Figure 4

Calculated versus measured resistivity and carrier density. (a) To facilitate the comparison between calculated (lines) and measured sheet resistance (\(\rho _{\Box }\sim A_2T^2\) – (opaque symbols)⁷ or in the crossover regime \(\rho _{\Box }\sim A_1T^1 + A_2T^2\) – (shaded symbols)⁴³) a temperature independent quantity \(\tau \rho _{\Box } \sim A_2/C_2\) is displayed, as a function of doping, for all three discussed compounds (see the Methods Section "Comparison with experimental data: resistivity" for details). Dashed and dotted lines for LSCO correspond to calculations with the same \(f_g\) as presented by dashed and dotted lines in Fig. 3b. The inset shows an extended doping range to \(p=0\) on a logarithmic scale for clarity. A small kink in the calculated doping dependence for LSCO at \(p\sim 0.18\) (dashed line) coincides with the Lifshitz-transition of the underlying FS. (b) Full and dashed lines show \(n_{\text{eff}}\) as inferred from resistivity measurements, which is the only input parameter for the performed calculation. In case of LSCO, an additional dotted line indicates \(n_{\text{eff}}\) obtained by adjusting \(f_g\) (i.e., arc-length) for a better fit of the Hall data. For Hg1201 and Tl2201 \(n_{\text{eff}}\) (lines) and \(n_{\text{H}}\) (symbols) coincide. This is not the case for LSCO, where \(n_{\text{H}}\) diverges at the Lifshitz transition. However, \(n_{\text{eff}}\) shows a similarly smooth crossover in LSCO as it does in Hg1201 and Tl2201. The calculated \(\sigma _{xx}\) [Eq. (3)] strongly depends on \(v_F\), whose value is usually not controlled in tight-binding fits to ARPES data. Therefore, normalization factors \(f_{\text{norm}}\) have been applied to \(\tau \rho _{\Box }\) of LSCO and Tl2201. The details of this normalization are in Supplementary Information 2.