Abstract We study the magnetic properties in the vicinity of a single carbon defect in a monolayer of graphene. We include the unbound σ orbital and the vacancy induced bound π state in an effective two-orbital single impurity model. The local magnetic moments are stabilized by the Coulomb interaction as well as a significant ferromagnetic Hund's rule coupling between the orbitals predicted by a density functional theory calculation. A hybridization between the orbitals and the Dirac fermions is generated by the curvature of the graphene sheet in the vicinity of the vacancy. We present results for the local spectral function calculated using Wilson's numerical renormalization group approach for a realistic graphene band structure and find three different regimes depending on the filling, the controlling chemical potential, and the hybridization strength. These different regions are characterized by different magnetic properties. The calculated spectral functions qualitatively agree with recent scanning tunneling spectra on graphene vacancies. I. INTRODUCTION The two π orbital sub-bands of a honeycomb lattice in pristine graphene realize perfect Dirac fermions [1, 2] with a linearly vanishing density of states (DOS) at the Dirac point. As a result, graphene is a semimetal with a half-filled conduction band [2, 3] . While vacancies in metals typically act as non-magnetic scattering centres on the conduction electrons, it has been shown [4] [5] [6] that removing a carbon atom induces a local magnetic moment that can undergo a Kondo screening at low temperatures [7] with substantial Kondo temperatures of order T K ∼ 50 K. A tunable Kondo resonance in combination with the possibility to manipulate the spin degrees of freedom [8, 9] may be of fundamental use for practical graphene based spintronics. I. INTRODUCTION Resistivity measurements [7] or the magnetic response of graphene [6] with vacancies induced by irradiation provide information on ensemble averaged impurity properties while a scanning tunneling microscopy (STM) gives direct access to single vacancy qualities. Since the graphene DOS vanishes linearly at the Dirac point such a system appears to be an ideal system for observing quantum criticality [10, 11] . While the Kondo effect is absent for arbitrary coupling at the Dirac point and particle-hole (PH) symmetry in a linearly vanishing pseudo-gap DOS [12, 13] , finite doping controlled by an external gate voltage or strong PH asymmetry can lead to Kondo screening at low enough temperatures. In addition, the Kondo physics for adatoms such as Co on graphene [11, [14] [15] [16] [17] has drawn some attention in recent years. I. INTRODUCTION While the 3d transition metal Co carries an intrinsic magnetic moment, the generation of spin moments at carbon vacancies in graphene is less clear. Locally, twoorbitals of the three broken sp 2 bonds form a spin singlet bond leaving one radical with free moment [6] . Since these σ orbitals are orthogonal to the π orbital subsystem responsible for the Dirac fermions, they do not couple to the conduction band prohibiting the Kondo effect. Furthermore, the carbon vacancy induces a bound π state in the vicinity of the defect that is energetically located at or close to the Dirac point [18] [19] [20] . In a tight-binding formulation for the itinerant π electrons, the exact position of the bound state depends on the nearest and next nearest neighbor hopping t and t ′ . A vanishing t ′ results in a symmetrical DOS and a Dirac Point that coincides with the bound state. Those bound states are orthogonal to the itinerant states forming the conduction electron continuum and it is unclear [11] whether these states are responsible for the experimentally observed magnetic moments [6] . Due to the local graphene curvature in the vicinity of the carbon vacancy, the σ orbitals and the neighboring π orbitals start to hybridize and the possibility of a Kondo effect emerges. I. INTRODUCTION In this paper, we investigate the two-orbital model for single vacancies in graphene originally proposed by Cazalilla et al. [20] using Wilson's numerical renormalization group (NRG) approach [21, 22] . We calculate local spectra as functions of the gate voltage controlled chemical potential as well as the hybridization strength which is connected to graphene's curvature in the vicinity of the vacancy. The model comprises one unbound σ orbital, the locally bound π state and the coupling to the remaining π continuum. I. INTRODUCTION Mitchell and Fritz used this model [16] as starting point and constructed an effective Kondo model in the local moment regime comprising of a spin-1/2 coupled to a logarithmic divergent effective conduction band density of states [20, 23] for a non-interacting π subsystem. The authors include the vacancy induced bound state in the DOS which is singular at the DP. Thus, they neglect the Coulomb interaction between σ and π orbital and treat the problem as an effective s = 1/2 single impurity Kondo problem. However, the divergent DOS at the DP is irrelevant for doping away from charge neutrality. Since in this paper we are interested in local fluctuations in particular and treat Coulomb interactions explicitly, we incorporate the bound state in the impurity as an effective orbital and use an effective DOS comprising only of the itinerant graphene π states. I. INTRODUCTION The scanning tunneling spectra (STS) [24] indicate that the system is located closely to local charge degeneracy points at which the local orbital occupation is changed by one indicating that the external gate voltage not only alters the filling in the graphene layer but also the local charge configuration. Charge fluctuations become important for understanding the STS questioning the applicability of the Schrieffer Wolff transformation in the experimentally relevant parameter space. Since a density functional theory (LDA) calculation [19] predicted a substantial ferromagnetic Hund's rule