Energy-scale cascade and correspondence between the Mott and Kondo lattice physics

We propose an energy-scale correspondence between the Mott physics and the Kondo lattice physics and construct a tentative phase diagram of their correlated electrons with two characteristic energy scales $\omega^*$ and $\Omega$ marking the upper boundary of a low-energy regime with well-developed long-range coherence and the lower boundary of localized spectral weight in the energy space, respectively. In between, there exists a crossover region with emergent but damped quasiparticle excitations. We argue that the presence of two separate energy scales is a generic property of correlated electrons on a lattice and reflects an intrinsic two-stage process of the quasiparticle dynamics to build up the lattice coherence. For the Hubbard model, they correspond to the kink and waterfall structures on the dispersion, while for the periodic Anderson model, they are associated with the indirect and direct hybridization gaps. Our work reveals a deep connection between the Mott and Kondo lattice physics and provides a basic ingredient for the study of many-body systems.

In correlated electron systems, a "kink" denotes an abrupt slope change of the dispersive band and has often been regarded as a fingerprint of the coupling between electronic quasiparticles and collective bosonic excitations. It has been intensively studied in cuprates [1][2][3][4][5][6], where its appearance at an energy scale of around 30-90 meV has been attributed to spin fluctuations or phonons, which may potentially participate in the superconducting pairing. Different from these "bosonic kinks" which represent the characteristic energy scales of the bosonic excitations, it has been shown later that kink may also arise purely due to electronic mechanism without involving bosonic excitations [7]. This "electronic kink" was first seen in the Hubbard model (HM) within the framework of the dynamical mean-field theory (DMFT) [8] and then extended to other more complicated models [9][10][11][12][13][14]. It might explain the so-called high energy kink observed over a wide energy range from 40 to 800 meV in cuprates and some other transition-metal oxides [15][16][17][18][19][20][21][22].
Theoretically, the "electronic kink" has been argued to mark an important energy scale of the Mott physics, namely the boundary of its Fermi liquid ground state [7]. Roughly speaking, the kink energy ω * constraints a lowenergy region of well-defined Landau quasiparticles and poses a threshold for the application of the Fermi liquid theory. A higher energy scale Ω has also been proposed as the boundary separating the many-body resonant state from the high-energy Hubbard bands. ω * and Ω together define a cascade of key energy scales in the HM. Their discovery has since stimulated many debates concerning their true physical origin. Some suggested that the kink should still be of bosonic origin due to emergent collective spin fluctuations [23], while some associated it with an effective Kondo energy scale [24]. A better understand-ing of its origin will deepen our knowledge of the manybody physics, but a consensus has yet to be reached. In particular, its fundamental importance and potential implications have not been fully recognized.
Here we point out their deep connection with the general physics of quasiparticle emergence and coherence and report a close correspondence between the energy scales of the Mott and Kondo lattice physics. While the two phenomena have mostly been studied separately in different families of correlated systems based on either the HM or the periodic Anderson model (PAM), there are increasing evidences supporting an intimate underlying connection [25,26]. The Kondo lattice physics has been argued to be a special case of the (orbital-selective) Mott transition [27][28][29][30], and the Mott physics in cuprates is itself a low-energy projection of the Kondo-Heisenberg model [31]. In the lately-discovered nickelate superconductors, both seem to exist and determine the properties of the ground state and low-energy excitations [32].
Our proposed correspondence is based on the simple observation that the two energy scales of the HM have the simple relationship ω * ∝ Z and Ω ∝ Z 1/2 with the renormalization factor Z [7,33]. In comparison, the PAM also has two well-known energy scales, namely the indirect and direct hybridization gaps [34][35][36][37], which, in the mean-field approximation, are related to an effective hy-bridizationṼ through ∆ ind ∝Ṽ 2 and ∆ dir ∝Ṽ . This is reminiscent of ω * and Ω in the HM. Since the hybridization gaps have well-defined physical meanings, it is natural to ask if this similarity can provide some insight on the physics of the HM or reveal certain generic features of both models. In this work, we will establish such a correspondence through systematic numerical studies and explore their origin and potential implica-FIG. 1: Illustration of different energy scales identified from the f electron DOS ρ(ω) = −ImG(ω)/π, the real part of the f electron self-energy ReΣ(ω) and its slope ReΣ ′ (ω), and the real part of the Green function ReG(ω) for (a) the HM with U = 3 and (b) the PAM with U = 5.6 and V 2 = 0.4. ω * and ωm are estimated from the kink in ReΣ(ω) and the maximum of ReG(ω), respectively. For the HM, Ω is defined from the maximum of ReΣ ′ (ω). For the PAM, ∆ ind , ∆ dir , and Ω are obtained from the dispersion in Fig. 3. The insets are the enlarged plots for the low energy scales: ω * , ωm, and ∆ ind .
