Direct observation of finite size effects in chains of antiferromagnetically coupled spins

Origin of the Unusual Ground-State Spin S = 9 in a Cr10 Single-Molecule Magnet

The molecular wheel [Cr₁₀(OMe)₂₀(O₂CCMe₃)₁₀], abbreviated {Cr₁₀}, with an unusual intermediate total spin S = 9 and non-negligible cluster anisotropy, D/k_B = -0.045(2) K, is a rare case among wheels based on an even number of 3d-metals, which usually present an antiferromagnetic (AF) ground state (S = 0). Herein, we unveil the origin of such a behavior. Angular magnetometry measurements performed on a single crystal confirmed the axial anisotropic behavior of {Cr₁₀}. For powder samples, the temperature dependence of the susceptibility plotted as χT(T) showed an overall ferromagnetic (FM) behavior down to 1.8 K, whereas the magnetization curve M(H) did not saturate at the expected 30 μ_B/fu for 10 FM coupled 3/2 spin Cr³⁺ ions, but to a much lower value, corresponding to S = 9. In addition, the X-ray magnetic circular dichroism (XMCD) measured at high magnetic field (170 kOe) and 7.5 K showed the polarization of the cluster moment up to 23 μ_B/fu. The magnetic results can be rationalized within a model, including the cluster anisotropy, in which the {Cr₁₀} wheel is formed by two semiwheels, each with four Cr³⁺ spins FM coupled (J_FM/k_B = 2.0 K), separated by two Cr³⁺ ions AF coupled asymmetrically (J₂₃/k_B = J₇₈/k_B = -2.0 K; J₃₄/k_B = J₈₉/k_B = -0.25 K). Inelastic neutron scattering and heat capacity allowed us to confirm this model leading to the S = 9 ground state and first excited S = 8. Single-molecule magnet behavior with an activation energy of U/k_B = 4.0(5) K in the absence of applied field was observed through ac susceptibility measurements down to 0.1 K. The intriguing magnetic behavior of {Cr₁₀} arises from the detailed asymmetry in the molecule interactions produced by small-angle distortions in the angles of the Cr-O-Cr alkoxy bridges coupling the Cr³⁺ ions, as demonstrated by ab initio and density functional theory calculations, while the cluster anisotropy can be correlated to the single-ion anisotropies calculated for each Cr³⁺ ion in the wheel.

New Dimension in Magnetism and Superconductivity: 3D and Curvilinear Nanoarchitectures

Traditionally, the primary field, where curvature has been at the heart of research, is the theory of general relativity. In recent studies, however, the impact of curvilinear geometry enters various disciplines, ranging from solid-state physics over soft-matter physics, chemistry, and biology to mathematics, giving rise to a plethora of emerging domains such as curvilinear nematics, curvilinear studies of cell biology, curvilinear semiconductors, superfluidity, optics, 2D van der Waals materials, plasmonics, magnetism, and superconductivity. Here, the state of the art is summarized and prospects for future research in curvilinear solid-state systems exhibiting such fundamental cooperative phenomena as ferromagnetism, antiferromagnetism, and superconductivity are outlined. Highlighting the recent developments and current challenges in theory, fabrication, and characterization of curvilinear micro- and nanostructures, special attention is paid to perspective research directions entailing new physics and to their strong application potential. Overall, the perspective is aimed at crossing the boundaries between the magnetism and superconductivity communities and drawing attention to the conceptual aspects of how extension of structures into the third dimension and curvilinear geometry can modify existing and aid launching novel functionalities. In addition, the perspective should stimulate the development and dissemination of research and development oriented techniques to facilitate rapid transitions from laboratory demonstrations to industry-ready prototypes and eventual products.

Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers

Magnetic molecules are prototypical systems to investigate peculiar quantum mechanical phenomena. As such, simulating their static and dynamical behavior is intrinsically difficult for a classical computer, due to the exponential increase of required resources with the system size. Quantum computers solve this issue by providing an inherently quantum platform, suited to describe these magnetic systems. Here, we show that both the ground state properties and the spin dynamics of magnetic molecules can be simulated on prototype quantum computers, based on superconducting qubits. In particular, we study small-size anti-ferromagnetic spin chains and rings, which are ideal test-beds for these pioneering devices. We use the variational quantum eigensolver algorithm to determine the ground state wave-function with targeted ansatzes fulfilling the spin symmetries of the investigated models. The coherent spin dynamics are simulated by computing dynamical correlation functions, an essential ingredient to extract many experimentally accessible properties, such as the inelastic neutron cross-section.

