three 2D Cr(II) aromatic heterocyclic MOF magnets with enhanced Curie temperature by bridging Cr(II) ions with pyrazine, 1,4-diphosphinine, and 1,4-diarsenin linkers, i.e., Cr(pyz)₂, Cr(diphos)₂, and Cr(diarse)₂, and using first-principles calculations. Our results show that Cr(pyz)₂, Cr(diphos)₂, and Cr(diarse)₂ are ferrimagnetic semiconductors. In particular, the Curie temperature of Cr(pyz)₂ is estimated to be about 344 K and could be enhanced to 512 and 437 K in Cr(diphos)₂ and Cr(diarse)₂ by strengthening the hybridization between Cr ions and organic linkers via d-π* direct exchange interaction. This study presents a prototype to obtain room-temperature magnetism in 2D Cr(II)-based MOF magnets for nanoscale spintronics applications.

Two-Dimensional Metal-Organic Framework on Superconducting NbSe2

The combination of two-dimensional (2D) materials into vertical heterostructures has emerged as a promising path to designer quantum materials with exotic properties. Here, we extend this concept from inorganic 2D materials to 2D metal-organic frameworks (MOFs) that offer additional flexibility in realizing designer heterostructures. We successfully fabricate a monolayer 2D Cu-dicyanoanthracene MOF on a 2D van der Waals NbSe₂ superconducting substrate. The structural and electronic properties of two different phases of the 2D MOF are characterized by low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS), complemented by density-functional theory (DFT) calculations. These experiments allow us to follow the formation of the kagome band structure from Star of David-shaped building blocks. This work extends the synthesis and electronic tunability of 2D MOFs beyond the electronically less relevant metal and semiconducting surfaces to superconducting substrates, which are needed for the development of emerging quantum materials such as topological superconductors.

Spin-orbital entangled state and realization of Kitaev physics in 3dcobalt compounds: a progress report

The realization of Kitaev's honeycomb magnetic model in real materials has become one of the most pursued topics in condensed matter physics and materials science. If found, it is expected to host exotic quantum phases of matter and offers potential realizations of fault-tolerant quantum computations. Over the past years, much effort has been made on 4d- or 5d-heavy transition metal compounds because of their intrinsic strong spin-orbit coupling. But more recently, there have been growing shreds of evidence that the Kitaev model could also be realized in 3d-transition metal systems with much weaker spin-orbit coupling. This review intends to serve as a guide to this fast-developing field focusing on systems withd⁷transition metal occupation. It overviews the current theoretical and experimental progress on realizing the Kitaev model in those systems. We examine the recent experimental observations of candidate materials with Co²⁺ions: e.g., CoPS₃, Na₃Co₂SbO₆, and Na₂Co₂TeO₆, followed by a brief review of theoretical backgrounds. We conclude this article by comparing experimental observations with density functional theory calculations. We stress the importance of inter-t_2ghopping channels and Hund's coupling in the realization of Kitaev interactions in Co-based compounds, which has been overlooked in previous studies. This review suggests future directions in the search for Kitaev physics in 3dcobalt compounds and beyond.

Electric Probe for the Toric Code Phase in Kitaev Materials through the Hyperfine Interaction

The Kitaev model is a remarkable spin model with gapped and gapless spin liquid phases, which are potentially realized in iridates and α-RuCl_{3}. In the recent experiment of α-RuCl_{3}, the signature of a nematic transition to the gapped toric code phase, which breaks the C_{3} symmetry of the system, has been observed through the angle dependence of the heat capacity. We here propose a mechanism by which the nematic transition can be detected electrically. This is seemingly impossible because J_{eff}=1/2 spins do not have an electric quadrupole moment (EQM). However, in the second-order perturbation, the virtual state with a nonzero EQM appears, which makes the nematic order parameter detectable by nuclear magnetic resonance and Mössbauer spectroscopy. The purely magnetic origin of the EQM is different from conventional electronic nematic phases, allowing the direct detection of the realization of Kitaev's toric error-correction code.

