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The utility of band theory in strongly correlated electron systems

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Rep. Prog. Phys. 79 (2016) 124501 (15pp) doi:10.1088/0034-4885/79/12/124501

Report on Progress

The utility of band theory in correlated electron systems

Gertrud Zwicknagl

Institut für Mathematische Physik, Technische Universität Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig, Germany

E-mail: g.zwicknagl@tu-bs.de

Received 9 November 2015, revised 15 April 2016 Accepted for publication 20 May 2016 Published 17 October 2016

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Abstract

This article attempts to review how band structure calculations can help to better understand the intriguing behavior of materials with strongly correlated electrons. Prominent examples are heavy-fermion systems whose highly anomalous low-temperature properties result from quantum correlations not captured by standard methods of electronic structure calculations. It is shown how the band approach can be modified to incorporate the typical many-body effects which characterize the low-energy excitations. Examples underlining the predictive power of this ansatz are discussed.

Keywords: electronic structure, /-electrons, Fermi surface, quasiparticles, heavy fermions, quantum phase transitions

(Some figures may appear in colour only in the online journal)

strongly

1. Introduction

Band structure calculations are an important tool in modern material science. Theory and simulation have been shown to provide useful guidelines for materials discovery, design, and optimization. Understanding the collective electronic properties of emergent materials with strong correlations, however, remains a great challenge to condensed-matter theory [1]. Important examples are transition metal oxides, metals containing lanthanide or actinide atoms and organic conductors. At low temperatures, these materials exhibit novel phenomena like metal-to-insulator transitions, heavy fermions, unconventional superconductivity and unusual magnetism which may eventually provide new functionalities. The complex behavior and the high sensitivity with respect to external fields result from the fact that the quantum mechanical (ground) states are determined by subtle compromises between competing interactions. It is not surprising that a theory assuming independent electrons has difficulties to capture these effects. Nevertheless, it has been shown that effective single-particle descriptions may be successfully applied to interpret observations and even predict results

in materials with strongly correlated electrons. The central scope of the present article is the knowledge on the utility of band structure calculations in strongly correlated electron systems that has been accumulated during the past decades. To be comprehensive, I will restrict myself to intermetallic lanthanide (4f) and actinide (5f ) compounds.

The electronic structure problem for a solid is one of many interacting fermions (electrons) moving in a periodic lattice potential. The properties of these systems are generally determined by the interplay of two types of influences. First, there is a tendency towards delocalization which is a consequence of the fact that wave functions at neighboring lattice sites overlap. It favors the formation of a conventional band spectrum and coherent one-electron Bloch states in which the itinerant electrons are distributed throughout the entire crystal. Second, the electrons interact via the repulsive Coulomb interaction which leads to localization. The probability for hopping from one site to another will depend on whether the final site is occupied or empty. A strong Coulomb repulsion between two electrons at the same lattice site would favor the formation of local magnetic moments which can be observed in the magnetic susceptibility. The general theoretical problem has proved too difficult

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for an exact analysis. Approximate solutions for macroscopic systems can be found only in limiting cases.

The traditional electron theory of metals which is based on the Sommerfeld-Bethe model [2] emphasizes the aspect of delocalization starting from the picture of free electrons whose eigenstates are characterized by wave number k and spin a. The ground state is determined by filling the plane wave states with the lowest energies according to the Pauli principle. A characteristic feature is the isotropic Fermi surface in £-space separating the occupied states from the empty ones. By construction, its volume is given by the number of electrons. The model successfully explains a wide range of thermodynamic properties including the electronic contribution to the specific heat at low temperatures which varies linearly with temperature C ^ jT... with a proportionality factor of order y ~ 1 mJ mole K-2 as well as the magnetic susceptibility which approaches a temperature-independent value reflecting the non-magnetic character of the ground state. The magnitude of the quantities is directly related to the density of low-energy excitations.

Band theory applies this conceptual framework to explicitly calculate observable properties of real materials. To achieve this goal the free-electron energies and plane wave states are replaced by the eigenvalues and eigenfunctions obtained from solving a Schrodinger equation with an appropriate static periodic lattice potential given by the atomic building blocks. For the energies, we usually encounter several bands with aniso-tropic dispersion separated by 'forbidden regions', the band gaps. As the many-electron ground state and the low-energy excitations are built in close analogy to the Sommerfeld-Bethe model the thermodynamic properties parallel those of their free-electron counterparts. The anisotropic Fermi surface may have several sheets, its total volume is given by the number of electrons in partially filled bands. The existence of parallel portions on the Fermi surface indicates instabilities which may lead to reconstructions when interactions are turned on. Band structure calculations provide a material-specific single-particle basis from which the many-particle states can be con-structed—at least in principle. In addition, expectation values and matrix elements can be used to extract a realistic para-metrization for many-body models. Band theory will always lead to an independent-particle description for the ground state: for N particles the N low-lying band states will be filled. Historically, the great triumph of band theory was that it provided a simple explanation why some materials are metallic while others turn out to be insulating (or semi-conducting) at low temperatures.

The 'free' electron picture was justified by Landau in his famous theory of Fermi liquids [3]. It is, in fact, referred to as the 'standard model' for the modern electron theory of metals. The fundamental conceptual basis is the notion of 'adiabatic continuity' which should prevail if the interaction is gradually turned on. As states of similar symmetry do not cross during this adiabatic process there should result a one-to-one correspondence between the well-known free fermion states and their unknown complex counterparts in the interacting system. In particular, the character of the ground states are the same, i.e. filled Fermi seas. The low-energy excitations can be described

to good approximation in terms of a single-particle spectrum. This important fact is a direct consequence of the Pauli principle. The single-particle excitations or 'quasi-particles' are characterized by momentum and spin and obey Fermi statistics. They can be visualized as composite objects consisting of the bare particle (electron) and some kind of polarization cloud which results from the Coulomb correlations. The many-body effects modify the energy dispersion relation and the interactions of the quasi-particles. Hallmarks of a Fermi liquid are the linear specific heat and the T-independent susceptibility as well as a ^-variation of the resistivity. To confirm, however, that a system indeed forms a Fermi liquid one has to show that the Fermi surface exists and that the quasiparticles exhaust the low-energy excitations, i.e. that the observed thermodynamic quantities are consistent with the effective masses observed, e.g. in quantum oscillations.

As expected, the Fermi liquid picture provides an excellent description of 'simple' metals. More subtle is the question whether it can be applied to the electrons of inner shells which are known to preserve some of their atomic character.

