Photoemission spectra from reduced density matrices: The band gap in strongly correlated systems

We present a method for the calculation of photoemission spectra in terms of reduced density matrices. We start from the spectral representation of the one-body Green's function $G$, whose imaginary part is related to photoemission spectra, and we introduce a frequency-dependent effective energy that accounts for all the poles of $G$. Simple approximations to this effective energy give accurate spectra in model systems in the weak as well as strong correlation regime. In real systems reduced density matrices can be obtained from reduced density-matrix functional theory. Here we use this approach to calculate the photoemission spectrum of bulk NiO: our method yields a qualitatively correct picture both in the antiferromagnetic and paramagnetic phases, contrary to mean-field methods, in which the paramagnet is a metal.

Figure 1

Spectral function for a 12-site Hubbard ring at 1/2 filling: exact (EX) vs MEET with exact RDMs (${\delta }^{\left(1\right)}$ and ${\delta }^{\left(2\right)}$), MEET with approximate RDMs $({\delta }_{\alpha }^{\left(1\right)},\alpha =0.5)$, DER method (with $\alpha =0.5$), and $GW$ method. Peaks are broadened with a Lorentzian of width $\eta =0.1$.

Figure 2

Spectral function for a 6-site Hubbard ring at 1/6 filling: exact (EX) vs MEET with exact RDMs (${\delta }^{\left(1\right)}$ and ${\delta }^{\left(2\right)}$). Peaks are broadened with a Lorentzian of width $\eta =0.1$.

Figure 3

Paramagnetic (left) and antiferromagnetic (right) bulk NiO: experimental photoemission spectrum [45] vs MEET spectrum $({\delta }_{\alpha }^{\left(1\right)},\alpha =0.65)$. The color map and the distribution $f\left(n_i\right)$ illustrate the occupation numbers $n_i$ that play a role into the spectrum for the reported energy range.

Figure 4

Paramagnetic bulk NiO: experimental photoemission spectrum [45] vs MEET spectrum (${\delta }_{\alpha }^{\left(1\right)}$, with the power functional, $\alpha =0.65$, and without screening $(\beta =1)$ and with screening $\beta =0.8$).

Figure 5

Deviation of the fundamental gap $\mathrm{\Delta }{E}_g$ from $U$ as a function of $U/(U+4t)$ for the 1D (top) and 2D (middle) infinite Hubbard models at 1/2 filling. Dashed and dotted lines are obtained using the MEET method $({\delta }_{\alpha }^{\left(1\right)},\alpha =0.5,0.6)$. The black solid line in the top panel is the exact result derived from the Bethe-ansatz solution [49]. The curve corresponding to $\mathrm{\Delta }{E}_g=0$ is reported as reference. The triangles, circles, and squares in the middle panel are finite-size exact calculations for the $2\times 2, 4\times 2$, and $4\times 3$ 2D Hubbard clusters, respectively. The squares are the extrapolated values to the infinite 2D system. In the bottom panel the ratio between the ${\delta }_{\alpha }^{\left(1\right)}$ band gap and exact band gap $(\mathrm{\Delta }{E}_g^{\text{MEET}}/\mathrm{\Delta }{E}_g^{\text{EX}})$ is reported as function of $U/(U+4t)$. As a guide for the eye we reported $\mathrm{\Delta }{E}_g^{\text{MEET}}/\mathrm{\Delta }{E}_g^{\text{EX}}=1$ with a thin dashed line.

Figure 6

Paramagnetic (left) and antiferromagnetic (right) bulk NiO: experimental photoemission spectrum [45] vs DER spectrum (with the power functional, $\alpha =0.65$). The color map and the distribution $f\left(n_i\right)$ illustrate the occupation numbers $n_i$ that play a role into the spectrum for the reported energy range.