Matching DMFT calculations with photoemission spectra of heavy fermion insulators: universal properties of the near-gap spectra of SmB6

Paramagnetic heavy fermion insulators consist of fully occupied quasiparticle bands inherent to Fermi liquid theory. The gap emergence below a characteristic temperature is the ultimate sign of coherence for a many-body system, which in addition can induce a non-trivial band topology. Here, we demonstrate a simple and efficient method to compare a model study and an experimental result for heavy fermion insulators. The temperature dependence of the gap formation in both local moment and mixed valence regimes is captured within the dynamical mean field (DMFT) approximation to the periodic Anderson model (PAM). Using the topological coherence temperature as the scaling factor and choosing the input parameter set within the mixed valence regime, we can unambiguously link the theoretical energy scales to the experimental ones. As a particularly important result, we find improved consistency between the scaled DMFT density of states and the photoemission near-gap spectra of samarium hexaboride (SmB6).


ISTIC TEMPERATURES
The basic understanding of the renormalized f states is substantially established in the local moment regime of the single impurity Anderson model (SIAM). When the local moments of localized f states are antiferromagnetically coupled with conduction electron states, they become quasiparticle states in the Landau Fermi liquid scheme. This local Fermi liquid phase emerges as a crossover, which smoothly happens without any phase transition, i.e. a crossover [1][2][3]. Moreover, the physical properties follow universal functions on one energy scale, which is the Kondo temperature T K . It implies that the physical properties can be written in terms of T /T K [4]. Other crossover parameters for the paramagnetic ground states in the periodic Anderson model (PAM) have also been studied when the screened local moments, which are positioned on a lattice, form coherent Bloch-like states [1,[5][6][7][8][9][10].
The coherent states can be clearly identified below the coherence temperature T coh , which is typically smaller than T K [7,8]. Because the PAM may result in different ground states and phases, there are different characteristic temperatures [3]. However, for a simple Fermi liquid quasiparticle state, the concept of T coh describes the emergence of the coherent quasiparticle band, especially for the mixed valence regime [1,2,10]. In this regime, the local moments are not clearly defined and charge fluctuations play an important role alongside spin fluctuations. Due to the charge fluctuations, spectral weight is redistributed across the gap, which has been experimentally and theoretically observed for SmB 6 [10][11][12].

II. EXPERIMENTAL PREREQUISITES
Note that the comparison of DMFT and PES results relies on the following three points: First, we have restricted the photon energy hν > 40 eV to emphasize the 4f character in the photoemission spectra, which takes into account the dominant spectral weight of f states near gap region in the model study. Second, we have considered only the state 6 H 5/2 appearing at the lowest energy below E F . Experimentally, the next lowest state 6 H 7/2 , which appears at the energy of E-E F ≈ −0.18 eV [13][14][15][16], shows much less changes in both shape and intensity than 6 H 5/2 , and thus shall be neglected for the contribution to low-temperature behavior. And, third, to make connection with the DMFT calculation, we have chosen model   [10,17] for details). Two main changes appear as one crosses the temperature scale T N , Fig. S1 (a) and (b). One is that the gap is getting more pronounced, and the other is that the f and d spectra becomes more similar to each other.
Note that the f features near w/t ∼ 0 is apparent even at high T , so it is not buried under the background intensity (incoherent spectral weight). Thus, at high T the f bands in the mixed regime is not the same as the Kondo resonance in the local moment regime, which completely disappears at high temperatures.
Close to E F , the f features are getting blurred at T > T N (Fig. S1 (a, right)), whereas they have reduced intensity at T < T N (Fig. S1 (b, right)). The reduced spectral weight in the gap region has been already observed in other PES investigations [18][19][20][21]. In addition, the band dispersions of heavy f states and itinerant d states change as a function of the temperature.
Above T N , d and f spectral weights are strongly dispersing and flat bands, respectively, whereas below T N they show very alike band dispersion. This is due to hybridization, which gives rise to the character mixing of f and d states. Hence, if one can selectively measure one of the orbital character by setting adequate experimental conditions, it will be possible to observe the change in the band dispersion with temperature. Moreover, we note that in the model calculations the hybridization between the f and d states is given by V V V (k k k)·σ σ σ, where σ σ σ is a vector of Pauli spin matrices. Time reversal symmetry requires V V V (k k k) to be odd, such that the hybridization vanishes at time reversal momenta. This explains in particular the loss of spectral weight at the high symmetry points (0, 0), (0, π) and (π, π) at low temperatures.