The general problem manifests itself in the transition metals whose ¿-electrons exhibit both aspects. This fact was the origin for the historical controversy between Slater [4] emphasizing the band picture and VanVleck [5] who advocated a description of the ¿-electrons in terms of local magnetic moments. The latter model explains various features found at elevated temperatures (or energies). Examples are the CurieWeiss susceptibility observed in ferromagnetic transition metals above the ordering temperature. On the other hand, typical low-temperature properties such as the relatively large Sommerfeld coefficient and the non-integral Bohr magneton number per atom of the observed spontaneous magnetization could only be understood when the ¿-electrons were treated as itinerant band states. Direct evidence for the delocalized character of the ¿-electrons, as far as the low-energy excitations are concerned, is the fact that they contribute to the Fermi surface (see e.g. [6]). There is no doubt that the ¿-electrons are delocalized despite the fact that some atomic features persist. The correlations restrict the energy or temperature range where the Fermi liquid picture is valid.

The fact that ¿-electrons in transition metal systems may exhibit both localized and itinerant character is a hallmark of strong correlations which cannot be captured by an independent-particle model. As a consequence, band theory has to be augmented by a many-body treatment of the essentially local, i.e. atomic-like interaction effects. This concept is the basis of the the dynamical mean-field theory (DMFT) [7]. The Fermi surfaces of transition metals, however, are usually well described within standard band theory based on density functional theory (DFT) [8, 9]. This scheme provides an efficient and rather accurate method to explicitly construct effective potentials required for material specific calculations and allows for parameter-free calculations of the ground state properties (see [10] and references therein). Combined with highly efficient numerical methods for solving the Schrodinger equation this ab initio method greatly helped to understand ordinary metals and their (ground state) properties. In addition, DFT provides the parametrization for the band-electron part

of realistic many-body model Hamiltonians which are solved within DMFT approach.

Having in mind the scenario encountered for the d-elec-trons we next turn to the case of the f-electrons. The corresponding wave functions are much more localized than the highest occupied atomic d-levels. The intra-atomic Coulomb repulsion is much larger than the estimated band widths which seems to favor a localized picture. In lanthanide-based heavy fermion systems (HFS) with partially filled 4f shells the local-moment behavior is indeed observed at 'high' temperatures T > T* with kBT* ~ 1 meV. In the high-temperature regime for T > T*, the degrees of freedom associated with the partially filled f-shells are conveniently modeled by local (magnetic) moments which weakly interact with the conduction electrons. The variation with temperature of the specific heat exhibits pronounced Schottky anomalies corresponding to crystalline electric field (CEF) excitations while the magnetic susceptibility is Curie-Weiss-like reflecting the magnetic moment of the partially filled f-shell.

The f-electrons should also be described within a band picture as delocalized states as far as the low-energy excitations in heavy-fermion compounds are concerned. The first hint came in 1975 from experiments on the intermetallic compound CeAl3 [11] where the specific heat and electrical resistivity were observed to vary according to C = jT + ... and p(T) = p0 + AT2 at low temperatures. The values of y and A, however, were enhanced by factors ~ 103 and ~ 106 over those known from simple metals. The discovery of superconductivity in CeCu2Si2 [12], UBeis [13], and UPts [14] showed that the degrees of freedom of the f-shells may give rise to a strongly renormalized Fermi liquid where the heavy quasipar-ticles can form Cooper pairs. A milestone was the measurement of the Fermi surfaces with heavy quasiparticles in UPt3 [15] where the cross sections were found in excellent agreement with DFT calculations [16, 17]. The difficulties these findings posed to theory were summarized in the the article [18] which emphasizes the need for new ideas and concepts. The observation of the de Haas-van Alphen (dH-vA) effect in CeSn3 [19] had demonstrated the existence of f-derived bands where the coherent Bloch states with well-defined wave vector and energy result from hybridization between conduction states and f-states. The measured Fermi surface cross sections had been found to agree with the ones obtained from standard band structure calculation within the DFT where the f-electrons were treated as ordinary band electrons. The band structure calculations, however, underestimated the effective masses. This finding shows that one can calculate the quasiparticle excitations for specific systems from a band approach which, however, has to be modified to incorporate many-body renormalizations. First attempts carrying out such calculations were those of Strange and Newns [20] for mixed-valent CeSn3 and of Sticht et al [21] for CeCu2Si2 where the latter was based on an approach devised by Razafimandimby et al [22] (see also [23] and references therein). The idea of heavy f-bands in lanthanide HFS was met with great skepticism given the fact that the high-temperature behavior of these materials clearly showed the existence of local moments [24]. It is rather obvious that at high temperatures the f electrons

should be excluded from the Fermi surface due to their apparent incoherent localized character. The latter, however, is lost at low temperatures. This hypothesis implies that the strong local correlations in Kondo lattices lead to an observable many-body effect, i.e. the change with temperature of the volume of the Fermi surface [25]. This unusual behavior and the validity of the Fermi liquid picture for HFS were confirmed experimentally 1992 by measurements of the dH-vA effect in CeRu2Si2 [26] which reproduced the Fermi surface cross sections and anisotropic effective masses predicted in 1989 by renormalized band calculations [27, 28].

The renormalized band method [29, 30] allows for material-specific calculations of strongly renormalized quasi-particles. The key idea is to introduce a small number of parameters to account for the essential many-body correlations. To select the appropriate parameters, a microscopic picture of the mechanisms leading to the formation of the Fermi liquid state is an indispensible prerequisite. In the case of lanthanide HFS we know that the heavy quasiparticles arise from the Kondo effect and a single parameter, the averaged effective mass, turned out to be sufficient. The resulting scheme proved to be a flexible tool with predictive power. Important applications are the prediction of spin density wave (SDW) instabilities and changes in the Fermi surface topology (Lifshitz transitions) induced by magnetic fields. The ansatz has been extended in a straightforward way to the calculation of f-spectral functions at finite temperatures. The results helped to interpret the data from high-resolution angle-resolved photoemission spectroscopy (ARPES).

In actinide HFS, there are indications that the underlying mechanism is partial localization of the 5f states due to the intra-atomic Hund's rule type correlations [31]. The central assumption is that only some of the 5f-orbitals lower their kinetic energy by forming extended Bloch states while the others remain localized forming multiplets to lower the Coulomb repulsion. Initially, the dual character has been conjectured for UPd2Al3 where the variation with temperature of the magnetic susceptibility points to the existence of CEF-split localized 5f states in a heavy-fermion system with 5f-derived itinerant quasi-particles. Direct experimental evidence for the co-existence of 5f-derived quasiparticles and local magnetic excitations is provided by recent neutron scattering and photoemission experiments. There is clear evidence that the presence of localized 5f states is even responsible for the attractive interaction leading to superconductivity. In addition the dual model could allow for a rather natural description of heavy-fermion superconductivity co-existing with 5f-derived magnetism. For a review of experimental facts see [32, 33] and references therein.

