Molecular nanomagnets: a viable path toward quantum information processing?

Molecular nanomagnets (MNMs), molecules containing interacting spins, have been a playground for quantum mechanics. They are characterized by many accessible low-energy levels that can be exploited to store and process quantum information. This naturally opens the possibility of using them as qudits, thus enlarging the tools of quantum logic with respect to qubit-based architectures. These additional degrees of freedom recently prompted the proposal for encoding qubits with embedded quantum error correction (QEC) in single molecules. QEC is the holy grail of quantum computing and this qudit approach could circumvent the large overhead of physical qubits typical of standard multi-qubit codes. Another important strength of the molecular approach is the extremely high degree of control achieved in preparing complex supramolecular structures where individual qudits are linked preserving their individual properties and coherence. This is particularly relevant for building quantum simulators, controllable systems able to mimic the dynamics of other quantum objects. The use of MNMs for quantum information processing is a rapidly evolving field which still requires to be fully experimentally explored. The key issues to be settled are related to scaling up the number of qudits/qubits and their individual addressing. Several promising possibilities are being intensively explored, ranging from the use of single-molecule transistors or superconducting devices to optical readout techniques. Moreover, new tools from chemistry could be also at hand, like the chiral-induced spin selectivity. In this paper, we will review the present status of this interdisciplinary research field, discuss the open challenges and envisioned solution paths which could finally unleash the very large potential of molecular spins for quantum technologies.

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.

Photon Condensation and Enhanced Magnetism in Cavity QED

A system of magnetic molecules coupled to microwave cavities (LC resonators) undergoes the equilibrium superradiant phase transition. The transition is experimentally observable. The effect of the coupling is first illustrated by the vacuum-induced ferromagnetic order in a quantum Ising model and then by the modification of the magnetic phase diagram of Fe_{8} dipolar crystals, exemplifying the cooperation between intrinsic and photon-induced spin-spin interactions. Finally, a transmission experiment is shown to resolve the transition, measuring the quantum electrodynamical control of magnetism.

Broad-band spectroscopy of a vanadyl porphyrin: a model electronuclear spin qudit

We explore how to encode more than a qubit in vanadyl porphyrin molecules hosting a S = 1/2 electronic spin coupled to a I = 7/2 nuclear spin. The spin Hamiltonian and its parameters, as well as the spin dynamics, have been determined via a combination of electron paramagnetic resonance, heat capacity, magnetization and on-chip magnetic spectroscopy experiments performed on single crystals. We find low temperature spin coherence times of micro-seconds and spin relaxation times longer than a second. For sufficiently strong magnetic fields (B > 0.1 T, corresponding to resonance frequencies of 9-10 GHz) these properties make vanadyl porphyrin molecules suitable qubit realizations. The presence of multiple equispaced nuclear spin levels then merely provides 8 alternatives to define the '1' and '0' basis states. For lower magnetic fields (B < 0.1 T), and lower frequencies (<2 GHz), we find spectroscopic signatures of a sizeable electronuclear entanglement. This effect generates a larger set of allowed transitions between different electronuclear spin states and removes their degeneracies. Under these conditions, we show that each molecule fulfills the conditions to act as a universal 4-qubit processor or, equivalently, as a d = 16 qudit. These findings widen the catalogue of chemically designed systems able to implement non-trivial quantum functionalities, such as quantum simulations and, especially, quantum error correction at the molecular level.

Chemical tuning of spin clock transitions in molecular monomers based on nuclear spin-free Ni(ii)

We report the existence of a sizeable quantum tunnelling splitting between the two lowest electronic spin levels of mononuclear Ni complexes. The level anti-crossing, or magnetic "clock transition", associated with this gap has been directly monitored by heat capacity experiments. The comparison of these results with those obtained for a Co derivative, for which tunnelling is forbidden by symmetry, shows that the clock transition leads to an effective suppression of intermolecular spin-spin interactions. In addition, we show that the quantum tunnelling splitting admits a chemical tuning via the modification of the ligand shell that determines the crystal field and the magnetic anisotropy. These properties are crucial to realize model spin qubits that combine the necessary resilience against decoherence, a proper interfacing with other qubits and with the control circuitry and the ability to initialize them by cooling.

