Area law and its violation: A microscopic inspection into the structure of entanglement and fluctuations

Quantum fluctuations of local quantities can be a direct signature of entanglement in an extended quantum many-body system. Hence they may serve as a theoretical (as well as an experimental) tool to detect the spatial properties of the entanglement entropy of a subsystem-more specifically, its scaling with the size of the subsystem itself. In the ground state of quantum many-body systems, this scaling is typically linear in the boundary of the subsystem (area law), with at most multiplicative logarithmic corrections. Here we propose a microscopic insight into the spatial structure of entanglement and particle-number fluctuations using the concept of contour, recently introduced to decompose the bipartite entanglement entropy of lattice free fermions between two extended regions A and B into contributions from single sites in A [ 20]. We generalize the notion of contour to the entanglement of any quadratic (bosonic or fermionic) lattice Hamiltonian, as well as to particle-number fluctuations. The entanglement and fluctuations contours are found to generally decay when moving away from the boundary between A and B. We show that in the case of free fermions the decay of the entanglement contour follows closely that of the fluctuation contour: this establishes a microscopic link between the scaling of entanglement and that of particle-number fluctuations, and it allows us to predict the presence (or violation) of entanglement area laws solely based on the density-density correlation function. In the case of Bose-condensed interacting bosons, treated via the Bogoliubov and spin-wave approximations, such a link cannot be established-fluctuation and entanglement contours are found to be radically different, as they lead to a logarithmically violated area law for particle-number fluctuations, and to a strict area law of entanglement. Analyzing in depth the role of the zero-energy Goldstone mode of spin-wave theory, and of the corresponding lowest-energy mode in the entanglement spectrum, we unveil a subtle interplay between the special contour and energy scaling of the latter, and universal additive logarithmic corrections to entanglement area law discussed extensively in the recent literature.