A combined analytical and numerical study is performed of the mapping between strongly interacting fermions and weakly interacting spins, in the framework of the Hubbard, t-J and Heisenberg models. While for spatially homogeneous models in the thermo
dynamic limit the mapping is thoroughly understood, we here focus on aspects that become relevant in spatially inhomogeneous situations, such as the effect of boundaries, impurities, superlattices and interfaces. We consider parameter regimes that are relevant for traditional applications of these models, such as electrons in cuprates and manganites, and for more recent applications to atoms in optical lattices. The rate of the mapping as a function of the interaction strength is determined from the Bethe-Ansatz for infinite systems and from numerical diagonalization for finite systems. We show analytically that if translational symmetry is broken through the presence of impurities, the mapping persists and is, in a certain sense, as local as possible, provided the spin-spin interaction between two sites of the Heisenberg model is calculated from the harmonic mean of the onsite Coulomb interaction on adjacent sites of the Hubbard model. Numerical calculations corroborate these findings also in interfaces and superlattices, where analytical calculations are more complicated.
One of the standard generalized-gradient approximations (GGAs) in use in modern electronic-structure theory, PBE, and a recently proposed modification designed specifically for solids, PBEsol, are identified as particular members of a family of funct
ionals taking their parameters from different properties of homogeneous or inhomogeneous electron liquids. Three further members of this family are constructed and tested, together with the original PBE and PBEsol, for atoms, molecules and solids. We find that PBE, in spite of its popularity in solid-state physics and quantum chemistry, is not always the best performing member of the family, and that PBEsol, in spite of having been constructed specifically for solids, is not the best for solids. The performance of GGAs for finite systems is found to sensitively depend on the choice of constraints steaming from infinite systems. Guidelines both for users and for developers of density functionals emerge from this work.
A density-functional formalism for superconductivity {em and} magnetism is presented. The resulting relations unify previously derived Kohn-Sham equations for superconductors and for non-collinear magnetism. The formalism, which discriminates Cooper
pair singlets from triplets, is applied to two quantum liquids coupled by tunneling through a barrier. An exact expression is derived, relating the eigenstates and eigenvalues of the Kohn-Sham equations, unperturbed by tunneling, on one side of the barrier to the proximity-induced ordering potential on the other.
We calculate the entanglement entropy of blocks of size x embedded in a larger system of size L, by means of a combination of analytical and numerical techniques. The complete entanglement entropy in this case is a sum of three terms. One is a univer
sal x and L-dependent term, first predicted by Calabrese and Cardy, the second is a nonuniversal term arising from the thermodynamic limit, and the third is a finite size correction. We give an explicit expression for the second, nonuniversal, term for the one-dimensional Hubbard model, and numerically assess the importance of all three contributions by comparing to the entropy obtained from fully numerical diagonalization of the many-body Hamiltonian. We find that finite-size corrections are very small. The universal Calabrese-Cardy term is equally small for small blocks, but becomes larger for x>1. In all investigated situations, however, the by far dominating contribution is the nonuniversal term steming from the thermodynamic limit.
We propose an experiment to obtain the phase diagram of the fermionic Hubbard model, for any dimensionality, using cold atoms in optical lattices. It is based on measuring the total energy for a sequence of trap profiles. It combines finite-size scal
ing with an additional `finite-curvature scaling necessary to reach the homogeneous limit. We illustrate its viability in the 1D case, simulating experimental data in the Bethe-Ansatz local density approximation. Including experimental errors, the filling corresponding to the Mott transition can be determined with better than 3% accuracy.