Heavy quasiparticles have been observed by dH-vA experiments in a number of U compounds. The experiments unambiguously confirm that some of the U 5f electrons must have itinerant character. It has been known for quite some time that the 5f-states in actinide intermetallic compounds cannot be considered as ordinary band states. Standard band structure calculations based on DFT fail to reproduce the narrow quasiparticle bands. The predicted bandwidths are too small to explain photoemission data [34]. Theoretical studies aiming at an explanation of the complex low-temperature

structures lay emphasis on the partitioning of the electronic density into localized and delocalized parts (see [35] and references therein). Concerning the low-energy excitations it has been shown that the dual model allows for a quantitative and parameter-free description of the heavy fermions in UPd2Al3 [36].

The paper is organized as follows: after a brief overview of standard DFT calculations in section 2, we demonstrate for exactly soluble models that the approximations underlying traditional ab initio band structure calculations fail to capture the characteristic features of the low-energy excitations in heavy-fermion systems in section 3. The renormalized band method is introduced in section 4 and applied to CeRu2Si2, CeCu2Si2, and YbRh2Si2 in section 5. In section 6, we summarize the calculation of heavy quasiparticles based on partial-localization. The formation of the heavy Fermi liquid in YbRh2Si2 as seen in high-resolution ARPES with decreasing temperature as well as consequences of these findings are briefly discussed in section 7. The paper closes with a summary and outlook in section 8 where we examine the relation between the renormalized band approach and the DFT-based dynamical mean-field approach (DFT-DMFT) which attempts at a realistic description of electronic properties of materials with strongly correlated electrons. In addition, we suggest methods and possibilities to estimate the characteristic heavy-fermion parameters based on information from first-principles in future calculations.

2. Standard band structure within density functional theory

Band structure calculations within DFT form the basis of the vast majority of electronic structure studies. A comprehensive overview of the results for lanthanide and actinide materials is given in [37]. The importance of DFT studies results from the fact that the calculations start from first-principles and hence provide a parameter-free material-specific picture. The central question is to which extent do these calculations capture the characteristic properties of strongly correlated electrons, i.e. how well can this scheme be applied to the discussion of the low-energy properties of heavy fermions. In general, questions addressed by DFT calculations refer to the 'nature' of the /-states and the characteristics of the ground state, in particular the shape of and the volume enclosed by the Fermi surface as well as its reconstruction following the formation of long-range order.

We begin this section by giving a brief summary of the DFT scheme which essentially maps the many-body Schrodinger equation onto an effective single-particle wave equation with an effective potential. We introduce the notation to be used in the subsequent section where we attempt to assess the validity of the scheme by applying it to exactly soluble models.

The density functional theory is based on the Hohenberg-Kohn theorem [8]: for a given two-particle interaction and given kinetic energy operator the properties of a non-degenerate ground state such as the ground state energy are determined by the ground state density n(r). The ground state energy F [n] is a unique functional of the density which is extremal

for the exact ground state density. An important assumption which finally allows to perform calculations is that the exact ground state density distribution of the interacting electrons can be represented by the density of ficticious non-interacting electrons moving in an external (local) potential. The density distribution is hence evaluated in terms of effective single-electron orbitals which are chosen to be orthogonal. Finally, the kinetic energy of the interacting system is approximated by that of its non-interacting counterpart F0 [n] which is easily calculated. This so-called non-interacting kinetic energy F0 and the usual contribution from the Coulomb interaction, the Hartree term £Hartree, are evaluated exactly. They both yield large contributions to the ground state energy which are are subtracted from the unknown total energy density functional leaving the exchange-correlation energy

Exc — F F EHartree (1)

which contains, in general, contributions both from the kinetic energy and the interaction term of the Hamiltonian. Stationarity of the total energy requires the single-electron orbitals to satisfy effective one-particle Schrodinger-like equations, the Kohn-Sham-equations [9] where the (non-trivial) many-body effects in the ground state are contained in the static exchange correlation potential defined as

Vxc = — [F — F0 — EHartree] . (2)

The eigenvalues and eigenfunctions of the effective Schrodinger equation are auxiliary quantities used to calculate the electron density and total energy. However, they are commonly interpreted as low-energy excitations of an interacting electron system. This is an unjustified ad hoc assumption which proved surprisingly successful for a great variety of systems. However, it does not correctly predict the small energy scale characterizing the /-derived low-energy excitations in to heavy fermion materials.

The origin of this failure will be discussed in the subsequent section. Here we concentrate on properties that we anticipate will only be weakly affected by the /-correlations. These are, in the first place, the conduction states which are only weakly correlated and hence can be described by conventional band theory. This has been shown by high-resolution ARPES studies (see e.g. [38]). The reason for this is that the electronic densities and the resulting potentials are adequately reproduced by the DFT. The interest in ab initio calculations of the conduction states originates from the attempts to construct realistic models for specific materials. They will serve as input for the renormalized band structure discussed here and for the DMFT. A second example is the shape of the Fermi surface that has been found to be accurately described by DFT for many heavy-fermion systems.

Let us now turn to the strongly correlated /-electrons and discuss the qualitative picture which emerges from a DFT calculation. In general, the strong Coulomb repulsion restricts the occupation of the /-shell. In the effective independent-particle picture, the partially occupied /-states are pinned to the Fermi energy. It is important to note that the Coulomb repulsion as described by the Hartree term is dominant which

implies that the choice of the exchange-correlation functional does not affect the general results. The f-density is rather localized and hence the corresponding wave functions overlap weakly with the conduction states. As a result, the effect of the hybridization between the conduction states and the f-states will be visible only in a rather narrow region in £-space. The ratio of the position of the effective f-level relative to the Fermi energy and the effective hybridization strength is related to the f-density which should be reasonably well predicted by DFT.

These findings help to understand why DFT yields rather good Fermi surfaces in many heavy-fermion compounds as delineated below for the case of CeRu2Si2. A more comprehensive discussion can be found in [29]. A typical property of CeRu2Si2 is the fact that many conduction bands are intersecting the Fermi energy. A similar behavior will always be encountered in inter-metallic compounds containing transition metal ions with partially filled d-states. In these systems, the f-states hybridize with many conduction bands of rather different symmetries and anisotropies resulting from CEF effects will play a minor role. As a result, the shape of the Fermi surface is mainly determined by the steep conduction bands. The total volume enclosed by the Fermi surface must change upon hybridization with the f-states corresponding to the number of itinerant fermions. The volume difference, however, results as sum from minor volume changes of several sheets resulting from the many conduction bands. In particular, there will be no metal-insulator transition associated with the formation of f-derived quasiparticles in this case. The distinctive feature in this argument is the number of conduction bands intersecting the Fermi energy. If there were a single conduction band dramatic changes like metal-to-insulator transitions could occur.

