Figure 1

(Color online) (a) We show the theoretical mean value of the staggered magnetization $S^z=\frac{1}{2}\left(n_{\uparrow }-n_{\downarrow }\right)$ obtained for LSCO and NCCO by both single-site DMFT $\left(\text{LDA}+\text{DMFT}\right)$ and cluster cellular DMFT $\left(\text{LDA}+\text{cDMFT}\right)$ of the three-band theory, and DMFT of the six-band theory. Experimental values $M\left(\delta \right)/M_0\left(M_0=M\left(\delta =0\right)\right)$ for NCCO (Ref. 51) and LSCO (Ref. 52) are also shown, and for comparison with DMFT, we assume $M_0=M_{\text{DMFT}}\left(\delta =0\right)$, where $M_{\text{DMFT}}$ is the magnetic moment at 0 doping obtained by single-site DMFT. The general trend compare well to our data, and for NCCO we obtain a qualitative agreement for the critical doping. For LSCO we find that the magnetization vanish at large doping within the single-site DMFT approximation but when the possibility of local singlet is allowed (cDMFT) the range of stabilization of the long-range order is much reduced. Note that the theoretical magnetic moment was obtained at constant temperature $T=89\text{K}$, and experimentally at 2 K for LSCO and 8 K for NCCO. (b) Magnetization versus temperature in the parent compounds of NCCO and LSCO obtained by single-site DMFT for the three-band theory. The extracted mean-field Neel temperature is about 1500 K for NCCO and 1000 K for LSCO.Reuse & Permissions

Figure 2

(Color online) We show the quasiparticle weight $Z$ for the three-band description of LSCO and NCCO. The quasiparticle weight of the PM is finite at integer filling for NCCO, which is a signature that the paramagnetic state of NCCO is a metal. In the ordered state of NCCO (AF), the quasiparticle weight is estimated by the specific heat of the $A$ and $B$ magnetic sublattices ${\gamma }_i\propto \frac{{\rho }_i\left({ϵ}_F\right)}{Z_i}$ and $i=A,B$, and the total quasiparticle weight $Z$ is given by $Z={\sum }_i{\rho }_i\left({ϵ}_F\right)/{\sum }_i\frac{{\rho }_i\left({ϵ}_F\right)}{Z_i}$. The specific heat $\gamma$ of the AF and PM states of NCCO are shown in the inset.Reuse & Permissions

Figure 3

(Color online) Frequency-dependent spectral weight $A\left(k,w\right)$ obtained by $\text{LDA}+\text{DMFT}$ of NCCO at (a) integer filling, (c) 10% and (e) 20% electron doping for NCCO. $A\left(k,w\right)$ was obtained along the usual path $\Gamma \text{-M-K-}\Gamma$ in the Brillouin zone (see inset of d). The solid lines are the rigid LDA bands. The partial density of states of the $d$ and $p$ orbitals are shown in (b), (d), and (f). The density of states is showing the UHB, the ZR singlet and the QP for the doped compounds. (a) At integer filling, we find for NCCO that the indirect gap is $\approx 1.2\text{eV}$, between $\left(\pi ,0\right)$ and $\left(\pi /2,\pi /2\right)$, as shown by the diagonal arrow, and the direct gap is about 1.5 eV, as indicated by the vertical arrow. (d) At 10% doping we observe a splitting of the quasiparticle peak due to magnetism. The splitting of the quasiparticle peak is associated with the magnetic pseudogap at $\left(\pi /2,\pi /2\right)$ in panel (c). The horizontal arrows pointing from (c) to (d) are guide to the eyes. The optical transitions at 10% doping occur around $\left(\pi ,0\right)$ within the quasiparticle band, as indicated by the vertical arrow. (e) At 20% doping, magnetism is destroyed and the pseudogap at $\left(\pi /2,\pi /2\right)$ is closed. The quasiparticle peak in panel (f) is clearly related to the spectral weight close to the Fermi surface at $\left(\pi ,0\right)$, as indicated by the horizontal arrow.Reuse & Permissions

