1. Laboratory for Neutron Scattering, ETH Zürich & Paul Scherrer Institut, 5232 Villigen PSI, Switzerland
2. Department of Chemistry and Biochemistry, Universität Bern, 3000 Bern 9, Switzerland
3. Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, France
4. BENSC, Hahn-Meitner-Institut, 14109 Berlin Wannsee, Germany

Correspondence to: Ch. Rüegg¹ Correspondence and requests for materials should be addressed to C.R. (Email: christian.rueegg@psi.ch).

Abstract

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.