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Quantum fluctuations of local quantities can be a direct signature ofentanglement in an extended quantum many-body system. Hence they may serve as atheoretical (as well as an experimental) tool to detect the spatial propertiesof the entanglement entropy of a subsystem. In the ground state of quantummany-body systems, its scaling is typically linear in the boundary of thesubsystem (area law). Here we propose a microscopic insight into the spatialstructure of entanglement and fluctuations using the concept of \emph{contour},recently introduced to decompose the bipartite entanglement entropy of latticefree fermions between two extended regions $A$ and $B$ into contributions fromsingle sites in $A$. We generalize the notion of contour to the entanglement ofany quadratic (bosonic or fermionic) lattice Hamiltonian, as well as toparticle-number fluctuations. The entanglement and fluctuations contours arefound 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 entanglementcontour follows closely that of the fluctuation contour. In the case ofBose-condensed interacting bosons, treated via the Bogoliubov and spin-waveapproximations, such a link cannot be established -- fluctuation andentanglement contours are found to be radically different, as they lead to alogarithmically violated area law for fluctuations, and to a strict area law ofentanglement. Analyzing in depth the role of the zero-energy Goldstone mode ofspin-wave theory, and of the corresponding lowest-energy mode in theentanglement spectrum, we unveil a subtle interplay between the special contourand energy scaling of the latter, and universal additive logarithmiccorrections to entanglement area law discussed extensively in the recentliterature.

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