Multipartite entanglement in quantum phase transitions

Quantum phase transitions are collective phenomena that take place in large interacting many-body systems when tuning a nonthermal control parameter across a critical value. They occur at zero temperature, driven by genuinely quantum fluctuations, and manifest as a qualitative change in the structure (symmetry or topology) of the ground state due to the competition between incompatible terms in the many-body Hamiltonian describing the system. A rich variety of signatures flags the presence of the quantum critical point, ranging from the singular behaviour in some properties of the low-lying spectrum to the persistence of quantum features even at finite temperature. The latter footprints permits to observe the effects of quantum phase transitions experimentally. In particular, the low-temperature region on the phase diagram where the critical point strongly affects the thermal behaviour of the system has been intensively studied in recent years but many issues about its nature and extension are still open. Generally speaking, the investigation of critical systems from the perspective of information science advances our understanding of criticality beyond standard approaches developed in statistical mechanics. Moreover, it sheds new light on the preparation and processing of useful resources for quantum technologies. In fact, the considerable amount of knowledge accumulated in the study of quantum systems during the last decades led to a fundamental as well as technological revolution: entanglement -- the quintessence of quantum world -- has turned from a troublesome source of paradoxes into a useful resource. It allows to outperform classical tasks in information processing or even accomplish tasks that cannot be carried out through classical means. This change of perspective justifies the growing attention entanglement has been attracting. The characterization of quantum phases and quantum phase transitions through entanglement measures and witnesses is an intriguing problem at the verge of quantum information and many-body physics. Current studies have mainly focused on bipartite or pairwise entanglement in the ground state of critical Hamiltonians: these studies have emphasized a growth of entanglement in the vicinity of quantum critical points. However, bipartite and pairwise correlations are hardly accessible in systems with a large number of particles, that are the preferred platforms for quantum sensors and the natural targets of quantum simulators. Moreover, they cannot fully capture the richness of multiparticle correlations and the complex structure of a many-body quantum state. Much less attention has been devoted to witnessing multipartite entanglement in critical systems and it has been mainly limited to spin models. Yet, multipartite entanglement among hundreds of particles has been detected experimentally in atomic ensembles so far, and a variety of witnesses are available in the literature. Among these witnesses, the quantum Fisher information has proved to be especially powerful: it extends the class of entangled states detectable by popular methods such as the spin squeezing, it can be extracted from experimental data and it has an appealing physical meaning in terms of distinguishability of quantum states under external parametric transformations. In our work, we merge the two concepts -- somehow abstract though experimentally accessible -- of multipartite entanglement and quantum phase transitions. We investigate the behaviour of multipartite entanglement as detected and quantified by the quantum Fisher information, both at zero and finite temperature, in a collection of benchmark models displaying quantum critical points. All the selected models describe exemplary systems in the field of condensed-matter or nuclear physics and can be realized and tested in the laboratory as well. We find that the multipartite entanglement at zero temperature is a reliable detector of criticality: its enhanced susceptibility to any minute change of the parameter driving the transition sharply marks the boundary between different phases, not only in symmetry-breaking first-order and second-order transitions, but also in topological transitions. Moreover, multipartite entanglement distinguishes between phases with different symmetries or different topologies in terms of different scaling with the system size. When witnessing multipartite entanglement in topological systems, where no local order parameter exists, we address an open problem in the literature: the extension of the standard protocol based on the quantum Fisher information to nonlocal probe operators. The study of the interplay between thermal and quantum fluctuations in the vicinity of quantum critical points reveals the existence of a universal decay law of multipartite entanglement at sufficiently low temperature. Thermal fluctuations and noise pose a significant threat to the preparation, protection and usage of highly-entangled states and constitute a big challenge for future quantum technologies. Our work offers insights on the robustness of quantum correlations against decoherence, especially for applications in the field of quantum metrology.