These considerations show, on the other hand, that the shape of the Fermi surface does not necessarily reflect the correlated nature of the system. In contrast to the shape of the Fermi surface, the effective masses of the heavy quasiparticles are not reproduced by DFT. They are underestimated by a factor of 20 in lanthanide systems and by a factor 5-10 in actinide materials which reflects the fact that dynamical renormalization is neglected.

At this point, I would like to mention that considerable progress has been made in including dynamical processes in DFT-based first-principles computations. The Green's function approaches to calculate the quasiparticle bands in periodic solids usually involve self-consistent perturbative approximations to the self-energies formulated in terms of the bare quantities, i.e. the unrenormalized energies, hybridizations and interactions. It is well-known, however, from microscopic studies of single impurities with strongly correlated electrons that these perturbative schemes based on random phase approximation (RPA) combined with ladder approximations fail to capture the Kondo features in the strong-coupling regime where divergences may appear. A possible new approach towards a first-principles treatment might come from the renormalized perturbation theory which will be briefly mentioned later.

3. Strong correlations from f electrons: model studies

The difficulties faced by traditional band structure calculations in reproducing the high effective masses in HFS can

be understood from simple exactly soluble models for the singlet formation via the Kondo effect in lanthanide and the mass renormalization due to intra-atomic Hund's rule type correlations in actinide compounds.

3.1. Kondo effect: molecular model

Much of the essential physics of Ce-based heavy-fermion systems is already contained in the molecular model [23, 39]

H = ec £ ¿ca + ef £ ff + Unf T nf i + V £(cO f + fl ca)

where the weakly correlated conduction states of the metal are replaced by a two-fold degenerate uncorrelated ligand state such as an s-orbital. The corresponding creation (annihilation) operators will be denoted by cl (ca) where the index a refers to the spin. The strong correlations are introduced by the strong Coulomb repulsion among the spin-degenerate localized f-states whose creation, annihilation and number operators are denoted by fl, fa and nfl = flfl, respectively. The orbital energies of the ligand state and thef-orbital are denoted by ec and ef, respectively, with ec = 0 and ef < ec = 0. The two subsystems are coupled through a weak hybridization V < | ef\ whereas the Coulomb repulsion in thef-state is large i.e. U > | e/|. For simplicity, we assume U ^ x>. We shall focus on the two-electron states of the model Hamiltonian in equation (3). In the absence of hybridization (V = 0), the ground state with E = ef and f-occupation nf = 1 is fourfold degenerate. The degeneracy is lifted by the hybridization between the ligand and the f-states. Since the latter conserves the spin it leaves the triplet states with S = 1 invariant but couples the two singlet states with S = 0. To

leading order in the small ratio x = -—^ the energy

E = £f - 2 I" ■ \ef I

of the singlet ground state is lowered by an amount kBT * = —-

which characterizes the low-energy excitations.

The ground state has predominantly f ^character but contains a small admixture of the f ^configuration ,JT—nf with an /-valence

nf = n-\ + n[= 1 — 2x2 + ...

close to unity where n-\, n are the occupancies of the two spin-degenerate f-states.

Let us now turn to the solution of the molecular model within DFT. The molecular model can be considered as a two-site problem where the density distribution is characterized by the f-occupation nf. The energy of the (effective) f-level plays the role of an external potential while the hybridization is equivalent to a kinetic energy. As the exact ground state of the model is known, it is possible to determine the exact exchange-correlation energy Exc as well as its derivative vxc which fully account for the correlations in the ground state.

According to the definition we subtract the potential contribution from the exact ground state energy thereby expressing the f energy ef in terms of the corresponding density distribution parameter n = nf and the hybridization V. This procedure yields

F[n] = -2-J2 V4n-J 1 - n for U^ to. (6)

The large Coulomb interaction U ^ to does not permit the f valence to exceed unity. The expression for the density functional exhibits a remarkable symmetry with respect to the densities of the f electrons n and of the conduction electrons 1 — n. This fact can be interpreted as some kind of particle-hole symmetry in the 'lower Hubbard band' where double occupancies are strictly excluded.

Starting from the exact density functional for the ground state we construct the exchange-correlation potential vxc by

subtracting the Hartree term ¿Hai-tree = 1 (« + n0 U (« + n0 and evaluating the kinetic energy for the non-interacting reference system. The latter is calculated from the eigenstates of two non-interacting particles in an external potential with the density distribution n

F0 [n] = -IVJn-Jl—i

which exhibits conventional particle-hole symmetry as it is expected in the absence of interactions. For a detailled derivation we refer to [39].

Combining these results, the exchange-correlation potential as defined in equation (2) is seen to contain a contribution proportional to the Coulomb repulsion U. In strongly correlated systems, vxc mainly acts so as to compensate the large Hartree term. The exchange-correlation potential cannot be considered as small in the limit U ^ to. In addition, the separation of the interaction effects into Hartree and exchange-correlation potentials does not emerge as a natural concept in systems with strong local correlations.

Following Kohn-Sham, the quasiparticle energies E^ are determined from the single-particle like Kohn-Sham equations where the effective potential at the f-site is a sum of the external potential, the Hartree contribution vHartree and the exchange correlation potential vxc. To illustrate the general structure of the excitation spectrum we do not have to explicitly evaluate the exchange-correlation functional. It is sufficient to know that the Kohn-Sham equations do exist. In the large U limit, we find

vDF = -VHartree + €f + 4x2V + ...; E^ = -V + 2x2V + ...

The results for the large U limit are summarized as follows:

• The exchange-correlation potential vxc mainly compensates the Hartree term and does not describe the many-body aspects of the problem.

• The position of the f-level is renormalized by the effective potential. One obtains an effective f-level above the ligand state.

• The eigenvalues and hence the excitation energies scale with the bare hybridization V.

Since the ligand energy corresponds to the Fermi energy of a metal these findings qualitatively explain the typical results of the standard band structure calculation in heavy-fermion metals: the f-bands are shifted to the Fermi energy but their bandwidth is much too broad. We should like to emphasize that this result is generally valid since the derivation did not refer to an explicit expression for vxc but only postulated its existence.

