• entanglement
• statistical physics
• correlated electrons
• condensed matter

Entanglement is a key aspect of quantum mechanics, and arguably the clearest manifestation of the non-locality that is so intimately written within the quantum theory.
The wave functions used to describe matter are, in fact, spatially extended; only when the position of a particle is measured the wave function collapses and the local particle description of matter can be used.
The counter intuitive nature of entanglement has intrigued physicists since the very beginning of the quantum era, and sparked controversies and paradoxes. Among these, we cite the famous paradox proposed by A.Einstein, B.Podolsky and N.Rosen 1935, in which a pair of entangled particles deceptively give the impression that information can be transmitted faster than the speed of light.

While in the past the concept of quantum entanglement was primarily applied to systems with few degrees of freedom, this has changed in the modern days. The developments of the theories of quantum decoherence, condensed matter and quantum information, together with the huge improvements in the experimental techniques of ultra-cold atoms, nano technologies and quantum optics, have fuelled a new interest on the entanglement of many-body systems.

An extensive literature has built up on the problem of measuring the entanglement of a part of an extended system with the rest. The interest in it was motivated, among other reasons, by the ideas of Bekenstein-Hawking black hole entropy and the holographic principle first proposed by 't Hooft. The development of measures such as the von Neumann and Rényi bipartite entanglement entropies are great theoretical achievements in these respects.

In thermodynamics, entropy is an extensive quantity, and hence scales with the volume of the system, i.e. $S_{therm}\sim L^d$, with $L$ the typical dimension of a system and $d$ its spatial dimension.
Entanglement entropies are somewhat peculiar in that they typically scale as the extension of the hyper-surface separating the two parts of the system one considers, i.e. $S_{ent}\sim L^{d-1}$. This is commonly referred in literature as the area law scaling of entanglement entropies. An extremely important application of area laws is their use as flags for quantum criticality: in fact, under reasonable assumptions, the scaling is logarithmically violated at quantum critical points, i.e. $S_{ent}\sim L^{d-1}\log(L)$.

Measuring entanglement is often not an easy task, as a quite deep understanding of the system is required.
Spin chain models are particularly useful to overcome these limitations, in that they are sufficiently complex to have real-world counterparts and at the same time sufficiently simple to allow a rigorous, albeit sometimes only approximate, analytical solution.
By exploiting the known solution to some models in low spatial dimensions, it was possible to analytically compute entanglement entropies, both for finite-size systems and in the thermodynamic limit.
This is even more the case for some one-dimensional models at quantum critical points, whenever the system is invariant under translations, rotations and scaling transformations. In these cases the system is usually invariant under the wider conformal group in 1+1 dimensions (the second dimension being the imaginary time obtained by a Wick rotation of the real time). The powerful methods of Conformal Field Theory confirmed and sometimes extended the results on the bipartite entanglement entropies of critical one-dimensional systems.
By using these methods, together with profound mathematical theorems, it was possible to also obtain the behaviour of the entanglement and the scaling of the leading corrections in dimensions higher than one.

In this thesis we review in detail the main aspects of quantum entanglement described above.
We then build upon exact results for systems of free spinless Fermi gases in a finite-size one-dimensional box.
From these, we derive analytical expressions for the entanglement entropies of free Fermi gases in higher dimensions for the particular strip-like geometry. The results are asymptotic for the limit of a large number of particles.
Contrarily to previously published results, in this work the subleading corrections are rigorously calculated and the next order corrections can be estimated.
The formulae are thoroughly verified against numerical results in two and three dimensions, and are in perfect agreement with the theoretical predictions.
Furthermore, an argument is found for the wide oscillation of entanglement entropies that are present in dimensions higher than one. These are found to be linked to the particular way in which the Fermi sea is filled as the particle number is increased. In two dimensions, it is numerically proven that the oscillation exactly correspond to the pseudo-periodicity of perfect squares.

Finally, the most recent developments in the field of entanglement entropies for Fermi gases are reviewed, with particular attention to dynamic situations with non-trivial interactions. These, together with known relations between entanglement entropies and particle densities, provide us with a complete picture that can be tested experimentally. Some known results of experiments with ultra-cold atoms in magneto-optic traps are reported and discussed.