Figure 4

(Color online) Fermi-surface maps obtained by DMFT calculations in the ordered state at 10% doping, (a) at the Fermi energy and (b)–(d) at lower energies ranging from $-0.05$ to $-0.15\text{eV}$. The Fermi-surface map at the Fermi level is shaped by the presence of magnetism, whereas the arc in the energy map at energy $-0.15\text{eV}$ is also present in the paramagnetic calculations. (e) comparison of $A\left(\omega \right)$ for a fixed $k$ point $k=\left(3\pi /4,\pi /4\right)$ (shown in the inset) obtained theoretically (lower curve) and experimentally from Ref. 60 (upper curve). The peak at $-0.03\text{eV}$ is due to magnetism, and the peak at lower energy $-0.2\text{eV}$ is associated to the arcs seen in (d).Reuse & Permissions

Figure 5

(Color online) Side-by-side comparison of $A\left(k,\omega \right)$, along the path as depicted in the inset, obtained theoretically (upper row) and experimentally (lower row), (a)–(c) at 15% electron doping from Ref. 61, and (d) at 13% doping from Ref. 60. (e) Comparison between DMFT (left side) and experiments of Ref. 62 (right side), along the nodal cut of the Brillouin zone at 17% electron doping. The agreement between DMFT and experiments is remarkable. The vertical dashed lines are guide to the eyes to illustrate the presence of a sharp kink in the dispersion (waterfall).Reuse & Permissions

Figure 6

(Color online) Side-by-side comparison of the Fermi surface obtained experimentally (reproduced from Ref. 63) and obtained by single-site DMFT calculations for NCCO in the ordered state. (a) Experimental results for doping 4%, (b) for 10% electron doping, and (c) 15% electron doping. The results are compared to DMFT calculations for similar dopings (d)–(g). The Fermi surface at low doping (a) is centered around $M=\left(\pi ,0\right)$ and is moving toward the Fermi arc shape (g) when magnetism is destroyed.Reuse & Permissions

Figure 7

(Color online) (a) Frequency-dependent spectral weight $A\left(k,w\right)$ obtained by $\text{LDA}+\text{DMFT}$ of a six-band model description of the parent compound of LSCO. (b) Partial density of states of the $d_{x2-y2}$, $d_{3z2-r2}$, $p_{\left(x,y\right)}$, and $p_{\pm z}$ orbitals. We observe a direct gap of 1.8 eV in LSCO. Notice that the spectral weight is very incoherent close to the Fermi energy in the lower band. (c) Partial density of states on a larger energy scales. The LHB is located at a very low energy $-10\text{eV}$, and the UHB is also shown. The $d_{3z2-r2}$ and $p_{\pm z}$ orbitals have a strong weight between $-4$ and $-1\text{eV}$.Reuse & Permissions

Figure 8

(Color online) Fermi surface obtained by single-site DMFT calculations for the three-band description of LSCO in the ordered state. The Fermi surface at low doping [(a) and (b)] is centered around $M=\left(\pi /2,\pi /2\right)$, and is moving toward the Fermi arc shape [(c) and (d)] when magnetism is destroyed.Reuse & Permissions

Figure 9

(Color online) [(a) and (c)] Frequency-dependent spectral weight $A\left(k,w\right)$ obtained by $\text{LDA}+\text{DMFT}$ of a six-band model description of doped LSCO. (b) and (d) Partial density of states of the $d_{x2-y2}$, $d_{3z2-r2}$, $p_{\left(x,y\right)}$, and $p_{\pm z}$ orbitals. (a) and (b) are obtained in the ordered state. (e) and (f) are obtained within a three-band description of doped LSCO and are shown for comparison. The optical transitions occur at different energies for the three-band and six-band theories (see vertical arrows in c and e). Also note that the weight of the $d_{3z2-r2}$ and $p_{\pm z}$ orbitals in (b) and (d) close to the Fermi level is consequent, which justifies that a six-band description is necessary. The solid lines in (a), (c), and (e) are the rigid LDA bands. (g) $A\left(k,w\right)$ along the nodal cut of the Brillouin zone for the paramagnetic six-band theory at 10% doping, compared side by side with (h) experimental data of Ref. 66. The dashed line are guide to the eyes to illustrate the sharp kink (waterfall) in the dispersion.Reuse & Permissions