3.2. Partial localization: two-site model

To illustrate the formation of heavy masses by intra-atomic correlations we consider a simple molecular model consisting of two actinide atoms at sites a and b. To model U compounds we assume that the total number of 5f electrons be five corresponding to an averaged f-occupation of 2.5 per U site. The dominant contribution to the ground state will be a linear combination of states |a; f3) |b; f2 ) and |a; f2) \b; f3 ), respectively. These two sets of states are coupled by the hopping term. Since both atoms have more than one electron in their 5f shells intra-atomic correlations come into play. The two sets of basis functions split into groups of states characterized by the total angular momenta J(a) and /(b), respectively, the energy differences being of the order of the 5f exchange constant, i.e. 1 eV. Since the spin-orbit interaction is large we use j — j-coupling and restrict ourselves to 5f states with j = 5/2. The adequacy of j — j — coupling for U compounds was recently shown in [40]. We are aiming at the low-energy subspace which is spanned by the states \a; f3, J (a) = 9/2)\b; f2, J(b) = 4)

and \a; f, J (a) = 4)\b; f, J (b) = 9/2), in close analogy to Hund's rules. Transferring a 5f-electron from site a to site b changes the local f occupation and the total angular momenta

\a; f, J (a) = 9/2)\b; f, J (b) = 4) ^ |a;f2, J'(a))\b;f3, J'(b))

and the resulting final state will usually contain admixtures from excited multiplets. The (effective) hopping of a 5f electron from site a to site b causes intra-atomic excitations against which the gain in kinetic energy has to be balanced. The crucial point is that the overall weight of the low-energy contributions to the final state depends upon (a) the orbital symmetry of the transferred electron, i.e. on jz and (b) on the relative orientation J(a) • J(b). The latter effect closely parallels the 'kinetic exchange' well-known from transition metal compounds. The requirement that the gain in energy associated with the hopping be maximal leads to orbital-selection. The dynamics in the low-energy subspace is described by an effective single-particle Hamiltonian where some of the transfer integrals are renormalized to zero while others are reduced yet remain finite.

These qualitative considerations are the basis for microscopic model calculations which proceed from the simple model Hamiltonian [41] H = Hband + HCoui where the local Coulomb repulsion

ln/21 / exact

Jz = 1/2

~\l5/2 = 3/2

0.6 0.8 1.0 Ul2 = l5/2 (eV)

Vj/l1/2 diagonal

Jz= 1/2

- 15/2 J

Q Jz= 3/2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

ti/2=t5/2 (eV)

Figure 1. Intra-atomic correlations and orbital selection as reflected in the ground state of the two-site model for partial localization: exact solution (left), and the diagonal approximation as used in LDA+U (right) [41, 42]. Reprinted with permission from [41, 42]. Copyright 2004 by the American Physical Society.

magnitude of the missing mass renormalization. For a discus-(10) sion of the performance of other approximation schemes we refer to [42].

is written in terms of the usual fermion operators c^(a)

(cj (a)) which create (annihilate) an electron at site a in the 4. Renormalized band method

5f-state with total angular momentum j and z-projection jz = -5/2,..., +5/2. The Coulomb matrix element U; ; ; ; are given in terms of the usual Clebsch-Gordan coefficients and the Coulomb parameters UJ where J denotes the total angular momentum of two electrons. The sum is restricted by the antisymmetry of the Clebsch-Gordan coefficients to even values J = 0, 2, 4. The low-energy sector is solely determined by the differences of the UJ-values which are of the order of 1 eV and thus (slightly) exceeding the typical bare 5f-band widths.

The kinetic energy operator describes the hopping between sites

Hband = - £ tj(Cj (a)c j(b) + h.c.).

We assume the transfer integrals tjz to be diagonal in the orbital index jz. In the calculations described below, we use the Ur parameters appropriate for UPt3 and study the ground state in the tji-space.

The ground state of the model is conveniently characterized by Jz = Jz(a) + Jz(b). Five different phases are visible in the phase diagram figure 1, two 'high-spin' phases with ferromagnetic site correlations and three 'low-' and 'intermediate-spin' phases with antiferromagnetic site correlations. In the isotropic case t\ /2 = t3= i5/2, the ground state is sixfold degenerate.

First-principles electronic structure calculations for acti-nides often keep only the diagonal interaction matrix elements (direct and exchange) in the many-body model Hamiltonian. This diagonal approximation breaks the rotational invariance of the interaction and hence it is not surprising that the isotropic line is absent in the phase diagram. In addition, the parameter space of phases with strong ferro- or antiferromagnetic alignment are strongly overemphasized. Generally, the density of low-energy excitations is lower in the diagonal approximation than in the exact solution. This is a direct consequence of the broken rotational symmetry which reduces the degeneracy of the ground state in the absence of hopping by a factor ~ 6. We should like to emphasize that this factor explains the order of

This sections gives a brief summary of the method, the underlying ideas and the implementation. For a detailed description we refer to [29, 30].

At this point we should like to add a technical remark. A number of different computational methods are available which solve band structure problems in a highly efficient way. The explicit form of the secular equation and, concomi-tantly, the choice of appropriate phenomenological parameters depend on the actual formulation of the problem as well as on the specific numerical procedure. Hence there are various different but equivalent schemes for introducing the renormalization.

The approach adopted here starts from a formulation of the band structure problem developed originally from scattering theory. The characteristic properties of a given material enter through the information about single-site scattering which can be expressed in terms of properly chosen phase shifts {nvi(E)} specifying the change in phase of a wave incident on site i with energy E and symmetry v with respect to the scattering center.

The central question encountered in this context is which quantities, in particular which phase shifts, reflect the many-body character of a specific system and, hence, have to be renormalized or, conversely, which phase shifts can be calculated directly from the ab initio potentials generated within a standard electronic structure calculation. Microscopic models for the scattering off magnetic sites in dilute alloys serve as useful guidelines. In the case of rare-earth compounds, the T-matrix for the band electron scattering is directly proportional to the Green's function of the localized states. As a consequence, we introduce renormalized phase shifts for waves which have the symmetry of the /-electron's spin-orbit ground state multiplet, i.e. 4/j = 5/2 for Ce systems with respect to the rare-earth sites.