Enhanced Molecular Spin-Photon Coupling at Superconducting Nanoconstrictions

We combine top-down and bottom-up nanolithography to optimize the coupling of small molecular spin ensembles to 1.4 GHz on-chip superconducting resonators. Nanoscopic constrictions, fabricated with a focused ion beam at the central transmission line, locally concentrate the microwave magnetic field. Drops of free-radical molecules have been deposited from solution onto the circuits. For the smallest ones, the molecules were delivered at the relevant circuit areas by means of an atomic force microscope. The number of spins N_eff effectively coupled to each device was accurately determined combining Scanning Electron and Atomic Force Microscopies. The collective spin-photon coupling constant has been determined for samples with N_eff ranging between 2 × 10⁶ and 10¹² spins, and for temperatures down to 44 mK. The results show the well-known collective enhancement of the coupling proportional to the square root of N_eff. The average coupling of individual spins is enhanced by more than 4 orders of magnitude (from 4 mHz up to above 180 Hz), when the transmission line width is reduced from 400 μm down to 42 nm, and reaches maximum values near 1 kHz for molecules located on the smallest nanoconstrictions.

Magnetic chains of Fe3 clusters in the {Fe3YO2} butterfly molecular compound

The "butterfly" molecule [Fe3Y(μ3-O)2(CCl3COO)8(H2O)(THF)3] (in brief {Fe3YO2}) includes three Fe3+ ions which build a robust Fe3 cluster with a strong intracluster antiferromagnetic exchange and a total spin S = 5/2. It represents the starting magnetic system to study further interactions with magnetic rare earths when Y is replaced with lanthanides. We present heat capacity and equilibrium susceptibility measurements below 2 K, which show that each cluster has a sizeable magnetic anisotropy pointing to the existence of intercluster interactions. However, no phase transition to a long-range magnetically ordered phase is observed down to 20 mK. The intercluster interaction is analysed in the framework of the one-dimensional Blume-Capel model with an antiferromagnetic chain interaction constant J/kB = -40(2) mK between Fe3 cluster spins, and a uniaxial anisotropy with parameter D/kB = -0.56(3) K. This is associated to single chains of Fe3 clusters oriented along the shortest intercluster distances displayed by the crystal structure of {Fe3YO2}. Ac susceptibility measurements reveal that the magnetic relaxation is dominated by a quantum tunnelling process below 0.2 K, and by thermally activated processes above this temperature. The experimental activation energy of this single chain magnet, Ea/kB = 3.4(6) K, can be accounted for by the combination of contributions arising from single-molecule magnetic anisotropy and spin-spin correlations along the chains.

A heterometallic [LnLn′Ln] lanthanide complex as a qubit with embedded quantum error correction

We show that a [Er–Ce–Er] molecular trinuclear coordination compound is a promising platform to implement the three-qubit quantum error correction code protecting against pure dephasing, the most important error in magnetic molecules. We characterize it by preparing the [Lu–Ce–Lu] and [Er–La–Er] analogues, which contain only one of the two types of qubit, and by combining magnetometry, low-temperature specific heat and electron paramagnetic resonance measurements on both the elementary constituents and the trimer. Using the resulting parameters, we demonstrate by numerical simulations that the proposed molecular device can efficiently suppress pure dephasing of the spin qubits.

We show that a [Er–Ce–Er] molecular trinuclear coordination compound is a promising platform to implement the three-qubit quantum error correction code protecting against pure dephasing, the most important error in magnetic molecules.

Quantum Monte Carlo simulations of a giant {Ni21Gd20} cage with a S = 91 spin ground state

The detailed analysis of magnetic interactions in a giant molecule is difficult both because the synthesis of such compounds is challenging and the number of energy levels increases exponentially with the magnitude and number of spins. Here, we isolated a {Ni₂₁Gd₂₀} nanocage with a large number of energy levels (≈5 × 10³⁰) and used quantum Monte Carlo (QMC) simulations to perform a detailed analysis of magnetic interactions. Based on magnetization measurements above 2 K, the QMC simulations predicted very weak ferromagnetic interactions that would give a record S = 91 spin ground state. Low-temperature measurements confirm the spin ground state but suggest a more complex picture due to the single ion anisotropy; this has also been modeled using the QMC approach. The high spin and large number of low-lying states lead to a large low-field magnetic entropy (14.1 J kg^-1 K^-1 for ΔH = 1 T at 1.1 K) for this material.

