Abstract
The properties of condensed matter are determined by singleparticle and collective excitations and their mutual interactions. These quantummechanical excitations are characterized by an energy, E, and a momentum, ℏk, which are related through their dispersion, E_{k}. The coupling of excitations may lead to abrupt changes (kinks) in the slope of the dispersion. Kinks thus carry important information about the internal degrees of freedom of a manybody system and their effective interaction. Here, we report a novel, purely electronic mechanism leading to kinks, which is not related to any coupling of excitations. Namely, kinks are predicted for any strongly correlated metal whose spectral function shows a threepeak structure with wellseparated Hubbard subbands and a central peak, as observed, for example, in transitionmetal oxides. These kinks can appear at energies as high as a few hundred millielectron volts, as found in recent spectroscopy experiments on hightemperature superconductors^{1,2,3,4} and other transitionmetal oxides^{5,6,7,8}. Our theory determines not only the position of the kinks but also the range of validity of Fermiliquid theory.
Main
In systems with strong electron–phonon coupling, kinks in the electronic dispersion at 40–60 meV below the Fermi level are well known^{9,10,11}. Therefore, the kinks that are detected at 40–70 meV in the electronic dispersion of hightemperature superconductors are taken as evidence of phonon^{12,13} or spinfluctuationbased^{14,15} pairing mechanisms. Collective excitations other than phonons, or even an altogether different mechanism, may be the origin of kinks detected at 40 meV in the dispersion of surface states of Ni(110) (ref. 16). Surface states of ferromagnetic Fe(110) show similar kinks at 100–200 meV (ref. 17), and even at 300 meV in Pt(110)—far beyond any phononic energy scale^{18}. Kinks at unusually high energies are also found in transitionmetal oxides^{5,6,7,8,19,20}, for example, at 150 meV in SrVO_{3} (ref. 7), where the Coulomb interaction leads to strong correlations. Very recently, kinks were reported at 380 and 800 meV for three different families of hightemperature superconductors^{1,2,3,4} and at 400–900 meV in graphene^{21}.
Interactions between electrons or their coupling to other degrees of freedom change the interpretation of E_{k} as the energy of an excitation with infinite lifetime. Namely, the interactions lead to a damping effect implying that the dispersion relation is no longer a real function. For systems with Coulomb interaction, Fermiliquid (FL) theory predicts the existence of fermionic quasiparticles^{22}, that is, exact oneparticle states with momentum k and a real dispersion E_{k}, at the Fermi surface and at zero temperature. This concept can be extended to k states sufficiently close to the Fermi surface (lowenergy regime) and at low enough temperatures, in which case the lifetime is now finite but still long enough for quasiparticles to be used as a concept.
Outside the FL regime, the notion of dispersive quasiparticles is, in principle, inapplicable as the lifetime of excitations is too short. However, it is an experimental fact that kresolved oneparticle spectral functions measured by angleresolved photoemission spectroscopy often show distinct peaks also at energies far away from the Fermi surface^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}. The positions of those peaks change with k, which means that the corresponding oneparticle excitations are dispersive, in spite of their rather short lifetime. It turns out that kinks in the dispersion relation are found in this energy region, which is located outside the FL regime.
We describe a novel mechanism leading to kinks in the dispersion of strongly correlated electrons, which does not require any coupling to phonons or other excitations, and which can occur at any energy inside the band. We begin with a discussion of the physics of this microscopic mechanism, which applies to a wide range of correlated metals. Consider first a weakly correlated system and imagine that we inject an electron into the partially filled band at an energy close to the Fermi surface. In this process the entire system becomes excited, leading to the generation of many quasiparticles and quasiholes. In view of their long lifetime, the Coulomb interaction with other quasiparticles or quasiholes modifies their dispersion which, according to FL theory, becomes E_{k}=Z_{FL}ε_{k}. Here, Z_{FL} is an FL renormalization factor and ε_{k} is the bare (noninteracting) dispersion. In contrast, an electron injected at an energy far from the Fermi level leads to excitations with only a short lifetime; their dispersion is hardly affected by the weak interaction, that is, E_{k}≈ε_{k} (see the Supplementary Information). The crossover from the FL dispersion to the noninteracting dispersion can lead to kinks near the band edges, which mark the termination point of the FL regime. However, for weakly correlated metals (Z_{FL}≲1), the slope of E_{k} changes only a little; hence the kinks are not very pronounced.