Figure 10

(Color online) Spectral functions for the six-band theory of LSCO in (a) the parent compound and at (b) 10% and (c) 20% doping and (d) 30% doping. The $d_{x2-y2}$, $d_{3z2-r2}$, $p_{\left(x,y\right)}$, and $p_{\pm z}$ components are shown. Note that the LHB at $-9\text{eV}$ is both of $d_{x2-y2}$ and $d_{3z2-r2}$ character. At finite doping (b) and (c), there is a QP close to the Fermi energy. Note that the $d_{3z2-r2}$ and $p_{\pm z}$ weight gets closer to the Fermi level for higher doping (c). These latter orbitals will therefore be important to describe high doping and/or high energy spectroscopy. (e) Ratio of the hole occupancy of the $d_{3z2-r2}$ and $d_{x2-y2}$ orbitals (squares) and of the $p_{\pm z}$ and $p_{\left(x,y\right)}$ orbitals (circles) obtained by DMFT, and (f) experimental data are also shown for comparison (Ref. 67).Reuse & Permissions

Figure 11

(Color online) (a) Theoretical optical conductivity of NCCO in ${\Omega }^{-1}{\text{cm}}^{-1}$ at integer filling (red line). We find that NCCO has an optical gap of about 1.5 eV, which is larger than the direct gap $\approx 1.2\text{eV}$. We also show data for doped NCCO (blue line). Note that for doping smaller than 20%, we observe a small peak at a small energy $\approx 0.035\text{eV}$ (see the inset). We also observe a second peak at larger energy scale $\approx 0.2\text{eV}$, which corresponds to optical transitions within the Zhang-Rice singlet [see Fig. 3c]. For comparison we also show the infrared optics of Refs. 72, 73 (dashed black line). (b) Optical conductivity of the six-band theory of LSCO for the parent compound (red line) and doped LSCO. For comparison we also show experimental data of Ref. 74 (short and long-dashed lines). (c) Optical conductivity of the three-band theory of LSCO in the parent compound (red line) and doped LSCO (yellow line), and experimental data of Ref. 74 (short and long-dashed lines). The vertical arrow in panel (c) emphasizes the disagreement between theory and experiments. Note also that at finite doping there is a peak at 0.8 eV in the optical conductivity of the three-band theory, which is absent from experimental data. This peak is related to the optical transitions shown in Fig. 9e (see the vertical arrows). All calculations were done in the ordered state.Reuse & Permissions

Figure 12

(Color online) (a) Theoretical optical conductivity of paramagnetic NCCO at 10% doping is shown. In the paramagnetic state of NCCO, there are no peaks at 0.035 and 0.2 eV which are observed in the magnetic state of NCCO at same doping [see Fig. 11a]. (b) We show the $k$-dependent spectral weight $A\left(k,\omega \right)$ for NCCO at 10% doping. (c) We show the corresponding total density of states. In the paramagnetic state of NCCO there is no splitting of the quasiparticle peak QP [see Fig. 3d].Reuse & Permissions