Strong Coupling of Microwave Photons to Antiferromagnetic Fluctuations in an Organic Magnet

Coupling between a crystal of di(phenyl)-(2,4,6-trinitrophenyl)iminoazanium radicals and a superconducting microwave resonator is investigated in a circuit quantum electrodynamics (circuit QED) architecture. The crystal exhibits paramagnetic behavior above 4 K, with antiferromagnetic correlations appearing below this temperature, and we demonstrate strong coupling at base temperature. The magnetic resonance acquires a field angle dependence as the crystal is cooled down, indicating anisotropy of the exchange interactions. These results show that multispin modes in organic crystals are suitable for circuit QED, offering a platform for their coherent manipulation. They also utilize the circuit QED architecture as a way to probe spin correlations at low temperature.

A scalable architecture for quantum computation with molecular nanomagnets

A proposal for a magnetic quantum processor that consists of individual molecular spins coupled to superconducting coplanar resonators and transmission lines is carefully examined. We derive a simple magnetic quantum electrodynamics Hamiltonian to describe the underlying physics. It is shown that these hybrid devices can perform arbitrary operations on each spin qubit and induce tunable interactions between any pair of them. The combination of these two operations ensures that the processor can perform universal quantum computations. The feasibility of this proposal is critically discussed using the results of realistic calculations, based on parameters of existing devices and molecular qubits. These results show that the proposal is feasible, provided that molecules with sufficiently long coherence times can be developed and accurately integrated into specific areas of the device. This architecture has an enormous potential for scaling up quantum computation thanks to the microscopic nature of the individual constituents, the molecules, and the possibility of using their internal spin degrees of freedom.

The classical and quantum dynamics of molecular spins on graphene

Controlling the dynamics of spins on surfaces is pivotal to the design of spintronic and quantum computing devices. Proposed schemes involve the interaction of spins with graphene to enable surface-state spintronics and electrical spin manipulation. However, the influence of the graphene environment on the spin systems has yet to be unravelled. Here we explore the spin-graphene interaction by studying the classical and quantum dynamics of molecular magnets on graphene. Whereas the static spin response remains unaltered, the quantum spin dynamics and associated selection rules are profoundly modulated. The couplings to graphene phonons, to other spins, and to Dirac fermions are quantified using a newly developed model. Coupling to Dirac electrons introduces a dominant quantum relaxation channel that, by driving the spins over Villain's threshold, gives rise to fully coherent, resonant spin tunnelling. Our findings provide fundamental insight into the interaction between spins and graphene, establishing the basis for electrical spin manipulation in graphene nanodevices.

Paramagnetic spherical nanoparticles by the self-assembly of persistent trityl radicals

The self-association of a water-soluble persistent trityl radical gives rise to paramagnetic nanoparticles: a new class of non-metallic high spin systems.

Magnetic relaxation versus 3D long-range ordering in {Dy₂Ba(α-fur)₈}n furoate polymers

A novel Dy-complex formulated as {[Dy2Ba(α-C4H3OCOO)8·(H2O)4]·2H2O}n, {Dy2Ba(α-fur)8}n, has been synthesized, structurally characterized, and magnetically and thermally investigated as a function of field and temperature, down to 85 mK. The α-furoate ligands consolidate 1D zig-zag chains formed by Dy2 dimers separated by Ba ions. Ab initio calculations were used to determine the easy anisotropy axis direction, the gyromagnetic tensor components and the energy levels of each Dy. The heat capacity and susceptibility measurements allowed us to conclude that intradimer and interdimer interactions are ferromagnetic and of the same order, J'/k(B) ≈ J''/k(B) = +0.55 K. In the absence of an applied magnetic field, the dynamic relaxation of the magnetization occurs through the fast (τ(T) ~ 10(-5) s) spin-reversal of each of the individual Dys through a quantum tunneling (QT) process. A long-range 3D ordered state is achieved at T(N) = 0.25 K, in which the ferromagnetically coupled zig-zag chains (J'/k(B) ≈ J''/k(B) = +0.528(1) K) running along the c-axis are antiferromagnetically coupled to the adjacent chains (J'''/k(B) = -0.021(1) K). Critical slowing down of the QT time constant is observed when the temperature approaches T(N). Under the application of a magnetic field, the QT relaxation is replaced by an Orbach process (with energy barrier U/k(B) = 68 K and τ0 ~ 10(-9) s at H = 2 kOe) and a very slow (τ(s) ∼ 0.2 s) relaxation process. We propose and demonstrate the proof of concept of a spintronic device, in which two different relaxation rates can be selected, and on/off switched by magnetic field biasing. The dynamical behavior of this compound is compared with another furoate to discuss the effect of competitive interactions.