Operationally, the renormalization procedure amounts to transforming the /-states of the spin-orbit ground state multiplet at the lanthanide site into the basis of CEF eigenstates\m) and introducing resonance-type phase shifts

Fully selfconsistent LDA bandstructure calculation Input: atomic potentials and structure information

Selfconsistent potentials

Phase shifts for the conduction states (Non-f states)

Renormalized f phase shifts Heavy masses CEF states

Renormalized bandstructure Heavy quasiparticle bands

Figure 2. Schematic summary of renormalized band calculation for metals with strongly correlated electrons (left panel). Comparison of typical quasiparticle DOS of Yb (upper right panel) and Ce (lower right panel) HFS. Since the number of 4f holes (Yb case) and electrons (Ce case) is slightly smaller than unity the corresponding band centers are below and above the Fermi energy, respectively.

nfm{E) ~ arctan (12)

where the resonance width Af accounts for the renormalized quasiparticle mass. The resonance energies efm = ef + 8m refer to the centers of gravity of the f-derived quasiparticle bands. Here ef denotes the position of the band center corresponding to the CEF ground state while the Sm are the measured CEF excitation energies. In the systems under consideration, the energy differences between the CEF ground state doublet and the excited states largely exceeds the resonance width. The latter is given by the characteristic energy kBT* where T* is of the order of the single-ion Kondo temperature. The low-temperature properties derived from the heavy quasiparticles will reflect the spatial symmetry of the CEF ground state wave function. One of the remaining two parameters, ef, is determined by imposing the condition that the charge distribution is not altered significantly by introducing the renormalization. This makes the renormalized band method a single-parameter scheme. As the 4f-count in Ce systems is slightly smaller than unity, ef will lie above the Fermi level. The free parameter, Af, is adjusted so as to reproduce the coefficient of the linear specific heat at low temperatures. The method has been shown to reproduce Fermi surfaces and anisotropies in the effective masses of a great variety of Ce-based compounds. In addition, it allows one to predict Fermi liquid instabilities. Typical results for Ce systems can be found in [32, 33, 43] and reference therein.

In calculating the coherent 4f-derived quasiparticle bands in Yb-based heavy-fermion compounds we have to account for the fact that Yb can be considered as the hole analogue of Ce. Operationally this implies that we have to renormal-ize the 4f j = 7/2 channels at the Yb sites instead of the 4f j = 5/2 states in the Ce case. As the 4f hole count is slightly less than unity the center of gravity ef will lie below the Fermi energy. In addition, we have to reverse the hierarchy of the

CEF scheme, i.e. ef < 0; efm = ef — 6m. The parameters are schematically summarized in figure 2.

The influence of an external magnetic field B is included via field-dependent parameters ?/(B) and Af (B). They are obtained from fits to field-dependent quasiparticle DOS of the single-impurity Anderson model calculated microscopically by means of the numerical renormalization group (NRG) or, more conveniently, by renormalized perturbation theory (RPT) (see [44] and references therein). This procedure properly accounts for the progressive de-renormalization of the quasiparticles with increasing magnetic field and the correlation-enhanced Zeeman splitting.

Having specified the parameters we next turn to a brief description of the actual implementation within the band structure scheme. The propagation from site i to site j of a state characterized by wave vector k is described by the structure constants S^Ck). They specify the contribution of the wave outgoing from center i' with symmetry v' to the incoming wave on center i with symmetry v. The structure constants are usually evaluated in the angular momentum (£, m)-basis; explicit expressions can be found e.g. in [45, 46]. Since spin-orbit splitting is large compared to the width of the quasiparticle bands, a Dirac-relativistic formulation of the scattering problem in terms of the eigenfunctions of the total angular momentum j = £ + 1/2 is more appropriate ((j, ^-representation). To account for CEF effects in heavy-fermion compounds we choose proper linear combinations (r, ja) of the 4j = 5/2 or 4j = 7/2 states which are adapted to the local symmetry of the lattice site. The abbreviation v may denote v = (£, m), v = (j, jz), v = (r, ja), respectively. The quasiparticle energies En(k) are solutions of the linear muffin tin orbital (LMTO) secular equation [45, 46] in the Dirac-relativistic generalization [47]

detGPl,(E}6vv'Sii' - sUk)) = 0 (13)

Figure 3. CeRu2Si2: Fermi surface sheets for quasiparticles withf-character. The labels ^ and a, 6, e and Z — 5 refer to the branches observed in dH-vA experiments (Lonzarich 1988). Left panel: hole surface centered around the Z-point of the Brillouin zone with effective mass m* ~ 100 m which dominates the specific heat 7- value (experiment [50] and theory [27]). Right panel: multiply-connected sheet. For localizedf-electrons at elevated temperatures, the hole surface expands extending to the boundaries of the BZ while the multiply connected electron- like surface shrinks. The expansion of ^ is confirmed by photoemission experiments [38]. Reprinted from [32]. Copyright 2004, with permission from Elsevier.

where the potential functions PV(E) are closely related to the scattering phase shifts.

The secular equation equation (13) of the renormalized band scheme can be cast into the more familiar form of a hybridization model [29]. The effective band Hamiltonian which reproduces the band structure in a narrow energy range surrounding the Fermi edge is a multi-band multi-orbital generalization of the model described by Hewson [48].

5. Heavy fermions in lanthanide-based HFS

To demonstrate the predictive power of the renormalized band method we discuss the heavy-fermion paramagnet CeRu2Si2 (Y ~ 350 mJ per mole K—2), the heavy-fermion superconductor CeCu2Si2 (y ~ 700 mJ per mole K—2, Tc ~ 0.7 K) and YbRh2Si2 which has emerged as prototypical system for the investigation of quantum phase transitions. For a review of the properties of this class of isostructural materials which crystallize in the tetragonal ThCr2Si2 structure see [49].

The renormalized band scheme correctly predicts the Fermi surface topology for CeRu2Si2 and thus consistently explains the measured dH-vA data [27-29]. These results are summarized in figure 3. The measured quasiparticle masses are consistent with the coefficient of the linear specific heat [26, 50, 51]. This proofs that the heavy quasiparticles exhaust the low-energy excitations associated with the f-states. There are no further low-energy excitations involved. For localized f-elec-trons at elevated temperatures, the hole surface expands extending to the boundaries of the BZ while the multiply connected electron- like surface shrinks. The expected change in volume of the Fermi surface upon heating was observed in recent photoemission experiments. Denlinger et al [38] have

shown that at temperatures around 25 K, the Fermi surface of CeRu2Si2, is that of its counterpart LaRu2Si2 which has no f electrons.