tions on the physics of correlated electrons. We will show that they represent two distinct groups of related energy scales and provide a good characterization of the energy boundaries separating the fully coherent and localized spectral weights with an itinerant but damped crossover region in between. This allows us to construct a tentative phase diagram in the energy space. We argue that the separation of the two energy scales reveals an instrinsic two-stage process of the quasiparticle dynamics for building up the long-range spatial coherence and represents a generic property of correlated electrons on a lattice.
In both models, the Hamiltonians include two parts: The potential energy has the form, where U is the onsite Coulomb interaction and n iσ is the density operator of the local orbital. There are two general ways to delocalize the f electrons. In the HM, one introduces a direct hopping between neighoring lattice sites, giving rise to a kinetic energy, are the creation (annihilation) operator of the f electrons. In the PAM, the f orbitals remain local and the delocalization is achieved by hybridizing with an additional conduction band such that conduction electrons and V is the bare hybridization parameter. For simplicity, we focus on the paramagnetic state in the half-filled one-band case with particlehole symmetry and use the numerical renormalization group (NRG) [38,39] as the impurity solver within the DMFT framework [40][41][42][43]. The local self-energy of the f electrons is calculated using Σ is the noninteracting density of states for a hypercubic lattice with the dimensionality d → ∞. We set t = 1 as the energy unit and choose the logarithmic discretization parameter Λ = 2 and a total number of stored states N s = 800 for the NRG calculations. Figure 1 illustrates how the different energy scales can be identified from the f electron density of states (DOS) ρ(ω) = −ImG(ω)/π, the real part of the self-energy ReΣ(ω) and its ω-derivative ReΣ ′ (ω), and the real part of the Green function, ReG(ω). For the HM, the DOS has a three-peak structure including two Hubbard peaks at higher energy and a quasiparticle peak around zero energy. The larger scale Ω is defined here from the maximum of ReΣ ′ (ω). In Ref. [7], it was defined from the minimum of ReΣ(ω), which, however, only follows the proposed scaling near the Mott transition with a small Z. Nonetheless, both reflect the same physics and separate roughly the quasiparticle peak and the Hubbard peaks in the DOS. The lower scale ω * can be defined from an additional slope change or the kink in ReΣ(ω) that constraints a low-energy region for quasiparticle excitations. A different scale ω m may be introduced from the maximum of ReG(ω). ω * and ω m are of the same origin and related through ω * ≈ ( √ 2−1)ω m for the HM [7]. A third definition may be found from the maximum (ω s ) of the local spin susceptibility as discussed later in Fig. 4. The relationship of these scales can be established if we tune the local Coulomb interaction U and plot their variations with the renormalization factor, The results are summarized in Fig. 2(a). We see that ω * ∝ ω m ∝ Z and Ω ∝ Z 1/2 over a wide range of Z. Thus, ω * and Ω represent two distinct groups of related energy scales. The different definitions originate from the crossover nature of the underlying physics.