Self-assembled trityl radical capsules – implications for dynamic nuclear polarization

The OX63 radical forms supramolecular capsules hosting tetraalkylammonium cations. Extensive self-association is also observed under standard dynamic nuclear polarization conditions.

Correction: Magnetic relaxation versus 3D long-range ordering in {Dy2Ba(α-fur)8}n furoate polymers

Correction for ‘Magnetic relaxation versus 3D long-range ordering in {Dy₂Ba(α-fur)₈}_n furoate polymers’ by E. Bartolomé et al., Dalton Trans., 2014, 43, 10999–11013.

Heterodimetallic [LnLn'] lanthanide complexes: toward a chemical design of two-qubit molecular spin quantum gates

A major challenge for realizing quantum computation is finding suitable systems to embody quantum bits (qubits) and quantum gates (qugates) in a robust and scalable architecture. An emerging bottom-up approach uses the electronic spins of lanthanides. Universal qugates may then be engineered by arranging in a molecule two interacting and different lanthanide ions. Preparing heterometallic lanthanide species is, however, extremely challenging. We have discovered a method to obtain [LnLn′] complexes with the appropriate requirements. Compound [CeEr] is deemed to represent an ideal situation. Both ions have a doubly degenerate magnetic ground state and can be addressed individually. Their isotopes have mainly zero nuclear spin, which enhances the electronic spin coherence. The analogues [Ce₂], [Er₂], [CeY], and [LaEr] have also been prepared to assist in showing that [CeEr] meets the qugate requirements, as revealed through magnetic susceptibility, specific heat, and EPR. Molecules could now be used for quantum information processing.

{Dy(α-fur)3}n: from double relaxation single-ion magnet behavior to 3D ordering

The synthesis and magnetostructural properties of a new low-dimensional magnetic system based on α-furoate ligands, {[Dy(α-C4H3OCOO)(μ-(α-C4H3OCOO))2(H2O)3]}n, abbreviated {Dy(α-fur)3}n, are reported. X-ray diffraction experiments results evidence the presence of two different Dy coordination environment types differing only in the position of one of the furoate ligands. The crystallographic structure is formed by polymeric chains along the c-axis, each composed of just one Dy type, coupled within the bc-plane with chains of the same Dy type. These planes, each of them containing only one Dy type, are randomly stacked along the a-axis. The magnetic behaviour was studied by magnetization, static and dynamic susceptibility, heat capacity measurements and ab initio simulations. The directions of the easy axes of magnetization, gyromagnetic values and energy level structures of the two Dy types were obtained from ab initio calculations. {Dy(α-fur)3}n exhibits slow magnetic relaxation dynamics below 10 K. The two Dy types with different coordination environments behave as single-ion magnets, with different thermal activation energies of 80.5(6) K and 32.4(5) K, until they reach, upon cooling, a quantum tunneling (QT) regime. Magnetic diluted samples, substituting Dy by Y, {Y(x)Dy(1-x)(α-fur)3}n, were prepared to study the effect of intercluster interactions. Decreasing the Dy interaction by dilution by 90-95% leaves the activation energy unchanged, but shifts the transition to the QT regime to lower temperatures. At T = 2.4 K the tunneling time constant has been shown to decrease weakly with the field in the x = 0 case, and more strongly for x = 0.9. As the external field increases, quantum tunneling is quenched and a new slow relaxation appears that is identified at high fields as caused by a direct relaxation process. As the temperature is decreased, interchain AF coupling becomes effective and gives rise to the occurrence of an antiferromagnetic 3D order transition at T(N) = 0.66 K. From all the evidence, it is concluded that within each bc-plane Dy ions arrange in chains along the c-direction, having weak uncompensated ferromagnetic spin-canted intrachain coupling and antiferromagnetic interchain coupling.