coupling J H ≈ 0.35 eV between the π and the σ orbital, combining the π subsystem into a single locally projected density of states (PDOS) of non-interacting degrees of freedom appears to be an oversimplification. I. INTRODUCTION Previously, single orbital Kondo and Anderson models with a graphene type pseudo-gap DOS have been studied using the NRG see also the reviews [11, 14] . Miranda et al. [25] focused on a disorder-mediated Kondo effect. Ruiz-Tijerina et al. [26] concentrated on the investigation of a single orbital Anderson impurity model for different geometries in the context of a dilute ensemble of atomic impurities in graphene. I. INTRODUCTION The inclusion of the ferromagnetic coupling [19] between both orbitals, the σ orbital and the zero-mode (ZM), can favor a local triplet formation in the n doped region leading to distinct local spin configurations in different parameter spaces, and consequently to different STS responses as function of the local graphene curvature that match the recent experimental findings [24] as we will show below. We have also explicitly taken into account the hybridization between the σ orbital and the locally bound π state as included in the starting point of Ref. [16] . This induces a level repulsion in the twoorbital model as function of the hybridization strength that turned out to be crucial for the development of different local charge and spin configurations. We investigate the competition between these different charge and spin states as function of the chemical potential and the hybridization caused by the graphene curvature. We are able to qualitatively explain the different experimental STS spectra obtained on different carbon vacancy locations using the same two-orbital model. I. INTRODUCTION The paper is organized as follows. After giving a brief overview on the experimental motivation and the STS data in Ref. [24] in Sec. II, we summarize the literature on the electronic configuration around carbon vacancies in graphene in Sec. III A and introduce a two-orbital model including our parameterization of the local curvature. First, we focus on the charge and spin configurations of the isolated impurity in Sec. III B in order to set the stage for the detailed NRG calculation presented in Sec. IV. Sec. III C is devoted to the basics of the Kondo effect in pseudo-gap systems such as graphene. We begin the result section IV by examining each of the three relevant hybridization parameter regimes independently -Sec. IV A, IV B, and IV C -before we combine a full parameter scan of hybridization and chemical potential into a finite temperature diagram in Sec. IV E. A detailed analysis of the Kondo temperatures is presented in Sec. IV G. We discuss the impact of parameter changes within the model on the spectral functions as well as possible extensions of the model in Sec. IV H and end with a summary (Sec. V) of our findings. II. EXPERIMENTAL MOTIVATION In a recent STM study [24] , Mao et al. have shown that carbon vacancies in graphene exhibit Kondo physics adjustable by an external gate voltage V G . They identified two different classes of vacancies clearly distinguishable by their characteristic scanning tunneling spectra (STS) as a function of V G . The gate voltage V G can be directly converted into a chemical potential µ separating the system by either p (µ < 0) or n (µ > 0) doping. II. EXPERIMENTAL MOTIVATION The STS of the first class of vacancies show a Kondo resonance for electron and hole doping with a vanishing T K close to charge neutrality as expected from the pseudo-gap DOS in graphene. For the second class of vacancies a Kondo peak is only visible in the hole doped regime and disappears upon approaching charge neutrality [24] . II. EXPERIMENTAL MOTIVATION We see the same characteristic behavior in our NRG calculations where we use the hybridization strength to distinguish both scenarios. Thus, we will refer to first scenario as the 'intermediate hybridization' and the second as the 'strong hybridization' regime throughout this paper. In our calculations, we also identified a third regime, where a Kondo peak is only found for µ > 0. This so-called 'weak hybridization' regime has been experimentally observed as well [24] . Note, that we are dealing with broad crossovers between these regimes and not sharp phase boundaries. Therefore, we will do without giving absolute values for the boundaries of the different regimes. II. EXPERIMENTAL MOTIVATION The goal of this paper is to give a physical explanation of all three regimes using the same two-orbital impurity model and calculate the spectral function ρ(ω) for those classes by small parameter changes related to the local curvature of the graphene sheet at the location of the impurity. III. THEORY A. Modeling vacancies by a two-orbital Anderson model Modeling the electronic degrees of freedom in the vicinity of a graphene vacancy has a long history [27] . Removing a carbon atom in a 2D monolayer of graphene leads to three dangling σ orbitals in the neighboring atoms. Two of them will form a new chemical bond with a doubly occupied orbital causing a Jahn-Teller like distortion of the local lattice. Electronically active remains only the third unpaired σ orbital [20] (see also Fig. 1 (left) ). Charge neutrality and a weakly screened local Coulomb interaction causes a free local moment formed in this chemical radical which can undergo a Kondo screening at low temperature. III. THEORY A. Modeling vacancies by a two-orbital Anderson model There is a long-standing debate about the possibility of Kondo screening of this moment by coupling to the π conduction electrons since the π Wannier states are orthogonal to the σ orbital in a flat 2D monolayer of graphene and, therefore, do not hybridize. This situation however changes in the presence of a local curvature which is naturally induced by either suspending the sample or by supporting it on a corrugated substrate such as SiO 2 . Due to the