# Many-Body Effects in FeN\({}_{4}\) Center Embedded in Graphene
## 3 Method  
Once all the parameters for the model are obtained, the problem can be recast and solved using the DMRG method. In order to do so, we mapped the problem onto an equivalent one-dimensional model by employing an exact canonical transformation, as presented in [88; 89] and reviewed in detail in [90] and closely related to the chain mapping used in DMFT [35]. An outline of the method is as follows. A conventional Hamiltonian for impurity problems will have the form:  
\[H=H_{l}+H_{imp}+V. \tag{3}\]  
Here, \(H_{l}\) denotes the single-particle tight-binding Hamiltonian for the lattice, which could include more than one band, and it can be obtained from DFT calculations. \(H_{imp}\) and \(V\) describe the impurity and the coupling between the impurity and the lattice, respectively. Note that this method is applicable regardless of the geometry or dimensionality of the non-interacting Hamiltonian. The key is to map \(H_{l}\) onto an equivalent one-dimensional chain. First, for simplicity, let us consider a single-impurity problem with one orbital per site. The more general case involving multiple orbitals and impurities will be discussed in the next section. The first step is to define the \"seed\" state to perform a Lanczos recursion as:  
\[\ket{\Psi_{0}}=c_{r_{0}}^{\dagger}\ket{0}\, \tag{4}\]  
where \(c_{r_{0}}^{\dagger}\) creates an electron at orbital \(r_{0}\) and \(\ket{0}\) is the vacuum state. For an Anderson-like impurity, which is the case here, we chose the seed to be the impurity orbital. The remaining states are then constructed with the following iterative procedure:  
\[\ket{\Psi_{1}}=H_{l}\ket{\Psi_{0}}-a_{0}\ket{\Psi_{0}} \tag{5}\] \[\ket{\Psi_{n}+1}=H_{l}\ket{\Psi_{n}}-a_{n}\ket{\Psi_{n}}-b_{n}^{ 2}\ket{\Psi_{n-1}}\] (6) \[a_{n}=\frac{\bra{\Psi_{n}}H_{l}\ket{\Psi_{n}}}{\bra{\Psi_{n}} \Psi_{n}}\quad b_{n}^{2}=\frac{\bra{\Psi_{n}}\Psi_{n}}{\bra{\Psi_{n-1}}\Psi_{n -1}}\quad. \tag{7}\]  
The equations for \(a_{n}\) and \(b_{n}\) can be obtained by requiring the states to be orthogonal. Note, however, that at this stage, the states are not normalized. After this transformation, \(H_{l}\) assumes a tri-diagonal form:  
\[H_{l}= \begin{pmatrix}a_{0}&b_{1}&0&0\\ b_{1}&a_{1}&b_{2}&0\\ 0&b_{2}&a_{2}&b_{3}\\ 0&0&b_{3}&\ddots\end{pmatrix} \, \tag{8}\]  
which corresponds to a one-dimensional Huckel Hamiltonian. Equivalently, in second quantization, it reads:  
\[H_{l}=\sum_{i=0}^{L}a_{i}\bar{n}_{i}+\sum_{i=0}^{L-1}b_{i+1}(\hat{c}_{i}^{ \dagger}\hat{c}_{i+1}+h.c.)\quad, \tag{9}\]  
where \(\hat{c}_{i}^{\dagger}\) and \(\hat{c}_{i}\) are the normalized creation and destruction operators, respectively, \(\bar{n}_{i}=\hat{c}_{i}^{\dagger}\hat{c}_{i}\) is the particle number operator, and \(L\) is the total length of the chain. The diagonal \(a_{n}\) terms are on-site potentials, while the \(b_{n}\)'s are the new hopping along the chain. This is an exact canonical transformation. The remaining missing orbitals correspond to different symmetry sectors of the Hamiltonian, are completely decoupled from the impurity, and can be safely ignored, which highlights the power of this change of basis. Coming back to the transition-metal complex, in our case, we had two orbitals (iron \(d_{xy}\) and \(d_{yz}\)) coupled to different sites (nitrogen) that will generate two orthogonal chains along the lines very similar to those described above, except that we would now require two seeds for the \(\pi\)-bonded nitrogen \(p_{z}\) orbitals. Labeling these as \(|\alpha_{0}\rangle\) and \(|\beta_{0}\rangle\), we chose these as:  
\[|\alpha_{0}\rangle=\frac{1}{2} \begin{pmatrix}1\\ 1\\ 1\\ 1\end{pmatrix} \quad|\beta_{0}\rangle=\frac{1}{2} \begin{pmatrix}1\\ -1\\ 1\\ -1\end{pmatrix} \, \tag{10}\]  
where the labeling refers to the nitrogen sites in Figure 1 (Note that due to symmetry, only these two -out of four- wave-functions couple to the \(d\) orbitals of the transition metal because the other degrees of freedom live in an orthogonal Hilbert space).