# Many-Body Effects in FeN\({}_{4}\) Center Embedded in Graphene
## 1 Introduction  
Porphyrins and metalloporphyrins attract a great deal of interest due to their crucial role in biological processes such as respiration and photosynthesis. These, and similar molecules such as phthalocyanine, are polyaromatic complexes that can accommodate a range of atoms at their centers to induce changes in magnetic and optical properties [1; 2; 3; 4]. As a result of their versatility, these complexes have found a range of exciting applications in spintronics [5; 6; 7; 8; 9; 10; 11; 12; 13; 14], optoelectronics [15; 16], solar cells [17; 18; 19; 20; 21; 22], and as building blocks of magnetic materials [23; 24; 25; 26; 27; 28; 29] or highly tunable qubits for quantum computing applications [30]. These complexes display important correlation physics, such as spin and orbital variants of the Kondo effect in phthalocyanine (FePc) molecules deposited on the (111) surface of noble metals [31; 32; 33; 34; 35; 36; 37; 38]. Porphyrin-like centers can be embedded in graphene and carbon nanotubes for oxygen reduction catalysis [39; 40; 41; 42; 43; 44; 45; 46; 47; 48]. Finally, transition metals can be a source of magnetism [49] to provide another knob for realizing unconventional functionalities. Despite the relative simplicity of graphene and a transition-metal atom, understanding the combined electronic structure of a transition-metal center embedded in the graphene matrixremains a challenge as the transition metal atom typically involves an incomplete \(d\) shell that gives rise to confinement-induced correlations and magnetism. Studies of iron porphyrins and other heme-like molecules have been carried out within the framework of density functional theory (DFT) [44; 50; 51; 52; 53; 54; 55; 56; 57], DFT+DMFT (dynamical mean-field theory) [58; 59; 35], quantum Monte Carlo [60; 61], coupled-cluster [51], molecular dynamics [62], and configuration interaction techniques [63]. Here, we discuss a new approach in which we use the DFT to construct a material-specific effective model Hamiltonian, which is then solved via the density matrix renormalization group (DMRG) [64; 65; 66; 67; 68] technique along with a unitary transformation to account for the many-body physics in a numerically exact way. Due to the large number of geometries, transition-metal centers, and axial ligands involved, a versatile method such as that discussed here to account for strong correlation effects will be useful for exploring the properties and possible applications of these complexes. Finding accurate approximations to the exchange-correlation functional represents the central problem of DFT. The simplest such functional is the local density approximation (LDA) [69]. LDA generally yields reasonable ground-state properties of 3d magnetic metals, including magnetic moments and Fermi surfaces [70]. LDA however does not correctly capture the phase diagrams of magnetic materials, including in particular the lowest energy crystal structure of elemental iron, where it predicts non-magnetic iron to be face-centered cubic and ferromagnetic iron to be body-centered cubic. The generalized gradient approximation (GGA) [71] provides a systematic improvement over the LDA with the added variational freedom by including the density gradient in the construction of the exchange-correlation functional. The GGA correctly predicts the relative stability of the ferromagnetic phase of pure Fe in addition to giving a good description of its ground state properties [72]. Further studies of iron porphyrin-type molecules based on the GGA+U scheme have revealed that interaction effects on the iron play an important role in determining the ground state magnetic moment [52; 58; 73]. In this paper, we consider a FeN\({}_{4}\) center embedded in a graphene lattice. For this purpose, we take a FeN\({}_{4}\)C\({}_{10}\)H\({}_{10}\) complex as the fundamental building block for this center, which was referred to as the \(D1\) center in [47]. \(D1\) centers have been realized experimentally in graphene and carbon nanotubes [40; 44].