**1. Introduction**

Strongly correlated Fermi systems such as heavy-fermion metals, graphene, and high-*Tc* superconductors exhibit the non-Fermi-liquid (NFL) behavior. Theoretical predictions [1–4] and experimental data collected on many of these systems show that at low temperatures a portion of their excitation spectrum becomes approximately dispersionless, giving rise to so-called flat bands and high-*Tc* superconductivity, see, e.g., [1,5–12]. The emergence of flat bands at low *T* indicates that the system is close to a special quantum critical point, namely a topological fermion condensation quantum phase transition (FCQPT), leading to the formation of flat bands dubbed the fermion condensation (FC). The flat bands are formed by the Landau interaction between quasiparticles, while a frustration and van-Hove singularities can facilitate the process. Flat bands have notable features, e.g., raising temperatures, and the superconducting phase transition makes them upward tilted [3,4,13–17]. These observations have been predicted [3,4,14,15,17] and are in accordance with experimental data, see, e.g., [13,16,18]. Moreover, the FC theory allows one to qualitatively and quantitatively evaluate the NFL and Landau Fermi liquid (LFL) behaviors of strongly correlated

**Citation:** Shaginyan, V.R.; Msezane, A.Z.; Japaridze, G.S. Peculiar Physics of Heavy-Fermion Metals: Theory versus Experiment. *Atoms* **2022**, *10*, 67. https://doi.org/10.3390/atoms 10030067

Academic Editors: Anatoli Kheifets, Gleb Gribakin and Vadim Ivanov

Received: 17 May 2022 Accepted: 21 June 2022 Published: 23 June 2022

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Fermi systems, and explain the crossover from one another [1,2,4,15,19,20]. We note that in our review we analyze strongly correlated Fermi systems formed by and located near their topological FCQPT and consider experimental observations that are collected on such systems. Consideration of systems located relatively far from their topological FCQPT is possible within the framework of the FC theory as well, see, e.g., [15,19,20]. We review and explain recent prominent experimental results that to our best knowledge have not found alternative explanations and that strongly sugges<sup>t</sup> that the topological FCQPT is a generic feature of many strongly correlated Fermi systems, being the universal cause of their non-Fermi-liquid behavior, and the fermion condensation theory is able to explain the extraordinary behavior of strongly correlated Fermi systems.

In our review we consider exciting experimental facts such as:

(1) Recent experimental findings of linear dependence on temperature *T* of the resistivity *ρ*(*T*) ∝ *T*, collected on high *Tc* superconductors (HTSC), graphene, heavy fermion (HF) and common metals reveal that the scattering rate 1/*τ* of charge carriers reaches the Planckian limit 1/(*<sup>T</sup>τ*) = *kB*/¯*h*, with 1/*τ* being the scattering rate and *kB* and *h*¯ being the Boltzmann and Plank constants, respectively [21–24]. Within the framework of the FC theory, we show that the quasi-classical physics is still applicable for describing the linear *T*dependence of resistivity of strongly correlated metals at their quantum criticality since flat bands, forming the quantum criticality, generate transverse zero-sound mode with the Debye temperature *TD* [25]. At *T* ≥ *TD*, the mechanism of the linear *T*-dependence is the same in both ordinary metals and strongly correlated ones and is represented by the electron– phonon scattering. Therefore, it is the electron–phonon scattering at *T* ≥ *TD* that leads to the near material-independence of the lifetime *τ* that is expressed as 1/(*τ<sup>T</sup>*) ∼ *kB*/¯*h*. As a result, we describe and explain recent exciting experimental observations of universal scattering rate related to the linear *T*-dependent resistivity of a large number of both strongly correlated Fermi systems and common metals [21–24]. We show that the observed scattering rate is explained by the emergence of flat bands formed by the topological FQCPT rather than by the so-called Planckian limit at which the assumed Planckian scattering rate occurs [25,26]. The Planckian limit then has to occur in common metals. Moreover, in magnetic fields, HF metals transit from the NFL to LFL behavior and *ρ*(*T*) ∝ *T* vanishes, being replaced by the LFL behavior *ρ*(*T*) ∝ *<sup>A</sup>*2*<sup>T</sup>*2, with *A*2 as the temperature-independent coefficient.

(2) Recent observations of the linear *T*-dependence, *ρ*(*T*) ∝ *T*, at low temperatures, *T* → 0, relate the slope of the linear *T*-dependent resistivity *ρ* to the London penetration depth *λ*0, indicating a universal scaling property

$$\frac{d\rho}{dT} \propto \lambda\_0^2 \tag{1}$$

for a large number of strongly correlated high-temperature superconductors [27]. This scaling relation spans several orders of magnitude in *λ*0, attesting to the robustness of the empirical law (1) [28].

(3) We also analyze recent challenging experimental findings of tunneling differential conductivity *dI*/*dV* = *<sup>σ</sup>d*(*V*) as a function of the applied bias voltage *V*, collected under the application of magnetic field *B* on the twisted graphene and the archetypical heavy-fermion metals YbRh2Si2 and CeCoIn5 [5,29,30]. We explain the emergence of the asymmetrical part Δ*σd* = *<sup>σ</sup>d*(*V*) − *<sup>σ</sup>d*(−*<sup>V</sup>*) and demonstrate that Δ*σd* vanishes in magnetic fields as predicted [31].

(4) We consider the recent outstanding experimental observation of the density *ns* of superconducting electrons that turns out to be much less than the total density *<sup>n</sup>ρ* of electrons at *T* → 0 [32] as predicted [33].

(5) We show that the transition temperature *Tc* is proportional to the superconducting coupling constant *g*,

$$T\_c \propto \mathcal{g}.\tag{2}$$

This fact, see Equation (2), leads to creating high-*Tc* superconductors [1,5–12]. This observation is supported by special features of high-*Tc* superconductivity based on flat bands, namely that *Tc* is proportional to the Fermi velocity *VF* ∝ 1/*Ns*(0) *VF* ∝ *Tc*, rather than *Ns*(0) ∝ 1/ *VF* ∝ *Tc* as stated in standard BCS-like theories [13,16] as predicted [17].

Our results are in good agreemen<sup>t</sup> with experimental data and demonstrate that the topological FCQPT is an intrinsic feature of strongly correlated Fermi systems, and the FC theory can be viewed as the universal agen<sup>t</sup> explaining the physics of strongly correlated Fermi systems.