Mn12 single molecule magnets deposited on μ-SQUID sensors: the role of interphases and structural modifications

Direct measurements of the linear ac susceptibility and magnetic relaxation of a few Mn12 monolayers deposited on a μ-SQUID sensor are reported. In order to integrate the molecules into the device, DPN has been the technique of choice. It enabled the structuration of the molecules on the most sensitive areas of the sensor without the need for any previous functionalization of the molecule or the substrate, while controlling the number of molecular units deposited on each array. The measurements reveal that their characteristic SMM behaviour is lost, a fact that is attributed to molecular distortions originated by the strong surface tensions arising at the molecular interphases.

Quantum interference oscillations of the superparamagnetic blocking in an Fe8 molecular nanomagnet

We show that the dynamic magnetic susceptibility and the superparamagnetic blocking temperature of an Fe(8) single molecule magnet oscillate as a function of the magnetic field H(x) applied along its hard magnetic axis. These oscillations are associated with quantum interferences, tuned by H(x), between different spin tunneling paths linking two excited magnetic states. The oscillation period is determined by the quantum mixing between the ground S=10 and excited multiplets. These experiments enable us to quantify such mixing. We find that the weight of excited multiplets in the magnetic ground state of Fe(8) amounts to approximately 11.6%.

Strong paramagnetism of gold nanoparticles deposited on a Sulfolobus acidocaldarius S layer

Magnetic properties of Au nanoparticles deposited on an archaeal S layer are reported. X-ray magnetic circular dichroism and superconducting quantum interference device magnetometries demonstrate that the particles are strongly paramagnetic, without any indication of magnetic blocking down to 16 mK. The average magnetic moment per particle is M(part)=2.36(7) μ(B). This contribution originates at the particle's Au 5d band, in which an increased number of holes with respect to the bulk value is observed. The magnetic moment per Au atom is 25 times larger than any measured in other Au nanoparticles or any other configurations up to date.

Fragmenting gadolinium: mononuclear polyoxometalate-based magnetic coolers for ultra-low temperatures

The polyoxometalate clusters with formula [Gd(W(5) O(18) )(2) ](9-) and [Gd(P(5) W(30) O(110) )](12-) each carry a single magnetic ion of gadolinium, which is the most widespread element among magnetic refrigerant materials. In an adiabatic demagnetization, the lowest attainable temperature is limited by the presence of magnetic interactions that bring about magnetic order below a critical temperature. We demonstrate that this limitation can be overcome by chemically engineering the molecules in such a way to effectively screen all magnetic interactions, suggesting their use as ultra-low-temperature coolers.

Gd-based single-ion magnets with tunable magnetic anisotropy: molecular design of spin qubits

We report ac susceptibility and continuous wave and pulsed EPR experiments performed on GdW10 and GdW30 polyoxometalate clusters, in which a Gd3+ ion is coordinated to different polyoxometalate moieties. Despite the isotropic character of gadolinium as a free ion, these molecules show slow magnetic relaxation at very low temperatures, characteristic of single molecule magnets. For T≲200  mK, the spin-lattice relaxation becomes dominated by pure quantum tunneling events, with rates that agree quantitatively with those predicted by the Prokof'ev and Stamp model [Phys. Rev. Lett. 80, 5794 (1998)]. The sign of the magnetic anisotropy, the energy level splittings, and the tunneling rates strongly depend on the molecular structure. We argue that GdW30 molecules are also promising spin qubits with a coherence figure of merit Q(M)≳50.

Molecular prototypes for spin-based CNOT and SWAP quantum gates

We show that a chemically engineered structural asymmetry in [Tb2] molecular clusters renders the two weakly coupled Tb3+ spin qubits magnetically inequivalent. The magnetic energy level spectrum of these molecules meets then all conditions needed to realize a universal CNOT quantum gate. A proposal to realize a SWAP gate within the same molecule is also discussed. Electronic paramagnetic resonance experiments confirm that CNOT and SWAP transitions are not forbidden.