local curvature, the neighboring π orbitals acquire a finite overlap with the localized σ orbital and start to hybridize with a hybridization strength that depends on the local curvature: the larger the curvature, the stronger the coupling. III. THEORY A. Modeling vacancies by a two-orbital Anderson model In addition, removing a carbon atom from the lattice also influences the π band in the vicinity of the impurity. Treating this as a single-particle scattering problem [18] has demonstrated that now the π subsystem comprises itinerant states responsible for the typical graphene density of states and one additional bound state weakly localized in the vicinity of the vacancy being orthogonal to the itinerant π states. This bound state is often referred to as zero-mode since it is located at the Dirac point in the tight-binding description [18] . Furthermore, density functional theory predicts a rather significant ferromagnetic coupling between the local π state and the σ orbital of J H ≈ 0.35eV [19] . III. THEORY A. Modeling vacancies by a two-orbital Anderson model When setting up an effective model for the vacancy one has to carefully review the local density of states of the effective non-interacting conduction band, particularly when hunting for very subtle changes of many-body wave function due to the Kondo effect. III. THEORY A. Modeling vacancies by a two-orbital Anderson model A first guess would be to consider the local density of state (LDOS) of the neighboring π orbitals of the dangling σ orbital. In this LDOS [18, 27] the bound π state causes a singularity at the Dirac point. Using such LDOS is somehow misleading, since (i) the zero-mode stems for a δ-distribution that (ii) is associated with an interacting orbital with a significant Hund's rule coupling [19] which is neglected in the LDOS description. III. THEORY A. Modeling vacancies by a two-orbital Anderson model Alternatively, this bound π state could be ignored by setting up a simplified single-orbital pseudo-gap Ander- son model. Such a model may be sufficient to qualitatively explore the limit of strong hybridization where only occupation fluctuations between a singly and a doubly occupied σ orbital is relevant -see below. However, it fails to capture the physics of the weak hybridization regime without major modifications and assumptions (e.g. hybridization dependent Coulomb interaction) which have to be added by hand into the model. This simplified model also does not address the issue of the bound state: the resonance resides at ω = 0 yielding a singular contribution to the DOS that must be properly dealt with. Furthermore, the ferromagnetic interaction between the π bound state and dangling σ orbital is expected to be quite strong and should still be taken into account explicitly. III. THEORY A. Modeling vacancies by a two-orbital Anderson model Cazalilla et al. [20] proposed the following two-orbital Anderson model III. THEORY A. Modeling vacancies by a two-orbital Anderson model for the description of a carbon vacancy in graphene that will be the starting point of this paper. The local twoorbital Hamiltonian is given by III. THEORY A. Modeling vacancies by a two-orbital Anderson model where ǫ d and ǫ π denotes the single-particle energy of the orbitals with the corresponding creation operators III. THEORY A. Modeling vacancies by a two-orbital Anderson model creating an electron in the π (respectively d) orbital with spin σ (we used the subscript d for the σ orbital to avoid interference with the spin index) U ππ and U dd label the intra-orbital Coulomb repulsion and U dπ the inter-orbital Coulomb interaction (see Fig. 1 (right) ). In order to ′ tight-binding model with t = 2.91 eV and t ′ = −0.16 eV (solid) in accordance to ab-initio calculations [29, 30] . We performed a summation over the first Brillouin zone and evaluated the well-known single-particle dispersion Eq. (4). The dashed curve shows a simplified approximate Γ(ω). The frequency is shifted in both cases such that the Dirac point lies at ω = 0 for µ = 0. III. THEORY A. Modeling vacancies by a two-orbital Anderson model ensure rotational invariance in the spin space [28] the Hund's rule coupling J H has to be augmented with a pair-hopping term with the same coupling strength. III. THEORY A. Modeling vacancies by a two-orbital Anderson model Since we have included the bound state of the π system in the effective impurity, we treat the π continuum by a simple tight-binding model for a bipartite honeycomb lattice as III. THEORY A. Modeling vacancies by a two-orbital Anderson model where ≪ i, j ≫ denotes the summation over a nextnearest neighbor hopping [3] . The operators a ( †) and b III. THEORY A. Modeling vacancies by a two-orbital Anderson model destroy (create) an electron on the respective sublattice A or B. Fitting of the hopping integrals to DFT calculations for the band structure yields the values t ≈ 2.91 eV and t ′ ≈ −0.16 eV [30] where the inclusion of a next-nearest neighbor hopping t ′ leads to an asymmetrical DOS and a shift of the Dirac point. III. THEORY A. Modeling vacancies by a two-orbital Anderson model Throughout the paper, µ = 0 refers to charge neutrality of the system where the Dirac point coincides with the Fermi energy. In STM experiments µ is taken as the zero of energy and its variation with the gate induced doping is monitored by the motion of the Dirac point relative to µ. Likewise, the energy spectrum is symmetrically discretized around the chemical potential in the NRG. Therefore the spectral functions are also calculated with respect to µ as zero of energy. III. THEORY A. Modeling vacancies by a two-orbital Anderson model The tight-binding Hamiltonian (3) can be easily diagonalized which yields two energy bands (cf. [2, 3] ) III. THEORY A. Modeling vacancies by a two-orbital Anderson model where τ = ± denotes the band index. The hybridization energy takes the standard form III. THEORY A. Modeling vacancies by a two-orbital