Probing resonating valence bond states in artificial quantum magnets

Designing and characterizing the many-body behaviors of quantum materials represents a prominent challenge for understanding strongly correlated physics and quantum information processing. We constructed artificial quantum magnets on a surface by using spin-1/2 atoms in a scanning tunneling microscope (STM). These coupled spins feature strong quantum fluctuations due to antiferromagnetic exchange interactions between neighboring atoms. To characterize the resulting collective magnetic states and their energy levels, we performed electron spin resonance on individual atoms within each quantum magnet. This gives atomic-scale access to properties of the exotic quantum many-body states, such as a finite-size realization of a resonating valence bond state. The tunable atomic-scale magnetic field from the STM tip allows us to further characterize and engineer the quantum states. These results open a new avenue to designing and exploring quantum magnets at the atomic scale for applications in spintronics and quantum simulations.

The resonating valence bond state is a spin-liquid state where spins continuously alter their singlet partners. Here Yang et al. use spin-1/2 atoms precision-placed by a scanning tunnelling microscope to create artificial quantum magnets exhibiting the resonating valence bond state.

Breaking the ring: 53Cr-NMR on the Cr8Cd molecular nanomagnet

An accurate experimental characterization of finite antiferromagnetic (AF) spin chains is crucial for controlling and manipulating their magnetic properties and quantum states for potential applications in spintronics or quantum computation. In particular, finite AF chains are expected to show a different magnetic behaviour depending on their length and topology. Molecular AF rings are able to combine the quantum-magnetic behaviour of AF chains with a very remarkable tunability of their topological and geometrical properties. In this work we measure the ⁵³Cr-NMR spectra of the Cr₈Cd ring to study the local spin densities on the Cr sites. Cr₈Cd can in fact be considered a model system of a finite AF open chain with an even number of spins. The NMR resonant frequencies are in good agreement with the theoretical local spin densities, by assuming a core polarization field A _C = -12.7 T μ _B ^-1. Moreover, these NMR results confirm the theoretically predicted non-collinear spin arrangement along the Cr₈Cd ring, which is typical of an even-open AF spin chain.

Pure Metal-Organic Framework Microlasers with Controlled Cavity Shapes

Metal-organic frameworks (MOFs) are an emerging kind of laser material, yet they remain a challenge in the controlled fabrication of crystal nanostructures with desired morphology for tuning their optical microcavities. Herein, the shape-engineering of pure MOF microlasers was demonstrated based on the coordination-mode-tailored method. The one-dimensional (1D) microwires and 2D microplates were selectively fabricated through changing the HCl concentration to tailor the coordination modes. Both the single-crystalline microwires and microplates with strong optical confinement functioned as low-threshold MOF microlasers. Moreover, distinct lasing behaviors of 1D and 2D MOF microcrystals confirm a typical shape-dependent microcavity effect: 1D microwires serve as Fabry-Pérot (FP) resonators, and 2D microplates lead to the whispering-gallery-mode (WGM) microcavities. These results provide a special pathway for the exploitation of MOF-based micro/nanolasers with on-demand functions.

Multimodeling Approach to Ferromagnetic Spin-Wave Excitations in the High-Spin Cluster Mn18Sr Observed by Inelastic Neutron Scattering

