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 coupled1 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 possible2. 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 predicted3 gapless Goldstone mode characteristic of the Bose–Einstein condensation of the triplet states.
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We thank M. Matsumoto, B. Normand, T. M. Rice, M. Sigrist, M. Müller and H.-J. Mikeska for discussions, M. Meissner, S. Kausche and P. Smeibidl for assistance during the V2 measurements, and F. Thomas for assistance during the IN14 measurements. This work was supported partially by the Swiss National Science Foundation.
The authors declare that they have no competing financial interests.
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Rüegg, C., Cavadini, N., Furrer, A. et al. Bose–Einstein condensation of the triplet states in the magnetic insulator TlCuCl3. Nature 423, 62–65 (2003). https://doi.org/10.1038/nature01617
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