Anderson model where the influence of the π band on the impurity dynamics can be fully encoded into the complex hybridization function III. THEORY A. Modeling vacancies by a two-orbital Anderson model This defines a local Wannier orbital created by c † 0σ and an effective hybridization strength V via III. THEORY A. Modeling vacancies by a two-orbital Anderson model Taking the imaginary part yields the coupling function III. THEORY A. Modeling vacancies by a two-orbital Anderson model where ǫ ± ( k) is the one-particle tight-binding solution Eq. (4). Since {c † 0σ , c 0σ ′ } = δ σσ ′ , V is determined by the equation III. THEORY A. Modeling vacancies by a two-orbital Anderson model that is related to the effective hybridization strength Γ 0 by the integral III. THEORY A. Modeling vacancies by a two-orbital Anderson model where we choose D = 8 eV for the bandwidth of graphene. In order to capture the essence of the pseudo-gap density of states, we also used the parameterization of Γ(ω) by the analytical expression (D eff = 3 eV pins the van Hove singularities in the DOS while D = 8 eV determines the edges) III. THEORY A. Modeling vacancies by a two-orbital Anderson model The calculations presented below are performed for both hybridization functions Γ(ω), i.e. (i) a direct k-space summation of the single-particle dispersion ǫ τ ( k)) and (ii) the approximation (12) . The different Γ(ω) are depicted in Fig. 2 . III. THEORY A. Modeling vacancies by a two-orbital Anderson model Γ 0 is used as a free parameter to adjust the vacancy specific hybridization introduced above. Thus, it serves as a direct parameterization of the local curvature in the vicinity of the impurity. B. Local scenarios and choice of parameters In order to set the stage for the full solution of the correlated two-orbital impurity problem, we will discuss the different local configurations in a model which will become relevant for the explanation of the three different regimes found in the experiments. B. Local scenarios and choice of parameters In this section, we restrict ourselves to H loc defined in Eq. (2) . At charge neutrality, the d orbital and the π bound state should be singly occupied. In the p doped region, the d orbital remains half-filled while the π bound state is unoccupied and thus the total local occupation is typically close to N imp = 1 for µ ≪ 0. B. Local scenarios and choice of parameters Three distinct local configurations are of particular interest for positive µ: A second electron can populate either the d or π orbital depending on the local parameters. In addition, a third electron will occupy the π bound state in some parameter regimes if the additional energy gain µ is larger than the differences in the impurity energies. This results in a doublet state where the π orbital is fully occupied and forms a local singlet whereas the d electron's spin is providing a local moment for a spin-1/2 Kondo effect. B. Local scenarios and choice of parameters First, consider the case N imp = 2. The total energy for the doubly occupied d orbital (singlet state) is given by B. Local scenarios and choice of parameters which competes with the triplet state where each orbital is singly occupied and whose energy reads B. Local scenarios and choice of parameters A local singlet ground state formed by two electrons on the d orbital prohibits the Kondo effect once conduction electrons are coupled to the orbitals. On the other hand, the local triplet state still leaves the possibility for the underscreened Kondo regime with a twofold degenerate ground state from the π orbital spin. For a total orbital occupation of three electrons, the energy of the doublet state reads B. Local scenarios and choice of parameters Here, the spin of the d electron may also be screened by the conduction electrons showing Kondo physics. Since the state |D contains three electrons and, therefore, its occupation is governed by ∆E = E D − 3µ in a decoupled impurity. This state becomes favored with increasing µ over the singlet and the triplet configuration with N imp = 2 and can push the system close to a charge instability. B. Local scenarios and choice of parameters The local scenario depends sensitively on the parameters of the local two-orbital Hamiltonian Eq. (2). The exchange interacting J H is crucial in order to stabilize the local triplet state. Ab-initio calculations predict a substantial ferromagnetic Hund's rules coupling of J H ≈ 0.35 eV [19] between the orthogonal π state and σ orbital. We propose that the difference between the two classes of impurities detected experimentally and labeled as intermediate and strong hybridization regime are related to a hybridization controlled level crossing between the local singlet state with energy E S and the local triplet state with E T . B. Local scenarios and choice of parameters DFT calculations [31, 32] and recent experimental studies [24] suggest that U dd = 2 eV [33] . We will adopt this value but address the consequence of possible smaller U in Sec. IV H 1. B. Local scenarios and choice of parameters For the remaining Coulomb matrix elements, we comply with the parameters stated in the supplementary material of Ref. [20] where U dπ ≈ 0.1 eV. In order to stabilize the triplet state in the strong hybridization regime, we slightly increase the onsite Coulomb repulsion on the π orbital, U ππ = 0.01 eV. B. Local scenarios and choice of parameters Our goal is to switch between the different scenarios by just a small change in parameters that are solely caused by the variations in the local curvature of graphene in the vicinity of the different impurities. We chose the impurity single-particle energies such that the system is located close to the local crossover between singlet and triplet state outlined above. B. Local scenarios and choice of parameters Without a microscopic ab-initio theory for each impurity configuration at hand, we assume that in the experiment only two parameters, Γ 