Emergent electrochemical functions and future opportunities of hierarchically constructed metal-organic frameworks and covalent organic frameworks

Designing spatial and architectural features across from the molecular to bulk scale is one of the most important topics in materials science which has received a lot of attention in recent years. Looking back to the past research, findings on the influences of spatial features denoted as porous structures on the applications related to mass transport phenomena have been widely studied in traditional inorganic materials, such as ceramics over the past two decades. However, due to the difficulties in precise control of the porous structures at the molecular level in this class of materials, the mechanistic understanding of the effects of spatial and architectural features across from the molecular level to meso-/macroscopic scale is still lacking, especially in electrochemical reactions. Further understanding of fundamental electrochemical functions in well-defined architectures is indispensable for the further advancement of key next-generation energy devices. Furthermore, creating periodic porosity in reticular structures is starting to be recognized as an emerging approach to control the electronic structure of materials. In this review, we focus on the investigations on preparing well-defined molecular-level crystalline porous materials known as metal-organic frameworks (MOFs) and covalent organic frameworks (COFs) into hierarchically constructed architectures from molecular structures lower than the reticular frameworks to meso-/macroscopic scale structures. By connecting well-defined nanosized porous structures in MOFs/COFs and additional length-scale space or shapes, emergent electrochemical functions towards emerging devices, such as beyond Li-ion batteries including all-solid-state rechargeable batteries, are expected to be obtained. By summarizing recent advancements in synthetic strategies of hierarchically constructed MOF/COF based materials and fundamental investigation of their structural effect in a wide spectrum of electrochemical applications, we highlight the importance and future direction of this developing field of hierarchically constructed MOFs/COFs, while emphasizing the required chemical stability of the MOFs/COFs which meet the use in the game-changing electrochemical devices.

Quasi-one dimensional magnetic interactions in the three-dimensional hyper-honeycomb framework [(C2H5)3NH]2Cu2(C2O4)3

The Cu(ii) ions in [(C2H5)3NH]2Cu2(C2O4)3 form a hyperhoneycomb lattice and show no indication of long-range magnetic order down to 60 mK. It has therefore been suggested that [(C2H5)3NH]2Cu2(C2O4)3 is a three dimensional quantum spin liquid. We construct a tight-binding model of [(C2H5)3NH]2Cu2(C2O4)3 from Wannier orbital overlaps. Including interactions within the Jahn-Teller distorted Cu-centered eg Wannier orbitals leads to a highly anisotropic effective Heisenberg model. We show that this anisotropy arrises from interference between different superexchange pathways. This demonstrates that when two (or more) orbitals contribute to the localised spin superexchange can be significantly richer than in the textbook single orbital case. The hyper-honeycomb lattice contains two symmetry distinct sublattices of Cu atoms arranged in coupled chains. We show that one sublattice is strongly dimerized, the other forms isotropic antiferromagnetic chains. Integrating out the strongest (intradimer) exchange interactions leaves extremely weakly coupled Heisenberg chains.

Strongly correlated superconductivity in a copper-based metal-organic framework with a perfect kagome lattice

Metal-organic frameworks (MOFs), which are self-assemblies of metal ions and organic ligands, provide a tunable platform to search a new state of matter. A two-dimensional (2D) perfect kagome lattice, whose geometrical frustration is a key to realizing quantum spin liquids, has been formed in the π - d conjugated 2D MOF [Cu₃(C₆S₆)] _n (Cu-BHT). The recent discovery of its superconductivity with a critical temperature T _c of 0.25 kelvin raises fundamental questions about the nature of electron pairing. Here, we show that Cu-BHT is a strongly correlated unconventional superconductor with extremely low superfluid density. A nonexponential temperature dependence of superfluid density is observed, indicating the possible presence of superconducting gap nodes. The magnitude of superfluid density is much smaller than those in conventional superconductors and follows the Uemura's relation of strongly correlated superconductors. These results imply that the unconventional superconductivity in Cu-BHT originates from electron correlations related to spin fluctuations of kagome lattice.

