Bose-Einstein condensation denotes the formation of a collective quantum ground state of identical particles with integer spin or intrinsic angular momentum. In magnetic insulators, the magnetic properties are due to the unpaired shell electrons that have half-integer spin. However, in some such compounds (KCuCl3 and TlCuCl3), two Cu2+ ions are antiferromagnetically coupled to form a dimer in a crystalline network: the dimer ground state is a spin singlet (total spin zero), separated by an energy gap from the excited triplet state (total spin one). In these dimer compounds, Bose-Einstein condensation becomes theoretically possible. At a critical external magnetic field, the energy of one of the Zeeman split triplet components (a type of boson) intersects the ground-state singlet, resulting in long-range magnetic order; this transition represents a quantum critical point at which Bose-Einstein condensation occurs. Here we report an experimental investigation of the excitation spectrum in such a field-induced magnetically ordered state, using inelastic neutron scattering measurements of TlCuCl3 single crystals. We verify unambiguously the theoretically predicted gapless Goldstone mode characteristic of the Bose-Einstein condensation of the triplet states.

We follow the evolution of the elementary excitations of the quantum antiferromagnet TlCuCl3 through the pressure-induced quantum critical point, which separates a dimer-based quantum disordered phase from a phase of long-ranged magnetic order. We demonstrate by neutron spectroscopy the continuous emergence in the weakly ordered state of a low-lying but massive excitation corresponding to longitudinal fluctuations of the magnetic moment. This mode is not present in a classical description of ordered magnets, but is a direct consequence of the quantum critical point.Although quantum fluctuations of both spin and charge degrees of freedom are the key to the essential physics of many challenging problems in condensed matter systems, the microscopic control of zero-point fluctuations has to date remained largely a theoretical abstraction. However, full control over the interaction parameters can now be effected in cold atomic condensates through the standing-wave amplitudes of the optical lattice. Similarly, in quantum magnets the exchange interactions can be controlled by the application of pressure, altering the effect of spin fluctuations. We follow this approach to investigate the physics of a quantum system whose fluctuations are "tuned" in a continuous way.The most dramatic manifestation of such control is the driving of a quantum phase transition [QPT, Fig. 1(a)] between two different ground states [1]. Structurally dimerized S = 1/2 spin systems offer a particularly clean realization both of the magnetic field-induced QPT, which has been studied extensively in a number of materials [2], and of the qualitatively different magnetic QPT driven by hydrostatic pressure [3]. The Hamiltonian

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KCuCl3 is a three-dimensional coupled spin-dimer system and has a singlet ground state with an excitation gap ∆/kB = 31 K. High-field magnetization measurements for KCuCl3 have been performed in static magnetic fields of up to 30 T and in pulsed magnetic fields of up to 60 T. The entire magnetization curve including the saturation region was obtained at T = 1.3 K. From the analysis of the mag-netization curve, it was found that the exchange parameters determined from the dispersion relations of the magnetic ex-citations should be reduced, which suggests the importance of the renormalization effect in the magnetic excitations. The field-induced magnetic ordering accompanied by the cusplike minimum of the magnetization was observed as in the iso-morphous compound TlCuCl3. The phase boundary was almost independent of the field direction, and is represented by the power law. These results are consistent with the magnon Bose-Einstein condensation picture for field-induced magnetic ordering. PACS number 75.10.Jm

SwitzerlandInelastic neutron scattering experiments have been performed on binuclear Cr 3 + clusters and trinuclear Fe 3+ clusters embedded in molecular complexes. The intensities of transitions within the exchange-split ground state observed at various transfers exhibit an oscillatory behavior due to interference effects which are a direct picture of the metal-metal distances in the cluster. From these measurements, precise magnetic-formfactor data of Cr 3+ and Fe 3+ have been obtained.Recently inelastic neutron scattering has been introduced for studying exchange interactions in isolated clusters of magnetic ions. 1 ' 2 In contrast to the classical methods (measurements of the bulk properties, magnetic resonance experiments), the neutron scattering technique yields direct information about the exchange interactions since the energies of transitions between the exchange-split ground-state levels can be directly determined. Neutron scattering is also superior to optical techniques, because optical spectroscopic data cannot, in most cases, be collected over a wide temperature range because of broadening and partial quenching of the emission at higher temperatures; moreover, the wavevector dependence of the intensities in the neutron experiment yields direct information about the form factor of the unpaired electrons and about the geometrical configuration of the magnetic ions in the cluster. The latter gives rise to a characteristic interference term in the neutron cross-section formula.In this Letter, we present the observation of interference effects in neutron scattering from polynuclear clusters of transition metals embedded in molecular complexes in which the interacting magnetic ions are effectively in S states, i.e., the ligand fields are sufficiently strong to quench the orbital angular momentum. Under these circumstances the neutron cross-section formula can easily be derived, 3 and no assumptions have to be made concerning the wave-vector dependence of the scattering. Thus the magnetic form factor can be determined directly from experiment, in contrast to neutron diffraction and inelastic neutron scattering from spin waves in magnetically ordered systems in which experi-ments the evaluation of the form factor is based on assumptions concerning both the magnetic structure and the ordered magnetic moment, and cooperative phenomena often create uncertainties in the interpretation of spin-wave data. These disadvantages do not exist in inelastic neutron scattering experiments performed on magnetic clusters in which well-defined transitions between the exchange-split ground-state levels are observed. As a consequence, rather precise form factor data can be obtained up to A^sinfl = 0.4, which is demonstrated in the present work.Experiments were performed on polycrystalline samples of deuterated acid rhodo chromium chloride, [(NH 3 ) 5 Cr(OH)Cr(NH 3 ) 5 ]Cl 5 -H 2 0, and deuterated a-metavoltine, K 10 [Fe 3 O(SO 4 ) 6 (H 2 O) 3 ] 2 • 12H 2 0, which contain binuclear Cr 3+ clusters and trinuclear Fe 3 * clusters,...

The condensation of magnetic quasiparticles into the nonmagnetic ground state has been used to explain novel magnetic ordering phenomena observed in quantum spin systems. We present neutron scattering results across the pressure-induced quantum phase transition and for the novel ordered phase of the magnetic insulator TlCuCl3, which are consistent with the theoretically predicted two degenerate gapless Goldstone modes, similar to the low-energy spin excitations in the field-induced case. These novel experimental findings complete the field-induced Bose-Einstein condensate picture and support the recently proposed field-pressure phase diagram common for quantum spin systems with an energy gap of singlet-triplet nature.

KCuCl3 is an S = 1/2 magnetic insulator with a singlet ground state and a finite spin excitation gap. Above the gap, dispersive triplet excitation modes propagate in the whole reciprocal space. From single-crystal inelastic neutron investigations the three-dimensional coupling scheme is rationalized in the framework of a dimer Heisenberg model, and related to the structural features of KCuCl3. The experimental and theoretical characterization presented completes earlier works on the compound under investigation, providing also higher-order expressions for the singlet-triplet dispersion relation. The latter may also be of relevance for the parent quantum systems TlCuCl3 and NH4CuCl3, albeit at different coupling ratios with respect to KCuCl3.

Abstract:Inelastic neutron scattering experiments were performed for the spin-trimer compounds Ca 3 Cu 3-x Ni x (PO 4 ) 4 (x=0,1,2) in order to study the dynamic magnetic do not exhibit long-range magnetic ordering down to 1 K, the x=2 compound shows antiferromagnetic ordering below T N =20 K, which is compatible with the molecular-field parameter λ=(0.63±0.12) meV derived by neutron spectroscopy.