Figure 13

(Color online) We show the dimensionless integrated optical conductivity $N_{eff}$ for the three-band study of $\text{LDA}+\text{DMFT}$ done on LSCO (Ref. 31) and NCCO, obtained with a cutoff $\Lambda =1.2\left(\Lambda =1.5\right)$ for NCCO (LSCO). Experimental data for LSCO [red circles (Ref. 74) and yellow squares (Ref. 78)] and NCCO (Ref. 79) (open diamonds) are shown. The increase in $N_{eff}$ is similar for both compounds. The dashed line indicates the results obtained by the Hartree-Fock approximation. We also show for comparison results obtained within the six-band theory (green squares). The agreement between theory and experiments is quantitative.Reuse & Permissions

Figure 14

(Color online) (a) Three-band theoretical normalized variation in $N_{eff}$, $\Delta N_{eff}=N_{eff}\left(T\right)-N_{eff}\left(T=89\text{K}\right)$, at 10% electron doping for NCCO (red circles, left scale). $N_{eff}$ is reaching a maximum value when the magnetization (orange squares, right scale) is destroyed by the thermal fluctuations. The decrease in $N_{eff}$ at low temperature can be explained by the opening of a pseudogap in the ordered phase. The data were obtained by single-site DMFT calculations in the ordered phase. The dashed line corresponds to experiments [see Fig. 7c of Ref. 79), where they measured $N_{eff}\left(\Lambda =0.03\text{eV}\right)$ (contribution due to the Drude peak) and $N_{eff}\left(\Lambda =0.3\text{eV}\right)-N_{eff}\left(\Lambda =0.2\text{eV}\right)$ (contribution due to the pseudogap). The dashed line corresponds to the sum of these two contributions. (b) Three-band single-site DMFT of the ordered phase of LSCO is shown for 5% doping. Note that the trend of $N_{eff}$ is opposite between LSCO AND NCCO, which is a signature that NCCO is a Slater insulator and LSCO a Mott insulator.Reuse & Permissions

Figure 15

(Color online) Three-band theoretical temperature dependence of the (a) kinetic energy ${\mathcal{H}}_t$ [Eq. (1)] and (b) Coulomb enery ${\mathcal{H}}_U$ [Eq. (2)] of NCCO at 10% doping, and (c) Kinetic and (d) Coulomb energy of LSCO at 5% doping. The red area highlights the temperature region where the solution is magnetic (AF), and the solution is PM in the blue area. (a) and (b) are showing that there is a kinetic-energy optimization when LSCO becomes an antiferromagnet, which is proper to the Mott insulator, and (c) and (d) show that NCCO is a typical Slater insulator, which optimizes the Coulomb (local on-site repulsion) energy when it becomes an antiferromagnet, at the expense of a worse kinetic energy. This is consistent with the theoretical $N_{eff}$ shown in Fig. 14f. We show the imaginary part of the self-energy at zero frequency $\text{Im}\Sigma \left(\omega =0\right)$ in function of the temperature. The scattering rate of LSCO $\lambda =-\text{Im}\Sigma \left(\omega =0\right)$ (red lines) is strongly reduced for $T<T_{Neel}$ ($T_{Neel}$ is highlighted by the red vertical arrow), which is consistent with the kinetic-energy reduction observed for LSCO at low temperature [Fig. 15c]. NCCO (blue lines) is showing a small scattering rate weakly dependent on the temperature, which is consistent with NCCO being more metallic than LSCO. All calculations were obtained by CTQMC in the ordered state.Reuse & Permissions

Figure 16

(Color online) Three-band theoretical energy differences between the antiferromagnetic and the paramagnetic phases. We show the doping dependence of the (a) kinetic energy ${\mathcal{H}}_t$ [Eq. (1)] and (b) Coulomb energy ${\mathcal{H}}_U$ [Eq. (2)] of NCCO and LSCO at fixed temperature $T=89\text{K}$. There is a kinetic-energy optimization when LSCO becomes an antiferromagnet, which is proper to the Mott insulator, and NCCO is a typical Slater insulator, which optimizes the Coulomb (local on-site repulsion) energy when it becomes an antiferromagnet, at the expense of a worse kinetic energy. This is consistent with the temperature dependence of the theoretical $N_{eff}$ shown in Fig. 14 and with the temperature dependence of the kinetic and Coulomb energies 15. All calculations were obtained by CTQMC.Reuse & Permissions

