*3.2. Magnetoresistance*

In the LFL state, the resistivity *ρ*(*<sup>T</sup>*, *B*) ∝ *<sup>A</sup>*2(*B*)*T*2. In the case of common metals, it is well known that *ρ*(*<sup>T</sup>*, *B*) increases with the increasing applied magnetic field *B* and is described by the Kohler's rule, see, e.g., [47]. In contrast, HF metals exhibit decreasing resistivity in magnetic fields when the metal in question transits from the NFL behavior to the LFL one, see, e.g., [48,49]. The *A*(*B*) coefficient, being proportional to the quasiparticle —quasiparticle scattering cross section is found to be *A* ∝ (*M*∗(*B*))2, as follows from Equation (11) [15,48]. Taking into account Equation (11), we obtain

$$A(B) \simeq A\_0 + \frac{D}{B - B\_{c0}},\tag{16}$$

where *A*0 and *D* are fitting parameters. Figure 5 displays experimental data for *A*(*B*) collected on two HF metals: YbRh2Si2 [48] and Tl2Ba2CuO6<sup>+</sup>x [49]. The solid curves represent our calculations, and the inset demonstrates that the well-known Kadowaki– Woods ratio [50] is conserved [48]. This experimental result is in good agreemen<sup>t</sup> with Equations (15) and (16).

**Figure 5.** The charge transport coefficient *A*(*B*) as a function of magnetic field *B* obtained in measurements on YbRh2Si2 [48] and Tl2Ba2CuO6<sup>+</sup>x [49]. The different field scales are clearly indicated. The solid curves represent our calculations based on Equation (16) [15]. The inset (adapted from [51]) shows that *A*(*B*) ∝ *χ*(*B*)<sup>2</sup> ∝ *γ*20 ∝ (*C*/*T*)2.

To further elucidate the scaling of *<sup>A</sup>*(*B*), we rewrite Equation (16) in the re-scaled variables *A*/*A*0 and *B*/*Bc*0. Such a recasting immediately reveals the scaling nature of the behavior of these two substances. Both of them are driven to common QCP related to the FCQPT and induced by the application of a magnetic field. As a result, Equation (16) takes the form

$$\frac{A(B)}{A\_0} \simeq 1 + \frac{D\_N}{B/B\_{c0} - 1'} \tag{17}$$

where *DN* = *<sup>D</sup>*/(*<sup>A</sup>*0*Bc*0) is a constant. From Equation (17), it is seen that upon applying the scaling to both coefficients *A*(*B*) for Tl2Ba2CuO6<sup>+</sup>x and *A*(*B*) for YbRh2Si2, they are reduced

to a function depending on the single variable *B*/*Bc*0, thus demonstrating the universal behavior. To support Equation (17), we plot both dependencies in the reduced variables *A*/*A*0 and *B*/*Bc*0 in Figure 6; the universal scaling nature of the coefficients *A*(*B*) of these two substances is immediately revealed. We note that the negative magnetoresistance of both Tl2Ba2CuO6<sup>+</sup>x and YbRh2Si2 results from diminishing *A*(*B*) under the application of a magnetic field as follows from Equation (11).

**Figure 6.** Normalized coefficient *<sup>A</sup>*(*B*)/*A*0 1 + *DN*/(*y* − 1) given by Equation (17) as a function of a normalized magnetic field *y* = *B*/*Bc*0 shown by squares for YbRh2Si2 and by circles for high-*Tc* Tl2Ba2CuO6<sup>+</sup>x. *DN* is the only fitting parameter.

The scaling behavior of the longitudinal magnetoresistance (LMR) collected on YbRh2Si2 [48] confirms our above conclusions. This scaling behavior is displayed in Figure 7. Clearly, our calculations are in good agreemen<sup>t</sup> with the experimental data. Thus, the fermion condensation theory explains both the negative magnetoresistance and the crossover from the NFL behavior to the LFL one under the application of magnetic fields.

**Figure 7.** Magnetic field dependence of the longitudinal normalized magnetoresistance LMR versus a normalized magnetic field. The LMR *ρN* was extracted from the LMR of YbRh2Si2 at different temperatures [48] listed in the legend. The solid line represents our calculations [15]. The arrows show the NFL behavior at *B T*, the inflection point and the LFL behavior at *B T*.

## *3.3. Schematic Phase Diagram*

Based on Equation (12) and Figures 3 and 4, we can construct the schematic *T* − *B* phase diagram of HF metals [52], reported in Figure 8. We assume here that at *T* = 0

and *B* = *Bc*0 the system is approximately located at the FCQPT. In the case of *Bc*0 = 0, the system is located at the FCQPT without tuning. At fixed temperatures, the system is driven by the magnetic field *B* along the horizontal arrow (from the NFL to the LFL parts of the phase diagram). In turn, at fixed *B* and increasing *T*, the system moves from the LFL to the NFL regime along the vertical arrow. The hatched area indicating the crossover between the LFL and the NFL phases separates the NFL state from the slightly polarized paramagnetic LFL state. The crossover temperature *TM*(*B*) is given by Equation (13).