The magnetism of the mixed-valence high-spin cluster [Mn₁₈SrO₈(N₃)₇Cl(MedhmpH)₁₂(MeCN)₆]Cl₂ (1) exhibiting intramolecular ferromagnetic interactions was studied using inelastic neutron scattering (INS), and reliable values for the exchange coupling constants were determined based on the quality of simultaneous fits to the INS and magnetic data. The challenge of the huge size of the Hilbert space (3 375 000) and many exchange coupling constants (7 assuming a C₃ symmetry) generally encountered in large spin clusters was resolved as follows: (a) The results of the restricted Hilbert space ferromagnetic cluster spin wave theory were compared to the experimental spectroscopic data. The observed INS transitions were thus assigned to spin wave excitations in a bounded ferromagnetic spin cluster and moreover could be visualized in a straightforward way based on this theory. (b) Simultaneously, Quantum Monte Carlo (QMC) calculations of the temperature-dependent magnetic susceptibility with the same parameter set were compared to the experimental data. Application of state-of-the-art QMC algorithms, as available in the open source ALPS package, in ferromagnetic clusters avoids the full Hamiltonian diagonalization without sacrificing calculation accuracy of the magnetic susceptibility down to the lowest temperatures, which was crucial for the successful analysis. The combined fits revealed two exchange-coupling models with equally good overall agreement to the data. Our preferred model was inspired by magnetostructural correlations and is consistent with them. The model involves three different exchange interactions, one describing the interaction between the core Mn^III spins J_a = 14.3(1.0) K and two interactions linking the core and the peripheral Mn^II spins: J_b = 8.3(4) K and J₆ = 3.6(4) K. The use of open-source QMC software and our systematic approach to fitting multiple sets of data obtained by different experimental techniques are described in detail and are generally applicable for understanding large ferromagnetically coupled clusters.

Size-driven quantum phase transitions

Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analyzing a sequence of finite-size cases? We present models for which such an approach gives completely misleading results: translationally invariant, local Hamiltonians on a square lattice with open boundary conditions and constant spectral gap, which have a classical product ground state for all system sizes smaller than a particular threshold size, but a ground state with topological degeneracy for all system sizes larger than this threshold. Starting from a minimal case with spins of dimension 6 and threshold lattice size [Formula: see text], we show that the latter grows faster than any computable function with increasing local spin dimension. The resulting effect may be viewed as a unique type of quantum phase transition that is driven by the size of the system rather than by an external field or coupling strength. We prove that the construction is thermally robust, showing that these effects are in principle accessible to experimental observation.

Mesoporous Silica Matrix as a Tool for Minimizing Dipolar Interactions in NiFe₂O₄ and ZnFe₂O₄ Nanoparticles

NiFe₂O₄ and ZnFe₂O₄ nanoparticles have been prepared encased in the MCM (Mobile Composition of Matter) type matrix. Their magnetic behavior has been studied and compared with that corresponding to particles of the same composition and of a similar size (prepared and embedded in amorphous silica or as bare particles). This study has allowed elucidation of the role exerted by the matrix and interparticle interactions in the magnetic behavior of each ferrite system. Thus, very different superparamagnetic behavior has been found in ferrite particles of similar size depending on the surrounding media. Also, the obtained results clearly provide evidence of the vastly different magnetic behavior for each ferrite system.

Portraying entanglement between molecular qubits with four-dimensional inelastic neutron scattering

Entanglement is a crucial resource for quantum information processing and its detection and quantification is of paramount importance in many areas of current research. Weakly coupled molecular nanomagnets provide an ideal test bed for investigating entanglement between complex spin systems. However, entanglement in these systems has only been experimentally demonstrated rather indirectly by macroscopic techniques or by fitting trial model Hamiltonians to experimental data. Here we show that four-dimensional inelastic neutron scattering enables us to portray entanglement in weakly coupled molecular qubits and to quantify it. We exploit a prototype (Cr₇Ni)₂ supramolecular dimer as a benchmark to demonstrate the potential of this approach, which allows one to extract the concurrence in eigenstates of a dimer of molecular qubits without diagonalizing its full Hamiltonian.

Noise, delocalization, and quantum diffusion in one-dimensional tight-binding models

As an unusual type of anomalous diffusion behavior, namely (transient) superballistic transport, has been experimentally observed recently, but it is not yet well understood. In this paper, we investigate the white noise effect (in the Markov approximation) on quantum diffusion in one-dimensional tight-binding models with a periodic, disordered, and quasiperiodic region of size L attached to two perfect lattices at both ends in which the wave packet is initially located at the center of the sublattice. We find that in a completely localized system, inducing noise could delocalize the system to a desirable diffusion phase. This controllable system may be used to investigate the interplay of disorder and white noise, as well as to explore an exotic quantum phase.