Figure 17

(Color online) (a) We show the jump in the chemical potential $\delta \mu$ in the ordered state of NCCO (blue circles) for other values of the charge-transfer energy ${ϵ}_d-{ϵ}_p$. The results are obtained for the three-band single-site DMFT. For comparison, we also show the jump in the chemical potential of the paramagnet (red squares). There is a quantum critical point ${\Delta }_{c2}$ for the paramagnetic state, with respect to the charge-transfer energy, corresponding to the metal to charge-transfer insulator transition. The physical value obtained by $\text{LDA}+\text{DMFT}$ for ${ϵ}_d-{ϵ}_p$ places NCCO below the boundary. (b) Density of states of the paramagnetic state of NCCO. The quasiparticle peak close to the Fermi energy (QP) is a signature that NCCO is paramagnetic metal. c) Density of states of the ordered state of NCCO.Reuse & Permissions

Figure 18

(Color online) (a) We show the jump in the chemical potential $\delta \mu$ in the ordered state of LSCO (blue circles) for other values of charge-transfer energy ${ϵ}_d-{ϵ}_p$. The results are obtained for the three-band single-site DMFT. For comparison, we also show the jump in the chemical potential of the paramagnet (red squares). There is a quantum critical point ${\Delta }_{c2}$ for the paramagnetic state, with respect to the charge-transfer energy, corresponding to the metal to charge-transfer insulator transition. The physical value obtained by $\text{LDA}+\text{DMFT}$ for ${ϵ}_d-{ϵ}_p$ place LSCO above the boundary. (b) Density of states of the paramagnetic state of LSCO, which shows the presence of a gap. This is a signature that LSCO is a Mott insulator (c) density of states of the ordered state of LSCO. The gap in the density of states is of similar size for the paramagnetic insulator and the ordered state as shown in details in Fig. 19.Reuse & Permissions

Figure 19

(Color online) Density of states of the three-band description of the parent compound of LSCO in the paramagnetic (dashed line) and ordered state (red area) obtained with different solvers: (a) ED, (b) CTQMC, and (c) DMRG. All data show that LSCO is a paramagnetic insulator, and that the size of the gap obtained by the density of states is similar for both the paramagnet and the ordered state (within $\approx 10\mathrm{\%}$). (d) We show the variation in the doping $\delta$ with respect to the chemical potential $\mu$ for both the ordered state (spin-density wave) and the PM state obtained by ED and CTQMC. There is a jump in the chemical potential $\delta \mu$ (plateau at $\delta =0$) of similar size for all data. This shows that the magnetic correlations do not strongly affect the insulating properties of LSCO.Reuse & Permissions

Figure 20

(Color online) We show calculations done on the one-band Hubbard model for various Coulomb repulsion $U$ and transfer integral $t$ ratios. In the one-band theory there is also a quantum critical point $U_{c2}$ that is the minimal repulsion driving a paramagnet to an insulator. Locating the compounds LSCO and NCCO by fitting the gap $\Delta$ [$\Delta =1.2\text{eV}$ and $t=0.42\text{eV}$ (Ref. 83) for NCCO and $\Delta =1.8\text{eV}$ and $t=0.43\text{eV}$ for LSCO], we place both LSCO and NCCO below $U_{c2}$ in the one band picture. It is worth noting that for large enough $U/t$ the size of the paramagnetic gap is close to the gap in the ordered state. Results were obtained by using single-site DMFT with exact diagonalization. For comparison we show theoretical calculations of Ref. 37, done for the paramagnet (long-dashed line) and the ordered state (short dotted line), and the theoretical calculations of Ref. 36 in the paramagnetic state (white circle) and in the ordered state (black circle).Reuse & Permissions