V. V. França

Entanglement in Spatially Inhomogeneous Many-Fermion Systems

We investigate entanglement of strongly interacting fermions in spatially inhomogeneous environments. To quantify entanglement in the presence of spatial inhomogeneity, we propose a local-density approximation (LDA) to the entanglement entropy, and a nested LDA scheme to evaluate the entanglement entropy on inhomogeneous density profiles. These ideas are applied to models of electrons in superlattice structures with different modulation patterns, electrons in a metallic wire in the presence of impurities, and phase-separated states in harmonically confined many-fermion systems, such as electrons in quantum dots and atoms in optical traps. We find that the entanglement entropy of inhomogeneous systems is strikingly different from that of homogeneous systems.

Simple parameterization for the ground-state energy of the infinite Hubbard chain incorporating Mott physics, spin-dependent phenomena and spatial inhomogeneity

Simple analytical parameterizations for the ground-state energy of the one-dimensional repulsive Hubbard model are developed. The charge dependence of energy is parameterized using exact results extracted from the Bethe-ansatz (BA). The resulting parameterization is shown to be in better agreement with highly precise data obtained from a fully numerical solution to the BA equations than previous expressions (Lima et al 2003 Phys. Rev. Lett. 90 146402). Unlike these earlier proposals, the present parameterization correctly predicts a positive Mott gap at half filling for any Uxa0>xa00. The construction is extended to spin-dependent phenomena by parameterizing the magnetization dependence of the ground-state energy using further exact results and numerical benchmarking. Lastly, the parameterizations developed for the spatially uniform model are extended by means of a simple local-density-type approximation to spatially inhomogeneous models, e.g. in the presence of impurities, external fields or trapping potentials. The results are shown to be in excellent agreement with independent many-body calculations, at a fraction of the computational cost.

Quantum Mechanics in Metric Space: Wave Functions and Their Densities

Hilbert space combines the properties of two different types of mathematical spaces: vector space and metric space. While the vector-space aspects are widely used, the metric-space aspects are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground states and to that of particle densities, which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground states, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities.

Fulde-Ferrell-Larkin-Ovchinnikov critical polarization in one-dimensional fermionic optical lattices

We deduce an expression for the critical polarization

Hubbard model as an approximation to the entanglement in nanostructures

{P}_{C}

Entanglement of strongly interacting low-dimensional fermions in metallic, superfluid, and antiferromagnetic insulating systems

below which the Fulde-Ferrell-Larkin-Ovchinnikov state emerges in one-dimensional lattices with spin-imbalanced populations. We provide and explore the phase diagram of unconfined chains as a function of polarization, interaction, and particle density. For harmonically confined systems, we supply a quantitative mapping, which also allows applying our phase diagram for confined chains. We find analytically and confirm numerically that the upper bound for the critical polarization is universal:

Uniqueness of density-to-potential mapping for fermionic lattice systems

{P}_{C}^{mathrm{max}}=1/3

Feasibility of approximating spatial and local entanglement in long-range interacting systems using the extended Hubbard model

for any density, interaction, and confinement strength.

Testing density-functional approximations on a lattice and the applicability of the related Hohenberg-Kohn-like theorem

We investigate how well the one-dimensional Hubbard model describes the entanglement of particles trapped in a string of quantum wells. We calculate the average single-site entanglement for two particles interacting via a contact interaction and consider the effect of varying the interaction strength and the interwell distance. We compare the results with the ones obtained within the one-dimensional Hubbard model with on-site interaction. We suggest an upper bound for the average single-site entanglement for two electrons in M wells and discuss analytical limits for very large repulsive and attractive interactions. We investigate how the interplay between interaction and potential shape in the quantum-well system dictates the position and size of the entanglement maxima and the agreement with the theoretical limits. Finally, we calculate the spatial entanglement for the quantum-well system and compare it to its average single-site entanglement.

Entanglement and exotic superfluidity in spin-imbalanced lattices

We calculate the entanglement entropy of strongly correlated low-dimensional fermions in metallic, superfluid and antiferromagnetic insulating phases. The entanglement entropy reflects the degrees of freedom available in each phase for storing and processing information, but is found not to be a state function in the thermodynamic sense. The role of critical points, smooth crossovers and Hilbert space restrictions in shaping the dependence of the entanglement entropy on the system parameters is illustrated for metallic, insulating and superfluid systems. The dependence of the spin susceptibility on entanglement in antiferromagnetic insulators is obtained quantitatively. The opening of spin gaps in antiferromagnetic insulators is associated with enhanced entanglement near quantum critical points.