For comparison, Fig. 1(b) shows the energy scales in the PAM. The indirect hybridization gap ∆ ind is determined by the DOS gap and coincides with the additional minimum of ReG(ω), while the direct hybridization gap ∆ dir cannot be clearly discerned in these quantities and is best seen in Fig. 3 from the spectral func- The plot implies two separated hybridization bands for the PAM. The indirect gap ∆ ind is the energy difference between the bottom of the upper band and the top of the lower band and marks the boundary for the low-energy gap. Although the PAM is in a Kondo insulating state, the self-energy below ∆ ind behaves as that of a Fermi liquid [46], reflecting the establishment of full coherence on the Kondo lattice [47,48]. By contrast, the direct hybridization gap ∆ dir measures the minimal energy difference between the two hybridization bands, below which the spectra are governed by the itinerant f character. The energy scales ω * and ω m can still be defined similarly as in the HM and contain the flat part of the hybridization bands. We obtain ω * ∝ ω m ∝ ∆ ind . On the other hand, the maximum of ReΣ ′ (ω) changes slightly with varying U and is no longer a meaningful quantity separating the boundary of the itinerant and localized regions. Rather, we find it is better to define the larger energy Ω from the deviation of the hybridization bands from the original conduction bands. In Fig. 3, it is seen that Ω sets roughly the outmost boundary of the itinerant f electron spectral weight. Both Ω and ∆ dir reflect the separation of the hybridized (itinerant) and unhybridized (localized) spectral regions of the f electrons in the energy space.
The scaling results for the PAM are summarized in Fig. 2(b). Similarly, we have ω * ∝ ω m ∝ ∆ ind ∝ Z and Ω ∝ ∆ dir ∝ Z 1/2 over a wide range of Z. The obtained scaling relations link ω * and Ω to ∆ ind and ∆ dir and suggest that they reflect similar physics in the PAM. However, unlike ω * and Ω which are in reality hard to measure, the indirect and direct hybridization gaps have unambiguous physical meaning and can be directly probed in experiment. Their scaling relations in the PAM can be derived from the pole properties of the Green function, G k (ω) = [ω − Σ(ω) − V 2 ω−ǫ k +iη ] −1 . If we make the lowest order approximation, ReΣ(ω) ≈ (1 − Z −1 )ω, the poles are determined by Z −1 ω − V 2 ω−ǫ k = 0. The direct gap is defined as the energy differences of the two poles at ǫ k = 0, while the indirect gap is the energy difference at the band edges (ǫ k = ±D). We have ∆ dir = Z 1/2 V and ∆ ind = ZV 2 /D, confirming their relationship with Z. The relative deviation of the hybridization bands from the conduction band may be estimated to be δ = (ω − ǫ k )/ω = ZV 2 /ω 2 . We have thus Ω ≈ Z 1/2 V /δ 1/2 for a chosen small cutoff δ, at which the f electron spectral weight on the dispersion is also reduced to the order of Zδ. Since all these energy scales diminish at V = 0, their presence reflects the delocalization of the f electrons on the lattice driven by the hybridization with conduction electrons. In this respect, the larger scale Ω constraints the major spectral region of delocalized f electrons, while ω * puts a further restriction for well-established long-range coherence.
Similar analyses may help to understand ω * and Ω in the HM. that ω * marks the boundary of well-defined Landau quasiparticles, above which the dispersion has a different renormalization factor, while Ω separates the lowenergy many-body states and the high energy incoherent region, beyond which the f electron weights are localized. The presence of both ω * and Ω is a reflection of the special cascade structure of ReΣ ′ (ω), which, as shown in the middle panel of Fig. 1(a), has a maximum at Ω but, with lowering energy, decreases first to a plateau (or local minimum) on the hillside before it eventually reaches the valley floor below ω * and ω m . Interestingly, Ω lies roughly at the waterfall structure connecting the quasiparticle and Hubbard bands [49,50]. This poses a question concerning the relation of the two properties. For this, we note that the waterfall is an abrupt change of the pole of the Green function with increasing ǫ k . It can only occur around the inflection point of ω − ReΣ(ω) if one wants to tentatively avoid multiple poles. This immediately implies ReΣ ′′ (Ω) ≈ 0 or a maximum (minimum) in ReΣ ′ (Ω). Thus, Ω defined here is indeed associated with the waterfall structure in the HM. We conclude that the two energy scales correspond to two features of the dispersion: ω * for the kink and Ω for the waterfall, which also reflect the boundaries for the low-energy fully coherent region and the high-energy localized region. The fact that Ω connects two regions with opposite energy dependence in the self-energy implies Z −1 Ω ∼ Ω −1 or Ω ∝ Z 1/2 , similar to that in the PAM.
