Shintaro Takayoshi 1, Quentin Faure 2, Virginie Simonet 3, Béatrice Grenier 4, Sylvain Petit 2, Jacques Ollivier 5, Pascal Lejay 3, and Thierry Giamarchi6
1 Department of Physics, Konan University, 658-8501 Kobe, Japan
2 Laboratoire Léon Brillouin, CEA, CNRS, Université Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France
3 Université Grenoble Alpes, CNRS, Institut Néel, 38042 Grenoble, France
4 Université Grenoble Alpes, CEA, IRIG/DEPHY/MEM/MDN, 38000 Grenoble, France
5 Institut Laue Langevin, 38042 Grenoble, France
6 Department of Quantum Matter Physics, University of Geneva, 1211 Geneva, Switzerland
(Quasi-)one-dimensional (1D) magnetic systems, where magnetic ions interact preferentially along one direction in space, are model systems to study the influence of quantum effects enhanced by low dimensionality and small spin values. Moreover, these systems can be described by models that have the particularity of being integrable, i. e. exactly analytically soluble. Among all the quantum manifestations of matter, quantum phase transitions – corresponding to an abrupt change in the ground state of a system at zero temperature due to quantum fluctuations when an external parameter such as a magnetic field or pressure is applied – are particularly exacerbated in 1D magnetic systems.
Using inelastic neutron scattering and numerical simulations, we have studied the anisotropic antiferromagnetic chain compound BaCo2V2O8 under longitudinal magnetic field, i. e. under a field applied along the axis of Ising-type anisotropy. Probing the evolution of magnetic excitations under field conditions, we observed that the magnetic field closes the energy gap caused by the anisotropy, inducing a transition from the ordered antiferromagnetic Néel state to an exotic incommensurate longitudinal spin density wave. This state is understood in the Tomonaga-Luttinger liquid (TLL) theory, describing interacting fermions in one dimension. If the field is increased further, another transition to a transverse antiferromagnetic phase takes place at 9 T, due to competition between correlations along and perpendicular to the field. Strikingly, although this last 3-dimensional ordered state is rather standard (staggered magnetic moments), the associated dynamics still reflects the underlying TLL one-dimensional incommensurate spin excitations. By combining experiments and numerics, we have shown that a model of anisotropic spin chains coupled through a weak interchain interaction reproduces these successive quantum phase transitions.
(a) Longitudinal magnetic field vs. temperature phase diagram for BaCo2V2O8. Below 19 T, three phases are visible: (b) a « longitudinal » antiferromagnetic Néel phase, i.e. with moments along the axis of anisotropy, here along the chain direction; (c) an incommensurable longitudinal spin density wave; (d) a « transverse » antiferromagnetic Néel phase, i.e. with magnetic moments perpendicular to the anisotropy axis. (e) Spectrum showing the dispersion of magnetic excitations along the chain axis obtained by inelastic neutron scattering measurements on IN5 at ILL at T = 50 mK and under a magnetic field of 10 T (in the transverse Néel antiferromagnetic phase). (f) Numerical calculation by iTEBD (infinite Time Evolve Block Decimation) reproducing the experimental data by a model of weakly coupled XXZ chains under longitudinal field.
 Q. Faure, S. Takayoshi, V. Simonet, B. Grenier, M. Mansson, J. White, G. S. Tucker, C. Rüegg, P. Lejay, T. Giamarchi, and S. Petit, « Tomonaga-Luttinger liquid spin dynamics in the quasi-one-dimensional Ising-like antiferromagnetic BaCo2V2O8 », Phys. Rev. Lett. 123, 027204 (2019).
 B. Grenier, V. Simonet, B. Canals, P. Lejay, M. Klanjsek, M. Horvatik, C. Berthier, « Determination of the high field magnetic phase in the Ising-like chain antiferromagnet BaCo2V2O8 », Phys. Rev. B 92, 134416 (2015).
 B. Grenier, S. Petit, V. Simonet, L.-P. Regnault, E. Canévet, S. Raymond, B. Canals, C. Berthier, P. Lejay, « Longitudinal and transverse Zeeman ladders in the Ising-like chain antiferromagnet BaCo2V2O8 », Phys. Rev. Lett 114, 017201 (2015) ; Phys. Rev. Lett 115, 119902 (2015).