0 and µ, are varied: The former by the vacancy location, i. e. the local curvature, the later by the external control parameter V G . B. Local scenarios and choice of parameters The effect of the hybridization between the π orbitals on the neighboring carbon sites [34] and the local d orbital is twofold: (i) It couples the d orbital to the π band continuum and (ii) generates a hopping term between the bound π state and the d orbital. B. Local scenarios and choice of parameters A single-particle Green function approach [19] has been employed for the tight-binding model H π−band with one carbon vacancy to determine the fraction z of the local π orbital contributing to the bound π state. The single particle wave function of this bound state decays only as ∝ r −1 . Due to the extended nature of the wave function, the z factor is relatively small and has been calculated as z ≈ 0.07 [19] . Using the hybridization parameter V and diagonalizing the impurity single-particle matrix B. Local scenarios and choice of parameters with the appropriate transformation matrix U determines the new eigenenergies ǫ (14), and Eq. (13) (left to right). Kondo screening is only possible for small and medium hybridization. B. Local scenarios and choice of parameters We also have considered a simplified linear shift interpolating the level energies by the equation B. Local scenarios and choice of parameters Here, Γ i,s 0 stands for the hybridization at which the system is either in the intermediate or strong hybridization regime. The disadvantage is obvious: There is no microscopic justification and we need to determine four reference parameters for this interpolation to arbitrary Γ 0 . B. Local scenarios and choice of parameters In the following, we omit the prime and implicitly use the shifted Γ 0 -dependent orbital energies. We also refer to the extended bound π state as π orbital in the following, since the physical π orbitals have been decomposed in their contribution to Γ(ω) and to the bound π state. Since the Coulomb matrix elements are not determined by an ab-initio approach, we assume that they remain constant as function of Γ 0 and are always defined with respect to the final orbital-basis to keep the modeling simple. B. Local scenarios and choice of parameters In addition to the previous considerations, the level positions are subject to a dynamical, Γ 0 -dependent shift stemming from the interaction with the conduction band continuum that is explicitly included in our numerical renormalization group approach. As a consequence, the local estimations for E T and E S can only be regarded as an approximate guideline for the level positions and the charge instability. B. Local scenarios and choice of parameters For p doping (µ < 0), the lower lying d orbital is singly occupied in all cases eventually resulting in Kondo screening. The exact value of T K strongly depends on Γ 0 and µ. Fig. 3 summarises the different situations for positive µ. For small Γ 0 , the double occupation of the bound π states becomes favorable, leaving a local moment s = 1/2 in a N imp = 3 ground state that can undergo a Kondo screening at large enough µ. Increasing Γ 0 will favor the spin-triplet formation in the local moment regime with s = 1 that can exhibit an underscreened Kondo regime at very low temperatures. Increasing Γ 0 further reduces the single-particle energy ǫ d enough that the doubly occupied d orbital singlet state is locally favored, and the possibility of a Kondo effect is excluded. B. Local scenarios and choice of parameters C. Review of Kondo effect in systems with a pseudo-gap density of states B. Local scenarios and choice of parameters The Kondo problem has been a major point of interest in sold-state theory for multiple decades. It is a prime example of a correlated many-body problem naturally arising in metallic hosts exhibiting small magnetic dilution. At sufficient low temperature T < T K the antiferromagnetic interaction between the impurity spin and the electrons' spin of the host gives rise to the formation of a singlet state which screens the magnetic moment of the impurity. The standard Kondo Hamiltonian [21, 35] takes the form B. Local scenarios and choice of parameters where J 0 is the Kondo coupling between the impurity spin S and the local conduction electron spin density s 0 , and V 0 accounts for an additional potential scattering term arising from particle-hole symmetry breaking in the local moment (LM) regime. For a constant metallic DOS ρ(ω) ≡ ρ 0 the Kondo temperature shows an exponential dependence on J 0 [21, 35] B. Local scenarios and choice of parameters For a pseudo-gap density of states, i.e. ρ(ε) ∝ |ε| r , Withoff and Fradkin were the first to point out the existence of a critical coupling [12] using a perturbative renormalization group argument. This model exhibits a wide variety of different phases for r > 0. These phases are characterized by different fixed points (FP) whose properties and occurrence depend on the bath exponent r as well as on particle-hole symmetry or the absence of it. For 0 < r < 1/2, there exists a critical coupling strength Γ c [13, [36] [37] [38] governing the transition between a LM phase for weak coupling and a strong-coupling (SC) phase for large coupling to the metallic host. B. Local scenarios and choice of parameters As discussed in section III A and depicted in Fig. 2 , the case r = 1 is relevant for graphene. In the following, we briefly summarized the results of the literature [10, 11, 13] . For µ = 0 and PH-symmetry the system will always flow to the stable LM fixed point independent of the Kondo coupling J 0 . For µ = 0, Kondo screening will arise at any coupling J 0 due to the finite density of states and we expect an exponentially suppressed Kondo scale ln(T K ) ∝ −1/|µ| r [10] . This picture changes in case of a broken particle-hole symmetry. At µ = 0, screening is possible above a critical coupling J c and for strong enough asymmetry V 0 > V