Further insight on ω * may be obtained if we consider that ∆ ind in the PAM is related to the spin screening, ω * ∝ ∆ ind ∝ T K where T K is the lattice Kondo temperature [51,52]. The effect of spin screening might be best seen from the local susceptibility, χ s (ω) = S z i ; S z i ω with S z i = 1 2 (n i↑ − n i↓ ) [53]. Figure 4 plots the imag- inary part of the local spin susceptibilities for the HM and PAM using the same parameters as in Fig. 1. Both exhibit a maximum at ω s , which, as shown in the inset, varies linearly with Z as those of ω * and ∆ ind . In fact, we find ω s ≈ ω * in both models (Fig. 3). Below ω s the local susceptibility is suppressed due to the spin screening. It is therefore natural to associate the kink with an effective spin screening scale as proposed in Ref. [24]. We should emphasize, however, that the presence of both ω * and Ω is a property of the lattice but not in the usual single-impurity Kondo model where only the Kondo scale exists. This implies that the lattice feedback is crucial for separating ω * and Ω and causing the cascade structure of ReΣ ′ (ω), although DMFT itself does not contain explicit spatial correlations. Thus, all energy scales can be classified into two categories, a lower one ω * ∝ Z and a higher one Ω ∝ Z 1/2 , which separate the f electron spectral weight into different regions of distinct properties. If we start from localized f electrons, we may conclude that their spin screening and consequential delocalization is an energydependent process marked by the two scales. While Ω sets the outmost boundary for the many-body resonant states and covers the full energy range with delocalized f electron spectral weight, ω * is a lower boundary for quasiparticle excitations with well-established long-range coherence. This is common for both PAM and HM and may be best seen if we construct a tentative phase diagram in Fig. 5 in the energy space based on the properties of the f electron spectra. For Z > 0, the ω * ∝ Z line indicates a screening scale and marks the upper boundary of a well-developed coherent region irrespective of the Fermi liquid or Kondo insulating state, while Ω ∝ Z 1/2 marks the lower boundary of an incoherent region with localized f spectral weight. In between, there exists a crossover region with itinerant but strongly damped excitations of the f character. With decreasing Z to zero, the system enters a Mott insulating state for the HM or an orbital-selective Mott state for the PAM, where the many-body resonance is suppressed and all f electrons turn into fully localized magnetic moments.
We remark again that the presence of the two energy scales represents two related but distinct features of the correlated electron physics. It is not just a matter of theory but has real implications and observable consequences in experiment. To see this, we may consider the gradual delocalization of the f electrons with lowering temperature. The two energy scales then turn into two temperature scales, as observed in recent pump-probe experiment on the heavy fermion compound CeCoIn 5 [47]. Quantum Monte Carlo simulations for the PAM have confirmed two distinct stages of hybridization dynamics, namely, the onset of precursor hybridization fluctuations and the eventual establishment of a long-range coherent (Kondo insulating) state [48]. In the HM, the two energy scales are also expected to represent two succeeding stages of the quasiparticle dynamics, namely, a Fermi liquid phase with well-established Landau quasiparticles and a higher crossover region with itinerant but damped quasiparticles (bad metal) [54][55][56]. Along this line of thought, we anticipate that the existence and separation of two energy scales reflect a generic two-stage development of the electronic coherence and is an intrinsic property of correlated electrons in a Mott system. Before the long-range coherence is eventually established, there exists a precursor stage where the electrons become partially delocalized with damped quasiparticle excitations.
To summarize, we report systematic analyses of the energy-scale cascade in the HM and the PAM and reveal a deep connection between the Mott and Kondo lattice physics. This allows us to construct a tentative phase diagram for correlated electrons based on two energy scales marking the upper boundary of the fully coherent lowenergy states and the lower boundary of the incoherent regime with localized spectral weight in the energy space. For the HM, these correspond to the kink and waterfall structures on the dispersion. For the PAM, they are associated with the indirect and direct hybridization gaps. The separation of two energy scales is an intrinsic property of the lattice models and reflects a two-stage dynamical process to build up the lattice coherence of itinerant quasiparticles. Our work clarifies the origin of these energy scales and reveals a potentially basic and generic property of correlated electrons on a lattice.