Magnetic dipolar ordering and quantum phase transition in an Fe8 molecular magnet

We show that a crystal of mesoscopic Fe(8) single-molecule magnets is an experimental realization of the quantum Ising model in a transverse field, with dipolar interactions. Quantum annealing has enabled us to explore the quantum and classical phase transitions between the paramagnetic and ferromagnetic phases, at thermodynamical equilibrium. The phase diagram and critical exponents that we obtain agree with expectations for the mean-field universality class.

Design of magnetic coordination complexes for quantum computing

A very exciting prospect in coordination chemistry is to manipulate spins within magnetic complexes for the realization of quantum logic operations. An introduction to the requirements for a paramagnetic molecule to act as a 2-qubit quantum gate is provided in this tutorial review. We propose synthetic methods aimed at accessing such type of functional molecules, based on ligand design and inorganic synthesis. Two strategies are presented: (i) the first consists in targeting molecules containing a pair of well-defined and weakly coupled paramagnetic metal aggregates, each acting as a carrier of one potential qubit, (ii) the second is the design of dinuclear complexes of anisotropic metal ions, exhibiting dissimilar environments and feeble magnetic coupling. The first systems obtained from this synthetic program are presented here and their properties are discussed.

Size-dependent properties of magnetoferritin

We report a detailed experimental study of maghemite nanoparticles, with sizes ranging from 1.6 to 6 nm, synthesized inside a biological mould of apoferritin. The structural characterization of the inorganic cores, using TEM and x-ray diffraction, reveals a low degree of crystalline order, possibly arising from the nucleation and growth of multiple domains inside each molecule. We have also investigated the molecular structure by means of atomic force microscopy in liquid. We find that the synthesis of nanoparticles inside apoferritin leads to a small, but measurable, decrease in the external diameter of the protein, probably associated with conformational changes. The magnetic response of the maghemite cores has been studied by a combination of techniques, including ac susceptibility, dc magnetization and Mössbauer spectroscopy. From the equilibrium magnetic response, we have determined the distribution of magnetic moments per molecule. The results show highly reduced magnetic moments. This effect cannot be ascribed solely to the canting of spins located at the particle surface but, instead, it suggests that magnetoferritin cores have a highly disordered magnetic structure in which the contributions of different domains compensate each other. Finally, we have also determined, for each sample, the distribution of the activation energies required for the magnetization reversal and, from this, the size-dependent magnetic anisotropy constant K. We find that K is enormously enhanced with respect to the maghemite bulk value and that it increases with decreasing size. The Mössbauer spectra suggest that low-symmetry atomic sites, probably located at the particle surface and at the interfaces between different crystalline domains, are the likely source of the enhanced magnetic anisotropy.

Note: Highly sensitive superconducting quantum interference device microsusceptometers operating at high frequencies and very low temperatures inside the mixing chamber of a dilution refrigerator

We report the experimental results that show the operation of superconducting quantum interference device (SQUID) microsusceptometers immersed in the (3)He-(4)He mixture inside the mixing chamber of a dilution refrigerator at high frequency (1 MHz) and down to very low temperatures (13 mK). The devices are based on highly sensitive and easy-to-use commercial SQUID sensors. The integrated susceptometers are fabricated by rerouting some connections of the SQUID's input circuit. Examples of measurements on molecular magnets Mn(12) and HoW(10) are shown.

Equilibrium susceptibilities of superparamagnets: longitudinal and transverse, quantum and classical

The equilibrium susceptibility of uniaxial paramagnets is studied in a unified framework which permits us to connect traditional results of the theory of quantum paramagnets, S = 1/2,1,3/2,..., with molecular magnetic clusters, S∼5,10,20 all the way up (S = 30,50,100,...,) to the theory of classical superparamagnets. This is done using standard tools of quantum statistical mechanics and linear-response theory (the Kubo correlator formalism). Several features of the temperature dependence of the susceptibility curves (crossovers, peaks, deviations from Curie law) are studied and their scalings with S identified and characterized. Both the longitudinal and transverse susceptibilities are discussed, as well as the response of the ensemble with anisotropy axes oriented at random. For the latter case a simple approximate formula is derived too, and its range of validity assessed, which could be used in the modelization of experiments.