c . Thus, there is an unstable quantum critical point separating an unscreened LM and screened ASC phase. For finite chemical potential and near J c the system will eventually arrive at a strong coupling fixed point but in a different way with regards to particle or hole doping [10] . As a result, close to the quantum critical point (QCP) but for p doping, the Kondo temperature scales as T K = κ|µ|, while for µ > 0 it is proportional to |µ| x , where x ≈ 2.6 and κ is a prefactor of order O(1) that depends only on the bath exponent r (see Fig. 4 in Ref. [10] ). D. NRG approach to quantum impurity systems In general, we are interested in the dynamics and the thermodynamics of an interacting quantum-mechanical many-body system with an approximate continuous quasi-particle excitation spectrum. The importance of a wide range of different energy scales, from several eV on the scale of the bandwidth to arbitrary small excitations between degenerate states, gives rise to infrared divergences in the perturbation theory for the antiferromagnetic Kondo problem or Anderson impurity models. One successful approach is the renormalization group (RG) that allows for a non-perturbative treatment of all energy scales. The RG forms the foundation of Wilson's Numerical Renormalization Group (NRG) [22] approach which has been first applied to solve the Kondo problem [21, 39, 40] . D. NRG approach to quantum impurity systems The Hamiltonian of such a quantum impurity problem takes the three-part form of Eq. (1) where the effect of the impurity (band) is completely encoded in the H loc (H band ) term. The bath is divided into intervals on a logarithmic mesh, characterized by the discretization parameter Λ, around the chemical potential where Λ → 1 reconstructs analytically the original problem. One then proceeds by mapping this star geometry via a Householder transformation onto a half-infinite tight-binding chain, the so-called Wilson chain. The hopping parameter between neighboring sites n and n + 1 decay exponentially t n ∝ Λ −n/2 owing to the logarithmic discretization. The problem is now solved in an iterative fashion where the starting point is the easily diagonalizable bare impurity devoid of any bath degree of freedoms. In each iteration one additional site of the Wilson chain is included, the new eigenvalue problem is solved numerically, and high-energy excitations are discarded in order to control the otherwise exponentially growth of the many-body Fockspace. Each iteration provides access to a smaller temperature scale due to the exponentially decreasing hopping parameters. The iterative procedure is then continued until the desired temperature is reached. In the end, the set of all eigenstates and -energies make up an approximate solution to the original many-body spectrum up until the final temperature. D. NRG approach to quantum impurity systems The NRG algorithm is not limited to equilibrium calculations and has been adopted successfully to nonequilibrium problems [41, 42] . The identification of a complete basis set is the foundation for a sum rule conserving formulation of the impurity Green's function [43, 44] and eventually for present non-equilibrium extensions. Equilibrium NRG results provide an almost exact agreement with analytical results if available and often serve as a benchmark for other methods. For a more indepth examination refer to state-of-the-art reviews such as [22] . IV. RESULTS This section starts out with presenting our results for the three different generic scenarios that are characterized by Γ 0 before we compile everything into a finite temperature as well as a T → 0 regime diagram. These diagrams will be presented in Sec. IV E. Fig. 3 which summarizes the local configurations for positive chemical potential and three different generic values of Γ 0 serves as a road map. The scenarios represent the different classes of vacancies identified by STM experiments [24] . IV. RESULTS We used a discretization parameter Λ = 1.81 and kept N s = 2000 states after each iteration in all our NRG calculations. The calculations were done for finite temperature T = 4.2 K unless stated otherwise to be comparable to experimental results. IV. RESULTS We employed three different approximations for Γ(ω) and the single-particle orbital energies: (i) the approximate Γ(ω) stated in Eq. (12) and a simple linear interpolation, (ii) the realistic Γ(ω) generated from the t, t ′ dispersion and a linear interpolation of ε d/π , and (iii) the realistic Γ(ω) as well as a re-diagonalization based on the z factor for the bound π state. In all cases, the single particle energies ǫ d and ǫ π are functions of the selected Γ 0 . IV. RESULTS The parameters used in the NRG calculations are mainly extracted from different sources in the literature, as discussed in Sec. III B and as summarized in table I. The question whether different sets of parameters, especially smaller values for U dd , results in similar experimental findings is addressed in Sec. IV H 1. In order to reach the desired temperature, we choose β = 1.225 and the number of NRG iterations is given by N = 34. IV. RESULTS If not stated otherwise, the spectral functions are for the d orbital only, i.e. ρ d (ω). The effect of the π contribution to the spectrum is discussed in Sec. IV D. IV. RESULTS A. Weak hybridization regime Fig. 4 shows the calculated spectral functions ρ d (ω) [43, 44] for the d orbital for a weak hybridization (Γ 0 ≈ 1eV ). The results are depicted in panel Fig. 4 (a) for µ < 0 and those for µ ≥ 0 in panel Fig. 4(b) and cover a range from µ = −100 meV to µ = 100 meV. For clarity, the spectral functions are shifted by a constant for each µ and normalized by ρ 0 being the maximum of all spectral functions. While ρ d (ω) remains featureless in the p doped regime and reflects the pseudo-gap DOS of the conduction band, Kondo resonances are visible in the spectra for µ > 60 meV. IV. RESULTS In order to relate the NRG results to