We consider local magnetic moments coupled to conduction electrons with on-site attraction in order to discuss the interplay between pairing and magnetic order. We probe the ground-state properties of this model on a one-dimensional lattice through pair binding energies and several correlation functions calculated by means of density-matrix renormalization group. A phase diagram is obtained ͑for fixed electron density 1/3͒ from which we infer that coexistence between magnetic order and superconductivity is robust at the expense of a continuous distortion of the magnetic arrangement of the local moments as evidenced by a strong dependence of the characteristic wave vector k ‫ء‬ on the coupling constants. This allows us to understand some trends of the coexistence, such as the influence of the rare earth on k ‫ء‬ , as observed experimentally in the borocarbides. The advent of high-temperature cuprate superconductors in the late 1980s singled out the interplay between superconductivity and magnetic order. This was followed by the discovery of robust coexistence between some degree of magnetic order and superconductivity in ternary and quaternary rare-earth compounds, 1-3 as well as in heavy fermion matter.4,5 Very recently, a new class of FeAs-based superconductors has attracted a lot of attention due to their ͑moder-ately͒ high critical temperature, and new experimental evidence has been gathered indicating that superconductivity coexists with a spin-density-wave state in some members of the ferropnictide family. 6,7 In spite of these experimental advances, microscopic modeling of coexistence between magnetic and superconducting orderings is still in its infancy, and considerable insight should be gained by investigating the competition between these two opposing tendencies.With this in mind, here we focus on a specific class of materials showing this coexistence, namely, the borocarbides, in which the rare-earth element provides local moments ͑through their f electrons͒, while superconductivity arises from phonon-mediated pairing of conduction electrons: singlet superconductors with either antiferromagnetic or modified ferromagnetic ͑i.e., spiral or domainlike͒ arrangements have been observed experimentally. 2 We assume that pairing of conduction electrons can be described by the attractive-Hubbard model 8 and that they are coupled to local moments through a Kondo-like term. 9,10 The Hamiltonian then reads aswhere, in standard notation, the sums run over lattice sites, with ͗i , j͘ denoting nearest-neighbor sites, t sets the energy scale ͑we set t = 1 from now on͒, U Ͼ 0 is the attraction strength, and J is taken positive, thus favoring an antiferromagnetic coupling between the conduction electron spin i ϵ͚ ␣,␤=Ϯ c i␣ † ␣␤ c i␤ ͑ ␣␤ denotes the Pauli matrix elements͒ and the localized spin S i ; for simplicity, we take S =1/ 2. The two competing tendencies are clear: as J increases, the Kondo-like coupling drives the conduction electrons to form singlets with the local moments at the expense of breaking the pair...

We have used the density-matrix renormalization group method to study the ground-state properties of the symmetric periodic Anderson model in one dimension. We have considered lattices with up to N s = 50 sites, and electron densities ranging from quarter to half filling. Through the calculation of energies, correlation functions, and their structure factors, together with careful extrapolations (toward N s → ∞), we were able to map out a phase diagram U vs n, where U is the electronic repulsion on f orbitals, and n is the electronic density, for a fixed value of the hybridization. At quarter filling, n = 1, our data is consistent with a transition at U c 1 2, between a paramagnetic (PM) metal and a spin-density-wave (SDW) insulator; overall, the region U 2 corresponds to a PM metal for all n < 2. For 1 < n 1.5 a ferromagnetic phase is present within a range of U , while for 1.5 n < 2, we find an incommensurate SDW phase; above a certain U c (n), the system displays a Ruderman-Kittel-Kasuya-Yosida behavior, in which the magnetic wave vector is determined by the occupation of the conduction band. At half filling, the system is an insulating spin liquid, but with a crossover between weak and strong magnetic correlations.