The situation is very different in strongly correlated metals, where Z_{FL} can be quite small such that kinks can be well pronounced. The strong interaction produces a strong redistribution of the spectral weight in the oneparticle spectral function. Namely, the conduction band develops socalled Hubbard subbands, whose positions are determined by the atomic energies. For metallic systems, a resonant central peak emerges around the Fermi level that lies between these subbands. The central peak of this socalled threepeak structure is often interpreted as a ‘quasiparticle peak’, but it will be shown below that genuine FL quasiparticles exist only in a narrow energy range around the Fermi level. Outside this FL regime, but still inside the central peak, we identify a new intermediateenergy regime, where the dispersion is given by E_{k}≈Z_{CP}ε_{k}. Here, Z_{CP} is a new renormalization factor, given by the weight of the central peak, which differs significantly from Z_{FL}. At these intermediate energies, which are much smaller than the interaction strength, an injected electron or hole is still substantially affected by the other electrons in the system. Therefore, its dispersion is neither that of a free system, nor that of the (strongly renormalized) FL regime, but rather corresponds to a moderately correlated system (Z_{FL}<Z_{CP}<1). As a consequence, a crossover occurs at an intermediate energy inside the central peak from Z_{FL} renormalization to Z_{CP} renormalization, which is visible as kinks in the dispersion. These observations apply to any correlated metal. As shown below, in a microscopic theory the position of those kinks are located at the termination point of the FL regime. We emphasize that this mechanism yields kinks but does not involve coupling of electrons and collective modes; only strong correlations between electrons are required.
For a microscopic description of these electronic kinks, we use the Hubbard model, which is the generic model for strongly correlated electrons, and solve it by manybody dynamical meanfield theory^{23,24,25,26} (DMFT), using the numerical renormalization group as an impurity solver. DMFT is known to provide the correct behaviour of local observables in the limit of large coordination numbers, and is used here to quantitatively support the physical mechanism discussed above. We focus on a single band with particle–hole symmetry and discuss the asymmetric case in the Supplementary Information. For the strongly correlated Hubbard model (interaction U≈ bandwidth), the dispersion relation is shown in Fig. 1 and the spectral function is shown in Fig. 2a. The dispersion relation, E_{k}, crosses over from the FL regime (blue line in Fig. 1) to the intermediateenergy regime (pink line in Fig. 1), as described above, and shows pronounced kinks at the energy scale . In some directions in the Brillouin zone these kinks may be less visible because the band structure is flat (for example, near the X point in Fig. 1). The behaviour of E_{k} is now analysed quantitatively.
The physical quantity describing properties of oneparticle excitations in a manybody system is the Green function or ‘propagator’ G(k,ω)=(ω+μ−ε_{k}−Σ(k,ω))^{−1}, which characterizes the propagation of an electron in the solid^{22}. Here ω is the frequency, μ is the chemical potential, ε_{k} is the bare dispersion relation and Σ(k,ω) is the selfenergy, a generally complex quantity describing the influence of interactions on the propagation of the oneparticle excitation, which vanishes in a noninteracting system. The effective dispersion relation, E_{k}, of the oneparticle excitation is determined by the singularities of G(k,ω), which give rise to peaks in the spectral function A(k,ω)=−Im G(k,ω)/π. If the damping given by the imaginary part of Σ(k,ω) is not too large, the effective dispersion is thus determined by E_{k}+μ−ε_{k}−Re Σ(k,E_{k})=0. Any kinks in E_{k} that do not originate from ε_{k} must therefore be due to slope changes in Re Σ(k,ω).
In many threedimensional physical systems, the k dependence of the selfenergy is less important than the ω dependence and can be neglected to a good approximation. Then, the DMFT selfconsistency equations can be used to express Σ(k,ω)=Σ(ω) as Σ(ω)=ω+μ−1/G(ω)−Δ(G(ω)), where is the local Green function (averaged over k) and Δ(G) is an energydependent hybridization function, expressed here as a function of G(ω). In DMFT, Δ(G) is determined by the requirement G(ω)=G_{0}(ω+μ−Σ(ω)), that is, G_{0}(Δ(G)+1/G)=G. Here, G_{0}(ω) is the local Green function in the absence of interactions. The hybridization function describes how the electron at a given lattice site is quantummechanically coupled to the other sites in the system. It plays the role of a dynamical meanfield parameter and its behaviour is strongly dependent on the electronic correlations in the system. Figure 2a shows a typical result for the integrated spectral function A(ω)=−Im G(ω)/π with the aforementioned threepeak structure. The corresponding real parts of the local propagator, G(ω), and selfenergy, Σ(ω), are shown in Fig. 2b and c, respectively.
