In the last decades, there has been an increasing focus on low-dimensional systems, in particular one-dimensional (1D) or quasi-one dimensional magnetic systems where the magnetic ions interact preferentially along a single direction in space. This interest is justified by the fact that the physics in one dimension leads to a very rich physics as the quantum effects are enhanced by the low-dimensionality. Moreover these systems can be described by models that have the particularity to be integrable, i.e. that are easily solvable analytically. Let us cite for instance the famous Bethe ansatz giving the exact solution of an antiferromagnetic (AF) 1D Heisenberg spin-1/2 chain, which was the first exact solution of a many-body quantum system. Therefore 1D magnetic systems allow a direct comparison with theoretical work. I will show in this presentation the study of BaCo2V2O8, a quasi-1D Ising-like antiferromagnet showing many quantum features . The first part will be devoted to the study of this system under a longitudinal magnetic field, i.e. parallel to the easy-axis of anisotropy. By combining neutron scattering experiments and numerical techniques, we have shown that the spin-dynamics above the quantum phase transition occurring at 4 T corresponds to the one expected by the Tomonoga Luttinger liquid model. Second, I will show our study of BaCo2V2O8 under a transverse magnetic field, i.e. perpendicular to the easy-axis of anisotropy. While the 1D AF Ising model is a paradigm of quantum phase transition [2,3], the transition occurring in BaCo2V2O8 has been found to be topologic in nature. Indeed, by using inelastic neutron scattering experiments combined with theoretical calculations, we have shown that this transition is well described by the so-called dual-field double sine-Gordon model, reflecting the competition of two dual topological objects .
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