The heavy-fermion superconductor CeCu2Si2 exhibits a highly complex phase diagram at low temperatures which reflects instabilities of the Fermi surface. The renormalized band calculation [52] predicts two separate Fermi surface sheets for heavy and light quasiparticles. The latter can be considered as weakly renormalized conduction electrons. Heavy quasiparticles of effective masses m*/m ~ 500 are found on a Fermi surface which mainly consists of columns parallel to the tetragonal axis and of small pockets. As shown in figure 4 there are parallel portions connected by a wave vector close to (1/4,1/4,1/2). As a consequence, the static susceptibility x(q) exhibits a maximum for momentum transfer q close to the nesting vector (figure 4) which suggests that the strongly correlated Fermi liquid should become unstable at sufficiently low temperatures. This prediction is confirmed by neutron scattering experiments [55] (figure 4) which show the formation of a a spin-density wave (SDW) below TN ~ 0.7 K with propagation vector Q close to (0.22, 0.22, 0.55) and the ordered moment amounts to ^ ~ 0.1^B. These findings show that the SDW in CeCu2Si2 arises out of the renormalized Fermi liquid state. The transition is driven by the nesting properties of the heavy quasiparticles.

The characteristic energies and, concomitantly, the widths of the heavy quasiparticle bands are small. As a consequence, magnetic fields can strongly affect the Fermi surfaces and induce Lifshitz transitions. It was conjectured [52] that the transition into the B-phase should be related to a change in the topology of the Fermi surface as shown in figure 5.

A series of magnetic-field-induced Lifshitz transitions was identified recently in YbRh2Si2 [56-58] by means of the

Figure 4. CeCu2Si2: the static susceptibility x(q) as calculated from the renormalized bands (a) exhibits a peak at the nesting vector t of the heavy Fermi surface (b). The experimental modulation vector appearing in the neutron diffraction intensity (c) below the A- phase transition temperature TA shows perfect agreement with the calculated maximum position of x(q).

Figure 5. B-T phase diagram of CeCu2Si2 for B || a [32] (left, Copyright 2004, reprinted with permission from Elsevier) and magnetic-field-induced Lifshitz transition [52] (right, Copyright 1993, reprinted with permission by Elsevier). Original version of phase diagram from Bruls et al [53], completed version from Weickert et al [54].

Figure 6. YbRh2Si2: major sheets of the quasiparticle Fermi surface. For B = 0 T (left) consists of a Z-centered hole surface (left panel) and a multiply-connected surface ('jungle gym') (right panel). The modulus of the Fermi velocity is color-coded). In finite magnetic fields, the Fermi surface splits into majority and minority surfaces (right panel). In addition, it undergoes Lifshitz transitions.

Figure 7. YbRh2Si2: left panel: large Fermi surface from renormalized band calculation. Center panel: Fermi surface from ARPES, reproduced with permission from [61] Copyright 2015 by the American Physical Society. CC BY 3.0. Right panel: Fermi surface of the non-f reference system LuRh2Si2, reproduced with permission from [62] Copyright 2013 IOP Publishing and Deutsche Physikalische Gesellschaft. CC BY 3.0.

renormalized band calculation [30, 59]. There are three bands intersecting the Fermi energy. In the following discussion, we shall neglect the small T-centered electron pocket and focus on the two bands giving rise to the two major sheets displayed in figure 6. The overall topology qualitatively agrees with LDA results of [60]. The dominant contribution to the quasiparticle DOS and, concomitantly, to the specific heat and magnetic susceptibility comes from the Z-centered hole surface which has predominantly /-character. We refer to this surface as 'doughnut' although it is singly-connected, i.e. it has no hole). The states forming the 'jungle gym' , on the other hand, are strongly hybridized.

The Lifshitz transitions in a magnetic field can be understood from simple qualitative considerations. The two major sheets, i.e. the doughnut and the jungle gym, can be described in terms of a hole surface centered at the Z point (see figure 6). The main distinguishing feature is the number of holes nh. At ambient magnetic field, nh is small for the doughnut and the Fermi surfaces in neighboring Brillouin zones (BZ)s are not connected. For the jungle gym, however, the Z-centered hole surfaces in different BZs are interconnected. In an external magnetic field, the number of holes increases in the minority bands while it decreases in their majority counterparts. At a critical value, the minority doughnut surface transforms into a jungle gym surface. In close analogy, the majority jungle gym surface will transform at some other critical field into a doughnut.

The series of Lifshitz transitions which occur at well-defined values of the magnetic field are reflected in anomalies which have been found in the transport properties. A direct consequence of the field-induced Lifshitz transitions is that it is difficult to extract the shape of the Fermi surface from quantum oscillation experiments. These measurements are usually done in a field range where the Fermi surface topology differs from the one of the unperturbed system. The Fermi surface at ambient field (figure 6 left) was recently confirmed by high-resolution ARPES data [61] as shown in figure 7.

6. Dual model for 5f electrons and heavy quasiparticles in UPd2Al3

Within the dual model the strongly renormalized quasipar-ticles in [/-based heavy-fermion compounds are described as itinerant 5/ electrons whose effective masses are dressed by

low-energy excitations of localized 5/ states. We refer to [32, 36] for a detailed description of the method which proceeds in three steps. The latter include (a) a band structure calculation to determine the dispersion of the bare itinerant 5/ states (b) a quantum chemical calculation which yields the localized 5/ multiplet states and, in particular, the coupling to their itinerant counterparts and, finally, (c) a standard (self-consistent) many-body perturbation calculation to determine the renormalized effective masses. We should like to emphasize, however, that we treat all 5/ electrons as quantum mechanical particles obeying Fermi anti-commutation relations. In referring to the dual model one has to keep in mind that the latter provides an effective Hamiltonian designed exclusively for the low-energy dynamics. As such it seems appropriate for typical excitation energies Hu below ~10 meV. In general, effective low-energy models are derived form the underlying microscopic Hamiltonians—to borrow the language of Wilson's renormalization group—by integrating out processes at higher energies. In the case of 5/ systems the conjecture is that the hybridization between the conduction electrons and the 5/ states effectively renormalizes to zero for some channels while staying finite for others. In an effective model, this translates into orbital-dependent renormalization of effective hopping.

The concept of correlation-driven partial localization in U compounds has been challenged by various authors. The conclusions are drawn from the fact that conventional band structure calculations within DFT which treat all 5/-states as itinerant can reproduce ground state properties like Fermi surface topologies, densities. The calculation of ground state properties, however, cannot provide conclusive evidence for the delocalized or localized character of the 5/-states in acti-nides. First, the presence of localized states can be simulated in standard band calculations by filled bands lying (sufficiently far) below the Fermi level. Second, the Fermi surface is mainly determined by the number of particles in partially filled bands and the dispersion of the conduction bands which, in turn, depends mainly on the geometry of the lattice. A change in the number of band electrons by an even amount does not necessarily affect the Fermi surface since a change by an even number may correspond to adding or removing a filled band. As such, the Fermi surface is not a sensitive test of the microscopic character of the states involved.