Kinks in Re Σ(ω) appear at a new small energy scale that emerges quite generally for a threepeak spectral function A(ω). Kramers–Kronig relations imply that Re [G(ω)] is small near the dips of A(ω), located at ±Ω. Therefore, Re [G(ω)] has a maximum and a minimum at ±ω_{max} inside the central spectral peak (Fig. 2b). This directly leads to kinks in Re Σ(ω) for the following reason. There are two contributions to Σ(ω): ω+μ−1/G(ω) and −Δ(G(ω)). The first contribution Re [ω+μ−1/G(ω)] is linear in the large energy window ω<Ω (Fig. 2d); this is due to Kramers–Kronig relations (see the Supplementary Information) and is not particular to DMFT. On the other hand, the term −Re [Δ(G(ω))] is approximately proportional to −Re [G(ω)] (at least to first order in a moment expansion), and thus remains linear only in a much narrower energy window ω<ω_{max}. The sum of these two contributions produces pronounced kinks in the real part of the selfenergy at , where is the energy where Re [G(ω)] has maximum curvature (marked by blue circles in Fig. 2c). The FL regime with slope ∂ Re Σ(ω)/∂ ω_{ω=0}=1−1/Z_{FL} thus extends only throughout a small part of the central peak (). At intermediate energies (), the slope is then given by ∂ Re Σ(ω)/∂ ω_{ω=0}=1−1/Z_{CP}. The kinks at mark the crossover between these two slopes. As a consequence there is also a kink at in the effective band structure E_{k}.
The above analysis also explains why outside the FL regime E_{k} still follows the uncorrelated dispersion, albeit with a different renormalization Z_{CP} and a small offset c. This behaviour is due to ω+μ−1/G(ω), the main contribution to the selfenergy inside the central peak for . In particular, our analysis explains the dependence of E_{k} on k that was observed in previous DMFT studies of SrVO_{3} (ref. 27; see theSupplementary Information).
The FL regime terminates at the kink energy scale , which cannot be determined within FL theory itself. The quantities , Z_{CP} and c can nevertheless all be expressed in terms of Z_{FL} and the bare density of states alone; explicitly, we find , where D is an energy scale of the noninteracting system, for example, D is approximately given by half the bandwidth (see the Supplementary Information for details). For weak correlations (Z_{FL}≲1), the kinks in E_{k} thus merge with the band edges and are almost undetectable, as discussed above. On the other hand, for increasingly stronger correlations (Z_{FL}≪1), the kinks at move closer to the Fermi energy and deeper inside the central peak, whose width diminishes only as (ref. 28).
The energy scale involves only the bare band structure, which can be obtained, for example, from bandstructure calculations, and the FL renormalization Z_{FL}=1/(1−∂ Re Σ(ω)/∂ ω)_{ω=0}≡m/m^{*} known from, for example, specificheat measurements or manybody calculations. We note that because phonons are not involved in this mechanism, shows no isotope effect. For strongly interacting systems, in particular close to a metal–insulator transition^{26}, can become quite small, for example, smaller than the Debye energy.
The mechanism discussed here applies to systems with partially occupied d or f orbitals, where the local interaction is strong. An analysis similar to the one presented above also holds for systems with strong hybridization such as the hightemperature superconductors, where the overlap between d and oxygen p states is important. The assumption of a kindependent selfenergy may also be relaxed: if a correlationinduced threepeak spectral function, A(k,ω), is present for a certain range of momenta, k, the corresponding selfenergies, Σ(k,ω), and effective dispersion, E_{k}, will also develop kinks, as can be proved formally using cluster extensions to DMFT. Kinks in the dispersion are thus a robust manybody feature of correlated metals with a threepeak spectral function, independent of the computational approach.
The energy of electronic kinks is a quantitative measure of electronic correlations in manybody systems; they mark the termination point of the FL regime and can be as high as several hundred millielectron volts. Angleresolved photoemission spectroscopy experiments at such high binding energies can thus provide new, previously unexpected information about strongly correlated electronic systems. Electronic kinks are a fingerprint of a strongly correlated metal and are expected to be observable in many materials, including hightemperature superconductors.
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Acknowledgements
We acknowledge discussions with V. I. Anisimov, R. Bulla, J. Fink, A. Fujimori and D. Manske. This work was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereiche 484 (K.B., M.K., D.V.) and 602 (T.P.) and the EmmyNoether program (K.H.), and in part by the Russian Basic Research foundation grants 050216301, 050217244, 060290537 as well as by the RAS Programs ‘Quantum macrophysics’ and ‘Strongly correlated electrons in semiconductors, metals, superconductors and magnetic materials’, Dynasty Foundation, Grant of President of Russia MK2118.2005.02, interdisciplinary grant UBSB RAS (I.N.). We thank the John von Neumann Institute for Computing, Forschungszentrum Jülich and the Norddeutsche Verbund für Hoch und Höchstleistungsrechnen for computing time.
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Byczuk, K., Kollar, M., Held, K. et al. Kinks in the dispersion of strongly correlated electrons. Nature Phys 3, 168–171 (2007). https://doi.org/10.1038/nphys538
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DOI: https://doi.org/10.1038/nphys538
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