Scattering Signatures of Bond-Dependent Magnetic Interactions

Bond-dependent magnetic interactions can generate exotic phases such as Kitaev spin-liquid states. Experimentally determining the values of bond-dependent interactions is a challenging but crucial problem. Here, I show that each symmetry-allowed nearest-neighbor interaction on triangular and honeycomb lattices has a distinct signature in paramagnetic neutron-diffraction data, and that such data contain sufficient information to determine the spin Hamiltonian unambiguously via unconstrained fits. Moreover, I show that bond-dependent interactions can often be extracted from powder-averaged data. These results facilitate experimental determination of spin Hamiltonians for materials that do not show conventional magnetic ordering.

Enantiospecific Response in Cross-Polarization Solid-State Nuclear Magnetic Resonance of Optically Active Metal Organic Frameworks

We report herein on a NMR-based enantiospecific response for a family of optically active metal-organic frameworks. Cross-polarization of the ¹H-¹³C couple was performed, and the intensities of the ¹³C nuclei NMR signals were measured to be different for the two enantiomers. In a direct-pulse experiment, which prevents cross-polarization, the intensity difference of the ¹³C NMR signals of the two nanostructured enantiomers vanished. This result is due to changes of the nuclear spin relaxation times due to the electron spin spatial asymmetry induced by chemical bond polarization involving a chiral center. These experiments put forward on firm ground that the chiral-induced spin selectivity effect, which induces chemical bond polarization in the J-coupling, is the mechanism responsible for the enantiospecific response. The implications of this finding for the theory of this molecular electron spin polarization effect and the development of quantum biosensing and quantum storage devices are discussed.

Topological evolution of correlated band structures and heavy-fermion-like behavior in a molybdenum-based metal organic framework C48S36Mo6

Owing to the fascinating properties of two dimensional transition metal dichalcogenides and the stability of ZIF-8 as the subclass of metal organic frameworks (MOFs), we propose a Mo-based MOFs material C₄₈S₃₆Mo₆ containing organic ligands [Formula: see text] connecting to each other by Mo⁴⁺ metal ions. We reveal heavy-fermion-like electronic behaviour that results from highly localized impurity-like Mo-d electrons and tiny energy difference (<0.4 meV/atom) between antiferromagnetic and ferromagnetic state, using the generalized gradient approximation and its combination with Coulomb correlation U. Considering thermal fluctuation and weak magnetic exchange energy, C₄₈S₃₆Mo₆ is a nonmagnetic metal at room temperature, and magnetic insulator at low temperature with a correlation-driven metal-insulator transition. Our Wannier functions analysis indicates that topological properties of energy bands around the Fermi level can be perturbed by correlation U, leading to moving of close nodal lines in three dimensional momentum space. Further introducing of spin-orbit coupling opens a small inverted energy gap of 3 meV at original nodal line due to bulk inversion symmetry. If a gate voltage of 48 meV is applied, the Fermi level will fall into the small energy gap of band inversion, and the system will realize a phase transition from Kondo metal to topological Kondo insulator.

Quantum Spin Liquid State in a Two-Dimensional Semiconductive Metal-Organic Framework

Two-dimensional metal-organic frameworks (2D MOFs) have attracted much attention, as they are the crystalline materials that exhibit both conductivity and microporosity. Numerous efforts have been made to advance their application as chemiresistive sensors or electrochemical capacitors. However, the intrinsic physical properties and spin states of these materials remain poorly understood. Most of these 2D MOFs possess a honeycomb lattice, with a Kagomé lattice arrangement of metal cations. These structural characteristics suggest that these MOFs would be candidates for geometrically frustrated spin systems with unprecedented magnetic phenomena. Herein, by performing magnetic susceptibility and specific heat measurements at an ultralow temperature down to 38mK on a 2D semiconductive MOF, Cu₃(HHTP)₂, a quantum spin liquid state that arises from the geometrical frustration was suggested. This result illustrates the potential of strongly correlated MOFs as systems with emergent phenomena induced by unusual structural topologies.

