Electronic spectral properties of the two-dimensional infinite-$U$ Hubbard model

A strong-coupling series expansion for the Green's function and the extremely correlated Fermi liquid (ECFL) theory are used to calculate the moments of the electronic spectral functions of the infinite-$U$ Hubbard model. Results from these two complementary methods agree very well at both low densities, where the ECFL solution is the most accurate, and at high to intermediate temperatures, where the series converge. We find that a modified first moment, which underestimates the contributions from the occupied states and is accessible in the series through the time-dependent Green's function, best describes the peak location of the spectral function in the strongly correlated regime. This is examined by the ECFL results at low temperatures, where it is shown that the spectral function is largely skewed towards the occupied states.

Figure 1

The first symmetric moment ${\varepsilon }_1^0\left(k\right)$ at $T=0.77$ vs momentum around the irreducible wedge of the Brillouin zone (the path is shown in the right inset). Lines are results from the series and symbols for $n\le 0.7$ are from ECFL calculations. Left inset: ${\varepsilon }_1^0\left(k\right)$ for $n=0.2$ at $k=(\pi /2,\pi /2)$ from the ECFL (diamonds), up to orders seven and eight of the series (labeled Series${}_7$ and Series${}_8$), and up to the eighth order after various Padé approximations, vs temperature on a logarithmic scale. The numbers in the subscripts of “Padé” labels represent the order of the polynomial in the numerator and in the denominator of the Padé ratio, respectively. “Avg.” denotes the average between Padé${}_{\{4,5\}}$ and Padé${}_{\{5,4\}}$. In the main panel, the results for the series are either the average between Padé${}_{\{4,5\}}$ and Padé${}_{\{5,4\}}$ or Padé${}_{\{5,5\}}$ and Padé${}_{\{5,4\}}$,[16] with the “error bars” defined as the differences between the two.[17]

Figure 2

(a) The first greater moment ${\varepsilon }_1^{>}\left(k\right)$ at $T=0.77$ vs momentum for the same path around the irreducible wedge of the Brillouin zone as in Fig. 1. Lines and symbols are also the same as in Fig. 1. (b) The bandwidth of ${\varepsilon }_1^{>}\left(k\right)$, defined as the difference between its maximum and minimum values at momenta shown in panel (a), vs density for $T=1.52,1.00$, and $0.77$. Panel (c) zooms in the results in panel (a) for $k$ along the nodal direction. The two methods more or less agree with each other, within the error bars, in this window for $n\le 0.7$, and therefore, we show only the ECFL results for the latter cases.

Figure 3

Comparison of the SP location ${\varepsilon }^{\mathrm{SP}}\left(k\right)$ (symbols) and the three moments from ECFL at $T=0.28$ and for (a) $n=0.2$, (b) $n=0.5$, and (c) $n=0.7$. Right panels show the corresponding spectral functions and their products to $\overline{f}\left(\omega \right)$ and $f\left(\omega \right)$ at $\Gamma$ for the same densities shown in the left panels. Dark (light) arrows show the values of ${\varepsilon }_1^0$ (${\varepsilon }_1^{>}$). At low densities, the SP location is estimated well by the first symmetric moment. At higher density, the spectral function is skewed and the greater moment, which is calculated for the spectral function after most of its weight in the negative frequency region is cut off, provides a better estimate.