Figure 1

(Color online) Momentum density, $n\left(\mathbf{k}\right)$ as calculated using Eq. (1), is shown for various dopings for NCCO in (a)–(c) and for LSCO in (d)–(f).Reuse & Permissions

Figure 2

(Color online) (a) $n\left(\mathbf{k}\right)$ for NCCO and LSCO at several representative dopings with dashed lines showing the discontinuous jump at $k_F$ (highlighted in inset). Gold arrows mark features from the shadow bands which cross $E_F$. (b) $Z_{\omega }$ at $E_F$ are shown along the antinodal (red) and nodal (blue) direction for NCCO. Filled (open) symbols give the main (shadow) bands, compared with their corresponding analytical approximation $Z_{\omega }^{\text{SDW}}$ plotted by solid (dashed) line of same color. ARPES result (green) is extracted from Ref. 16. (c) Same as (b) but for LSCO along the nodal direction. These results are compared with PRT’s calculations for hole doping (Ref. 27) and ARPES results (Ref. 3) for LSCO along the nodal direction. All the experimental data and PRT data in (b) and (c) are normalized to highlight their doping evolution. Brown dashed line shows that if there is nanoscale phase separation in LSCO then $Z_{\omega }$ would scale linearly with doping in the extreme underdoped region.Reuse & Permissions

Figure 3

(Color online) (a) Fermi velocity $v_F$ along the antinodal direction in NCCO and nodal direction in LSCO is compared with ARPES data on LSCO (Refs. 3, 39) and with PRT’s calculation (Ref. 27). The blue dashed (solid) line gives the corresponding LDA (MFT) results for LSCO. (b) Dispersion renormalization $Z_d$ for NCCO along antinodal direction, compared with an approximate analytical formula for the dispersion renormalization $Z_d^{\text{SDW}}$ (solid line). (c) Same as (b) but for LSCO along the nodal direction. The results are compared with experimental data (Ref. 1).Reuse & Permissions

Figure 4

(Color online) (a) Specific-heat coefficient $\gamma \left(x\right)$ for different theoretical calculations (various lines), and the experimental results on NCCO (red squares) (Ref. 7) and PLCCO (red circles) (Ref. 8). (b) Same as (a) but for LSCO [red squares (Ref. 3) and triangles (Ref. 9)]. Results are compared with $n_s$ [filled green squares (Ref. 3) and open green squares (Ref. 39)] and $\chi$ (blue) (Ref. 43), all normalized at the VHS. The red dashed line shows that in underdoped LSCO $\gamma$ scales linearly with doping. Theoretical $\gamma$ has been scaled by a factor of 1.1, consistent with a weak electron-phonon renormalization.Reuse & Permissions

Figure 5

(Color online) (a) Average renormalization factor $Z$ decreases linearly with doping, very much like $Z_{\omega }^0$ in Figs. 2b, 2c. (b) Our computed doping dependence of selfconsistent values of $U$ and $ZU$ is compared with earlier mean-field results (Refs. 31, 58).Reuse & Permissions

Figure 6

(Color online) (a) Spectral intensity as a function of $\omega$ along the high-symmetry lines for underdoping (at temperature $T=0$). Blue to red color map gives the minimum to maximum intensity. The yellow dashed line gives the underlying LDA dispersion where the gold lines represent the renormalized magnetic bands (${\Sigma }_0$ dressed). (b) The QP-GW DOS (blue lines) is compared with ${\Sigma }_0$-dressed DOS (red line) calculated at $T=0$. The green lines show the DOS without the vertex correction.Reuse & Permissions

Figure 7

(Color online) Real and imaginary part of the self-energy as expanded in the tight-binding form of Eq. (A11).Reuse & Permissions

Figure 8

(Color online) (a) The spectral intensity of NCCO at $x=0$, plotted (in logarithmic scale) along the high-symmetry lines, is compared with variational calculations (Ref. 26) in (b). [(c)–(f)] The computed DOS at various dopings are compared with the corresponding QMC results (blue lines) for $x=0.0$ (Ref. 59) and for $x=0.05–0.20$ (Ref. 6).Reuse & Permissions