Materials design of Kitaev spin liquids beyond the Jackeli-Khaliullin mechanism

The Kitaev spin liquid provides a rare example of well-established quantum spin liquids in more than one dimension. It is obtained as the exact ground state of the Kitaev spin model with bond-dependent anisotropic interactions. The peculiar interactions can be yielded by the synergy of spin-orbit coupling and electron correlations for specific electron configuration and lattice geometry, which is known as the Jackeli-Khaliullin mechanism. Based on this mechanism, there has been a fierce race for the materialization of the Kitaev spin liquid over the last decade, but the candidates have been still limited mostly to 4d- and 5d-electron compounds including cations with the low-spind5electron configuration, such as Ir4+and Ru3+. Here we discuss recent efforts to extend the material perspective beyond the Jackeli-Khaliullin mechanism, by carefully reexamining the two requisites, formation of thejeff= 1/2 doublet and quantum interference between the exchange processes, for not onlyd- but alsof-electron systems. We present three examples: the systems including Co2+and Ni3+with the high-spind7electron configuration, Pr4+with thef1-electron configuration, and polar asymmetry in the lattice structure. In particular, the latter two are intriguing since they may realize the antiferromagnetic Kitaev interactions, in contrast to the ferromagnetic ones in the existing candidates. This partial overview would stimulate further material exploration of the Kitaev spin liquids and its topological properties due to fractional excitations.

Novel Chiral Quantum Spin Liquids in Kitaev Magnets

Quantum magnets with pure Kitaev spin exchange interactions can host a gapped quantum spin liquid with a single Majorana edge mode propagating in the counterclockwise direction when a small positive magnetic field is applied. Here, we show how under a sufficiently strong positive magnetic field a topological transition into a gapped quantum spin liquid with two Majorana edge modes propagating in the clockwise direction occurs. The Dzyaloshinskii-Moriya interaction is found to turn the nonchiral Kitaev's gapless quantum spin liquid into a chiral one with equal Berry phases at the two Dirac points. Thermal Hall conductance experiments can provide evidence of the novel topologically gapped quantum spin liquid states predicted.

Metal-Organic Frameworks in Modern Physics: Highlights and Perspectives

Owing to the synergistic combination of a hybrid organic-inorganic nature and a chemically active porous structure, metal-organic frameworks have emerged as a new class of crystalline materials. The current trend in the chemical industry is to utilize such crystals as flexible hosting elements for applications as diverse as gas and energy storage, filtration, catalysis, and sensing. From the physical point of view, metal-organic frameworks are considered molecular crystals with hierarchical structures providing the structure-related physical properties crucial for future applications of energy transfer, data processing and storage, high-energy physics, and light manipulation. Here, the perspectives of metal-organic frameworks as a new family of functional materials in modern physics are discussed: from porous metals and superconductors, topological insulators, and classical and quantum memory elements, to optical superstructures, materials for particle physics, and even molecular scale mechanical metamaterials. Based on complementary properties of crystallinity, softness, organic-inorganic nature, and complex hierarchy, a description of how such artificial materials have extended their impact on applied physics to become the mainstream in material science is offered.

Strong magnetic exchange and frustrated ferrimagnetic order in a weberite-type inorganic-organic hybrid fluoride

We combine powder neutron diffraction, magnetometry and ⁵⁷Fe Mössbauer spectrometry to determine the nuclear and magnetic structures of a strongly interacting weberite-type inorganic-organic hybrid fluoride, Fe₂F₅(H taz). In this structure, Fe²⁺ and Fe³⁺ cations form magnetically frustrated hexagonal tungsten bronze layers of corner-sharing octahedra. Our powder neutron diffraction data reveal that, unlike its purely inorganic fluoride weberite counterparts which adopt a centrosymmetric Imma structure, the room-temperature nuclear structure of Fe₂F₅(H taz) is best described by a non-centrosymmetric Ima2 model with refined lattice parameters a = 9.1467(2) Å, b = 9.4641(2) Å and c = 7.4829(2) Å. Magnetic susceptibility and magnetization measurements reveal that strong antiferromagnetic exchange interactions prevail in Fe₂F₅(H taz) leading to a magnetic ordering transition at T_N = 93 K. Analysis of low-temperature powder neutron diffraction data indicates that below T_N, the Fe²⁺ sublattice is ferromagnetic, with a moment of 4.1(1) µ_B per Fe²⁺ at 2 K, but that an antiferromagnetic component of 0.6(3) µ_B cants the main ferromagnetic component of Fe³⁺, which aligns antiferromagnetically to the Fe²⁺ sublattice. The zero-field and in-field Mössbauer spectra give clear evidence of an excess of high-spin Fe³⁺ species within the structure and a non-collinear magnetic structure. This article is part of the theme issue 'Mineralomimesis: natural and synthetic frameworks in science and technology'.