the scenarios presented in Fig. 3 , we plot the local orbital occupations as function of µ in Fig. 5(a) . In the weak hybridization regime, the lower d orbital remains half-filled and nearly independent of µ while the occupation of the π orbital depends on the chemical potential. For µ < 0 the π orbital is also half-filled due to the energy gain by the Hund rule coupling J H forming a local triplet. Increasing µ to µ > 50 meV adds another electron to the π orbital since the U ππ interaction is weak and a local moment with s = 1/2 remains on the d orbital. Therefore, we identify an underscreened (undercompensated [45] ) Kondo problem for µ < 0, while we find a conventional s = 1/2 pseudo-gap Kondo problem for µ > 0 since n π = 2. The observation of a lower T K for a underscreened s = 1 Kondo problem in recent NRG calculations (see Ref. [46] ) is compatible with our findings where T K ≪ T = 4.2K IV. RESULTS For the approximate hybridization the Kondo effect sets in at higher µ and is not shown for visibility. All used parameters are listed in Tab. I. The calculations were done for T = 4.2 K. All curves are divided by ρ0 being the maximum of all ρ d (ω). IV. RESULTS for µ < 0. IV. RESULTS Additionally, the influence of the chemical potential on the Kondo temperature is asymmetric even for a fixed s = 1/2 Kondo problem [10, 11] with respect to the sign change of µ. B. Intermediate hybridization regime Increasing Γ 0 moves the system into the intermediate hybridization regime. Here, a Kondo peak is found in ρ d (ω) for both strong p doping or n doping as clearly depicted in Fig. 6 . The spectral functions are normalized as before for clarity. For small bias voltage, the Kondo effect breaks down since Γ(ω) vanishes linearly at ω = 0 suppressing screening close to the Dirac point. The Kondo temperature as function of µ is extracted using a Goldhaber-Gordon fit [47] of the zero bias conductivity G(T ) B. Intermediate hybridization regime where s = 0.22 and G 0 = G(T = 0). T K (µ) shows an exponential dependence on the chemical potential close to |µ| → 0, albeit with different slope for p and n doping (not shown). B. Intermediate hybridization regime We have chosen the impurity parameters in the intermediate hybridization regime in such a way that the occupation of the π orbital changes around µ = 0 [19] as depicted in Fig. 5(b) . The d orbital remains roughly half-filled as function of µ while the occupancy of the π orbital changes rapidly from zero to single occupation. There is only a very small difference in the results obtained from the approximated Γ(ω), Eq. (12), and the realistic hybridization function emphasizing the generic character of the power law approximation of Γ(ω). B. Intermediate hybridization regime Since the local occupancy changes from N imp = 1 to N imp = 2, the underlying Kondo effect is fundamentally different for both types of doping. For µ < 0 the local d spin moment is screened by the bath forming a conventional Kondo singlet. For positive µ, however, the two spins form a local triplet state due to the strong ferromagnetic Hund's rule coupling J H . Only a fraction of the resulting s = 1 moment can be screened by the bath resulting in an underscreened Kondo effect and a dangling s = 1/2 moment [45] . C. Strong hybridization regime When Γ 0 is increased further compared to the intermediate regime, eventually the strong hybridization regime is reached. C. Strong hybridization regime For µ < 0, it essentially resembles the intermediate regime. The single occupied d orbital provides an unpaired spin that is screened to a Kondo singlet, and ρ d (ω) shows a Kondo resonance (Fig. 7(a) ). The π orbital, however, remains mostly empty for all chemical potentials considered here and the physics is entirely governed by the d orbital; this regime could also be described by a single orbital Anderson model [24] . C. Strong hybridization regime Starting around µ = 0 and increasing µ, the d orbital gets slightly filled such that the total occupation approaches N imp ≈ 1.2 -see also Fig. 5(c) . Therefore, the system approaches the intermediate valence (IV) FP governed by local charge fluctuations [40, 48] . A resonance develops in ρ d (ω) close to the DP that moves towards lower energies as µ increases as shown in Fig. 7(b) . This coincides with an increase of the d orbital occupation. The Kondo resonance for µ < 0 evolves continuously to a charge fluctuation resonance close to µ = 0 since the local N imp = 1 and N imp = 2 charge configurations become energetically degenerate for µ ≈ 0. While a doubly occupied d orbital is predicted in a purely local picture, the substantial hybridization favors the itinerancy of the impurity electrons. The local occupation of the d orbital is shown in Fig. 5 (c) and approaches n dσ ≈ 2/3 indicating the IV FP. D. Zero-mode peak We only dealt with the spectral functions for the d orbital in the previous sections. However, the tunneling STS current measured in the experiments [24] results from a superposition of three parts: The d orbital, the π orbital as well as the substrate. The d orbital is responsible for the Kondo physics but the π orbital contributes to the total experimental dI/dV curves in form of the zero-mode. The orbital occupation Fig. 5 gives a rough estimate for the position of the zero-mode peak relative to the Fermi energy in the different regimes: a vanishing occupation implies a zero-mode that rests above E F . We used the same parameters as in the previous sections. The spectral functions depicted in Fig. 8 are calculated by Lehmann representation of the NRG data [43, 44] using a logarithmic Gaussian broadening b -see the NRG review for details [22] . D. Zero-mode peak The π-spectral functions in the intermediate hybridization regime, Γ 0 = 1.21 eV, are shown for varying the broadening parameter b [22] and µ = −60 meV in Fig. 8(a) . This demonstrates the very sharp nature of the ZM excitation whose width is artificially broadened by