Hybrid Fibrillar Xerogels with Unusual Magnetic Properties

We report on the preparation of a hybrid nanomaterial made up of 1D filaments of an antiferromagnetic self-assembling bicopper complex encapsulated in polymer nanofibrils. The encapsulation process is achieved through the heterogeneous nucleation of the growth of polymer fibrils obtained by thermoreversible gelation as shown by calorimetry experiments. Neutron scattering experiments confirm that the filaments of a bicopper complex retain their 1D character after encapsulation in the fibrils. Superconducting quantum interference device experiments show that the bicopper complex, originally in the gapped spin state in the 3D bulk mesophase, displays a gapless behavior once encapsulated. Extended absorption fine structure and infrared results further highlight the difference in the molecular arrangement of the bicopper complex between the bulk mesophase and the encapsulated state, which may account for the magnetic behavior. This material, which is largely disordered, differs totally from the usual magnetic systems where this effect is observed only on highly crystalline systems with long-range order. Also, this hybrid material is very easy to prepare from its basic constituents and can be further processed in many ways.

Emergent Interacting Spin Islands in a Depleted Strong-Leg Heisenberg Ladder

Properties of the depleted Heisenberg spin ladder material series (C_{7}H_{10}N)_{2}Cu_{1-z}Zn_{z}Br_{4} have been studied by the combination of magnetic measurements and neutron spectroscopy. Disorder-induced degrees of freedom lead to a specific magnetic response, described in terms of emergent strongly interacting "spin island" objects. The structure and dynamics of the spin islands is studied by high-resolution inelastic neutron scattering. This allows us to determine their spatial shape and to observe their mutual interactions, manifested by strong spectral in-gap contributions.

Comparison of spin dynamics and magnetic properties in antiferromagnetic closed and open molecular Cr-based rings

We present magnetization and (1)H nuclear magnetic resonance (NMR) measurements performed in both closed Cr8 and open Cr8Zn antiferromagnetic molecular rings in the temperature range 1.65  <  T  <  300 K at different external magnetic fields. The magnetization measurements on Cr8Zn are consistent with a small decrease of the exchange constant J(Cr-Cr) and a much smaller gap between the singlet ground state and the first magnetic excited state when compared with the same properties of the closed ring Cr8, in agreement with previous inelastic neutron scattering results. The temperature dependence of the (1)H NMR nuclear spin lattice relaxation rate (NSLR), 1/T1(T), was found to be similar in both open and closed rings with a magnetic field dependent peak centered at a temperature of the order of the corresponding exchange constant J(Cr-Cr). Such main peak in the NSLR could be fitted with a single correlation frequency ω(c1) as in most molecular magnets. At low temperature T  <  4 K, a new feature not observed in previous NMR measurements on antiferromagnetic rings and consisting in a smaller peak of 1/T1(T) which is well resolved only in Cr8Zn, was singled out. This low-T peak indicates the presence of a second correlation frequency ω(c2) of the magnetization, found to be quite different between the two rings and thus possibly reflecting the different low temperature level structure associated with the different spin topology. The presence of ω(c2) is confirmed by the NMR spin-spin relaxation rate enhancement, which generates a two-steps wipe-out effect of the NMR signal intensity.

Coherent Spin Dynamics in Molecular Cr8Zn Wheels

Controlling and understanding transitions between molecular spin states allows selection of the most suitable ones for qubit encoding. Here we present a detailed investigation of single crystals of a polynuclear Cr8Zn molecular wheel using 241 GHz electron paramagnetic resonance (EPR) spectroscopy in high magnetic field. Continuous wave spectra are well reproduced by spin Hamiltonian calculations, which evidence that transitions in correspondence to a well-defined anticrossing involve mixed states with different total spin. We studied, by means of spin echo experiments, the temperature dependence of the dephasing time (T2) down to 1.35 K. These results are reproduced by considering both hyperfine and intermolecular dipolar interactions, evidencing that the dipolar contribution is completely suppressed at the lowest temperature. Overall, these results shed light on the effects of the decoherence mechanisms, whose understanding is crucial to exploit chemically engineered molecular states as a resource for quantum information processing.