Magnetic-Field Tuning of Hydrogen Bond Order-Disorder Transition in Metal-Organic Frameworks

The ordering of polar hydrogen bonds may break space inversion symmetry and induce ferroelectricity or antiferroelectricity. This process is usually immune to external magnetic fields so that magnetic control of hydrogen bonds is very challenging. Here we demonstrate that the ordering of hydrogen bonds in the metal-organic frameworks [(CH_{3})_{2}NH_{2}]M(HCOO)_{3} (M=Fe, Co) can be manipulated by applying magnetic fields. After cooling in a high magnetic field, the order-disorder transition of hydrogen bonds shifts to a lower or higher temperature, depending on antiferroelectricity or ferroelectricity induced by hydrogen bond ordering. Besides, the order-disorder transition leads to a giant thermal expansion, exceeding ∼3.5×10^{4} and ∼2×10^{4}  ppm for M=Fe and Co, respectively, which is much higher than that of inorganic ferroelectrics. The influence of magnetic field on hydrogen bond ordering is discussed in terms of the magnetoelastic coupling.

Dimensionality controlled by light exposure: 1D versus 3D oxalate-bridged [CuFe] coordination polymers based on an [Fe(C2O4)3]3− metallotecton

The studied heterometallic [CuFe] compounds, based on an [Fe(C₂O₄)₃]³⁻ building block and containing a 3D network or 1D ladder-like chains, were synthesized depending on whether the test tube with the same reaction layers was exposed to daylight or not.

Emergent SU(4) Symmetry in α-ZrCl_{3} and Crystalline Spin-Orbital Liquids

While the enhancement of spin-space symmetry from the usual SU(2) to SU(N) is promising for finding nontrivial quantum spin liquids, its realization in magnetic materials remains challenging. Here, we propose a new mechanism by which SU(4) symmetry emerges in the strong spin-orbit coupling limit. In d^{1} transition metal compounds with edge-sharing anion octahedra, the spin-orbit coupling gives rise to strongly bond-dependent and apparently SU(4)-breaking hopping between the J_{eff}=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an SU(4)-symmetric Hubbard model. In the strong repulsion limit at quarter filling, as realized in α-ZrCl_{3}, the low-energy effective model is the SU(4) Heisenberg model on the honeycomb lattice, which cannot have a trivial gapped ground state and is expected to host a gapless spin-orbital liquid. By generalizing this model to other three-dimensional lattices, we also propose crystalline spin-orbital liquids protected by this emergent SU(4) symmetry and space group symmetries.

Models and materials for generalized Kitaev magnetism

The exactly solvable Kitaev model on the honeycomb lattice has recently received enormous attention linked to the hope of achieving novel spin-liquid states with fractionalized Majorana-like excitations. In this review, we analyze the mechanism proposed by Jackeli and Khaliullin to identify Kitaev materials based on spin-orbital dependent bond interactions and provide a comprehensive overview of its implications in real materials. We set the focus on experimental results and current theoretical understanding of planar honeycomb systems (Na₂IrO₃, α-Li₂IrO₃, and α-RuCl₃), three-dimensional Kitaev materials (β- and γ-Li₂IrO₃), and other potential candidates, completing the review with the list of open questions awaiting new insights.