b. In addition, a Hund's rule mediated excitation at slightly higher energies ω ≈ 0.35 eV emerges for small b which results from the exchange coupling between the d orbital and the π orbital. D. Zero-mode peak The zero-mode peak exhibits a simple µ dependence as shown in Fig. 8(b) : A higher chemical potential shifts the peak towards ω = 0. This is the exact same behavior observed in the experiments (Fig. 2(a) in [24] ). D. Zero-mode peak Since we are mainly interested in the Kondo physics which is only contained in the d orbital spectral function we will not discuss the π spectral function in the following. The parameters are listed in Tab. I. We used the realistic Γ(ω) and the Z factor to determine the level positions for arbitrary hybridization strength Γ0. For the color coding we used the residual entropy times the impurity occupation. D. Zero-mode peak E. Mapping out the parameter space Regimes at T = 4.2 K After presenting the d orbital spectral functions for three different characteristic values for Γ 0 representing the three different classes of vacancies found in the STM experiments, we combine the results into a more elabo-rate scan of Γ 0 from weak to strong hybridization in a single regime diagram where µ and Γ 0 are the only free parameters. Regimes at T = 4.2 K We used the realistic hybridization Γ(ω) calculated from the t, t ′ DOS and the single-particle orbitals energies calculated for a given Γ 0 via Eq. (16) in all NRG calculations in this section. For each data point (Γ 0 , µ), we obtained the impurity entropy S imp [22] and total orbital occupation N loc from an individual NRG run. The calculations were performed at a fixed finite temperature T = 4.2 K to make contact to the STM experiments. The diagram depicted in Fig. 9 shows the color-coded product of the residual entropy times the total impurity occupation. Regimes at T = 4.2 K In Fig. 9 three different areas are separated by two naturally appearing lines that define a sharp, but temperature broadened crossover region characterized by N loc S imp = const. In the left part of the figure, N loc rapidly drops from three to two, while simultaneously the entropy increases from ln(2) to ln(3). The second line occurs where N loc = 2 → 1 and S imp /k B = ln(3) → ln(2). Regimes at T = 4.2 K We identify seven different regimes in Fig. 9 . Increasing Γ 0 in the p doped region (µ < 0), we found three local moment and one strong coupling regime: s = 1/2 (LM-a), s = 1 (LM-b), s = 1/2 (LM-c), and s = 1/2 (SC-a). For positive µ, we find two strong coupling FPs (s = 1 SC-b and s = 1/2 SC-c) as well as a frozen impurity regime (FI). Note that the LM regimes mentioned above also extend partially to µ > 0 due to the vanishing Γ(ω) at the Dirac point. Regimes at T = 4.2 K For the smallest hybridization Γ 0 in the range of 0.5eV ≤ Γ 0 < 0.7eV, the π orbital is doubly occupied for positive µ, and we find a half-filled d orbital. This regime is labeled as the LM-a since the entropy tends towards S imp /k B ≈ ln(2) and the square effective magnetic moment [21] is given by µ 2 eff ≈ 0.25. This regime extends into p doping and crosses over into the SC-c regime upon simultaneous increase of µ and Γ 0 . Regimes at T = 4.2 K Increasing Γ 0 at a fixed µ enhances the level splitting between the two local orbitals: the doubly occupied π orbital becomes less favorable and the system crosses over to the s = 1 (LM-b) regime. The thick red line marks the crossover from a local occupation of N imp = 3 to N imp = 2 which also distinguishes LM-a and LM-b regimes. Thus, the entropy in the LM-b regime tends to S imp /k B ≈ ln(3) while the effective moment takes the value µ 2 eff ≈ 0.66. The ground state is a triplet state formed by both electrons which are coupled due to the strong ferromagnetic interaction J H . Regimes at T = 4.2 K Increasing the hybridization further delocalizes another impurity electron, and the system crosses over to the LM-c region where only a single d orbital spin is locally present. The spectral function already shows the onset of a Kondo peak (see Fig. 6 ) even though the entropy and the effective moment still signifies a LM FP: S imp /k B ≈ ln(2) and µ the SC-a regime is observed. The entropy as well as the square effective magnetic moment µ 2 eff tends to zero due to the formation of a Kondo singlet state in this regime. Now we discuss the diagram for µ > 0 starting from the upper right area of Fig. 9 . For strong enough hybridization and large µ, a doubly occupied d orbital and an empty π orbital are locally favored. Although that regime is labeled as frozen impurity (FI) regime, it is closer related the IV FP since N imp = 1.2 − 1.3. Interestingly enough, the FI ground state calculated by the NRG corresponds to a double occupied state while the impurity occupation is closer to N imp ≈ 1.3. Since the ground state is a local singlet state, the impurity decouples effectively from the conduction band. The coupling to the conduction band, however, leads to a gain of kinetic energy and partially delocalizes the d orbital electrons. The spectral function shows a pronounced excitation peak which shifts linearly with the chemical potential as depicted in Fig. 7(b) . Regimes at T = 4.2 K Upon the reduction of Γ 0 , the local spin triplet state becomes energetically favorable. As in the p doped region, we see a broad crossover regime where entropy and magnetic moment still take their LM fixed point values although an underdeveloped Kondo peak is already visible -see Fig. 6 (b) at µ = 60 meV. Both local electrons align ferromagnetically, and the hybridization is sufficient to initiate Kondo screening of the d electron spin. For sufficiently large Γ(µ), the underscreened Kondo fixed point (SC-b) is found with a dangling π electron spin. The SC-b regime is characterized by and impurity entropy of ln(2) and square effective moment of 0.25. Regimes at T = 4.2 K At small Γ 0 -upper left
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