Figure 8. Fermi surface and effective masses of UPd2Al3. Notation for FS sheets and experimental values from [63]. Theoretical values calculated within the dual approach [36]. The main cylinder part has effective masses with m* = 19-33 m, the highest masses are found on the corrugated (green) torus. right panel: comparison of experimental dH-vA frequencies (black symbols) and calculated frequencies from the dual approach. Large parabola corresponds to the main FS cylinder. Effective masses in UPd2Al3 for H ||c.

Figure 9. YbRh2Si2: temperature-dependent Fermi surface evolution [61]. Left panel: the T dependence was studied looking at the characteristic segments of the BZ which are schematically depicted in (a). (b), (c) 'diagonal' and (d)-(f) 'neck' direction. Right panel: 4f spectral functions corresponding to the crystal electrical field ground state calculated for temperatures T = 10, 30, 60, 90, and 120 K around the 'neck' feature of the BZ. The necks of the BZ remain open until well above 120 K, in accordance with the experimental result. To better visualize the changes in the distribution of 4f spectral weight, the spectra were normalized to the maximum value at the temperature under consideration. Reproduced figure with permission from [61]. Copyright 2015 by the American Physical Society.

YbR h2Si2 i-1—.........■—........

Figure 10. YbRh2Si2: B-T phase diagram indicating the regions with 'large' Fermi surface. Reproduced by permission from Macmillan Publishers Ltd: [64], copyright 2008.

An unambiguous proof of the dual character can be provided by an analysis of the spectral function. Of particular importance are characteristic high-energy features associated with transitions into excited local multiplets [65-67].

The scheme has been successfully applied to UPd2Al3 and UPt3 as can be seen from the comparison between the calculated and measured dH-vA frequencies and the effective masses in figure 8. It is important to note that the data are derived from a parameter-free calculation.

7. Finite temperatures: spectral function

The evolution with temperature of the electronic band structure and the shape of the Fermi surface of YbRh2Si2 have been investigated by high-resolution ARPES [61]. The results displayed in figure 9 clearly show that the characteristic feature of the 'large' Fermi surface, i.e. the 'open neck' at the XX-point persists in a rather wide temperature range from T = 1 K up three to four times the Kondo temperature TK as can be seen from figure 9.

The observed behavior is reproduced by the calculated variation with temperature of the ^-resolved /-spectral function p(k; w). The latter is given in terms of the /-Green's function

Gf(k; w) = Gf;,oc(k; w) - (W(k; w) - w(w)) (14)

where k and w denote wave vector and frequency, Gf;loc(w) is the fully renormalized local /-Green's function of a Kondo impurity and W, w account for the hybridization with the conduction states (For the notation see [23].).

This observation, i.e. that the ARPES data seem to suggest coherent band-like /-states in YbRh2Si2 for temperatures T > 1 K raises questions concerning the explanation of the anomaly line T* in figure 10. The renormalized band results clearly show that YbRh2Si2 forms a strongly renormalized Fermi liquid in finite magnetic fields B > 1 T, in agreement with the quadratic variation with temperature of the resistivity. The field-induced anomalies which are observed in transport properties result from Lifshitz-transitions of the 'large' Fermi surface. High-resolution ARPES data clearly show that

the Fermi surface is 'large' at T > 1 K in a vanishing magnetic field. The characteristics of the heavy quasiparticles are observed in the band dispersion up to rather temperatures. This behavior is consistent with theoretical calculations.

Considering the fact that we have a large Fermi surface on both the temperature and the magnetic-field axis leads us to question the Kondo break-down scenario for the T *-line. This ansatz conjectures that the anomalies observed along the T*-line are associated with the change in Fermi surface volume due to the break-down of the Kondo effect. On the other hand, our results seem to support the recent alternative 'critical quasiparticle' scenario which assumes the quasiparticles to be robust though modified by scattering from critical spin-fluctuations [68].

8. Summary and outlook

In conclusion, the renormalized band method as presented here provides a convenient and reliable scheme for calculating the narrow quasiparticle bands in heavy-fermion compounds. The central idea is to start from a standard fully selfconsis-tent first-principles band structure Hamiltonian and introduce a single parameter—essentially the linear coefficient of the measured specific heat—to account for the many-body renor-malization not captured by DFT. It would be, however, desirable to estimate the phenomenological parameter, i.e. the dynamical renormalization, from first principles.

A possible method to calculate this renormalization would be DFT-DMFT (see e.g. [69]), the state-of-the art scheme for realistic material-specific studies of materials with strongly correlated electrons. In close analogy to the renormalized band approach, this method proceeds in several steps starting from a standard DFT calculation to model the relevant conduction electron degrees of freedom. The strong local Coulomb repulsion is accounted for by a many-body model Hamiltonian. It is important to note that the Coulomb interaction among the correlated electrons is not (yet) calculated from first-principles. It is rather parametrized by plausible estimates for selected Coulomb matrix elements including intra- and inter-band repulsion, exchange coupling, etc. DFT-DMFT and the renor-malized band scheme both use first-principles Hamiltonians for the conduction states, both schemes, however, introduce parameters for the local many-body part. While the renor-malized band method introduces a single parameter for the renormalized band widths, DFT-DMFT introduces parameters for the 'bare' Coulomb interaction from which the renormalization is calculated selfconsistently. Concerning the low-energy quasiparticles, the main difference is that the DFT-DMFT scheme allows for changes in the charge distribution due to the coupling of the local strongly correlated electrons to the extended band states. This effect is certainly important in transition metal compounds or in intermediate-valent lanthanide and actinide systems where charge excitations are relevant also at low temperatures. In lanthanide-based heavy-fermion materials, however, the 4/-density remains (almost) unchanged by hybridization and selfconsistency with respect to hybridization should not play an important role.

A direction of future research will be to combine the renor-malized perturbation theory and renormalized band method to explain the magnitude of the parameter, i.e. the effective narrow width of the quasiparticle bands and to finally arrive at a microscopic description of the low-energy excitations in strongly correlated electron systems.

Acknowledgments

It is a great pleasure to thank E Abrahams, NE Christensen, J Floquet, S Friedemann, S-i Fujimori, P Fulde, C Geibel, AC Hewson, S Julian, G Knebel, K Kummer, C Laubschat, P Oppeneer, H Pfau, A Pourret, P Rourke, E Runge, Q Si, F Steglich, P Thalmeier, D Vyalikh, S Wirth, and P Wölfle for many fruitful discussions. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915.

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