Anatomy of the self-energy

The general problem of representing the Greens function $G(k,z)$ in terms of self-energy in field theories lacking Wick's theorem is considered. A simple construction shows that a Dyson-like representation with a self-energy $\Sigma (k,z)$ is always possible, provided we start with a spectral representation for $G(k,z)$ for finite-sized systems and take the thermodynamic limit. The self-energy itself can then be iteratively expressed in terms of another higher order self-energy, capturing the spirit of Mori's formulation. We further discuss alternative and more general forms of $G(k,z)$ that are possible. In particular, a recent theory, by the author, of extremely correlated Fermi liquids at density $n$, for Gutzwiller projected noncanonical Fermi operators, obtains a new form of the Greens function: $G(k,z)=[(1-\frac{n}{2})+\Psi (k,z)]/[z-{\widehat{E}}_k-\Phi (k,z)],$ with a pair of self-energies $\Phi \left(z\right)$ and $\Psi \left(z\right)$. Its relationship with the Dyson form is explored. A simple version of the two-self-energies model was shown recently to successfully fit several data sets of photoemission line shapes in cuprates. We provide details of the unusual spectral line shapes that arise in this model, with the characteristic skewed shape depending upon a single parameter. The energy distribution curve (EDC) and momentum distribution curve (MDC) line shapes are shown to be skewed in opposite directions, and provide a testable prediction of the theory.

Figure 1

The density $n=0.85$, temperature $T=600$ K, ${\Delta }_0=0.0786,$ and parameters are set I of Eq. (54). At this rather high temperature, we can see the details of the spectral shape clearly. The vertical line is at $x^{*}=E_{k_F}^{*}$, this energy is the location of the peak of the physical spectral function ${\rho }_G$ as marked. Its leftward (i.e., red) shift relative to the Fermi-liquid peak at the chemical potential is clearly seen. The two horizontal lines specify the magnitude of the $\Re eG(0,x)$ at $x=0$ (H1) and $x=E_{k_F}^{*}$ (H2). The line H1 is at height $n^2/\left(4{\Delta }_0\right)$ and H2 is at height $n^2/\left(4{\Delta }_0\right)(1-Z_k/2)$.

Figure 2

The density $n=0.85$, $T=300$ K, ${\Delta }_0=0.0678,$ and parameters are from set I in Eq. (54). The various dimensionless variables shown against $E_k^{\mathrm{FL}}$ are the peak ratio from Eq. (50), the variable $u_k$ (scaled by ten) from Eq. (44), the skew asymmetry factor $\kappa \left({\xi }_k\right)$ from Eq. (65), and the variable $Q_k$ from Eq. (61).

Figure 3

The ECFL (top left) and the auxiliary FL spectral functions (bottom left) at density $n=0.85$, $T=180$ K, ${\Delta }_0=0.12$ eV, $\eta =0.12$ eV, and the other parameters are from Set II in Eq. (54). Here, $\xi$ and $x$ are in units of eV. In the ECFL curve on left, it is seen that the excitations near the Fermi energy become broad and dissolve into the continuum at an energy $\sim$$-0.2$ eV, and reappear as sharp modes at a deeper binding energy. In the auxiliary FL, the excitations near the Fermi energy remain sharp and extend to lower energies than in the ECFL curves. The contour plots of the same functions in the right panel (top ECFL and bottom auxiliary FL) give a complementary perspective of the spectrum. The two superimposed solid lines at top right are from curves I and II of Fig. 5 and at bottom right curves III and IV of Fig. 5.

Figure 4

The contour plots of the aux-FL (left) and the ECFL model from Fig. 3 (right) with the same parameters as in Fig. 3 but over a smaller energy window. We superimpose the constant-wave-vector dispersion and MDC dispersion, with a value of $\eta =0.12$ common to the contour plots. The energy scale of the feature near the chemical potential is considerably reduced in the ECFL, and the “jump” in the EDC dispersion occurs at roughly half the corresponding energy in the aux-FL.

Figure 5

Energy dispersion curves in the ECFL and the aux-FL models. Here, the parameters are from set II in Eq. (54), with $n=0.85$ and $T=180$ K. With $\eta =0.12$, curves I and II have the peaks in constant-wave-vector and constant-energy scans of the spectral function (37), and curves III and IV are corresponding figures for the aux-FL in Eq. (39). The inset compares Eq. (48) (the truncated curve) with the exact locus found by numerical maximization.

Figure 6

MDC line shapes at different values of energy $x$ displayed in each curve. Here, the parameters are from set II in Eq. (54), with $n=0.85$, $T=180$ K, and $\eta =0.12$. Panel (a) corresponds to $x$ close to the chemical potential. It is interesting to note that the curves are skewed to the right, thus mirror imaging the leftward skew seen in the constant-$\xi$ (EDC) scans below Fig. 7a, in a comparable range of energies and wave vectors. Panel (b) corresponds to the midenergy range, within the reentrant range of $x$ from Figs. 5 or 3, with the counterintuitive movement of the shallow peak to the right with increasing $x$. Panel (c) corresponds to the second set of maxima in Fig. 3 far from the chemical potential, where the curves are quite symmetric.

Figure 7

EDC line shapes at different values of energy $\xi$ displayed in each curve. Here, the parameters are from set II in Eq. (54), with $n=0.85$, $T=180$ K, and $\eta =0.12$. Panel (a) corresponds to $\xi$ close to the chemical potential. Note that the curves are skewed to the left, i.e., a mirror image of the rightward skew seen in the constant-$x$ MDC scans above Fig. 6, in a comparable range of energies and wave vectors. Panel (b) corresponds to the higher energy range, and we see that only one broad maximum is found at a given $\xi$. The inset in (b) shows the aux-FL constant-$\xi$ scans for the same range; here, each $\xi$ results in a pair of maxima, originating from the functional form of the self-energy in Eq. (37).

Figure 8

Top left panel: density $n=0.85$, temperature $T=300$ K, ${\Delta }_0=0.0678,$ and other parameters are from set I in Eq. (54). From left to right ${\rho }_G(\xi ,x)$ for energies in units of eV: $\xi =-0.1,-0.075,-0.05,-0.025,0,0.025,0.05$. Top right panel: spectral function ${\rho }_g(\xi ,x)$ from Eq. (39) corresponding to the same $\xi$ as in the left panel. The difference in the line shapes becomes clear when we examine the Dyson self-energy that produces these curves. Bottom left panel: the panel shows the spectral function at $T=300$ K for the inferred Dyson self-energy ${\rho }_{\Sigma }\left(x\right)$ from Eqs. (23), (38), and (37) for the same energies. The dashed line is the input Fermi-liquid spectral function ${\rho }_{\Phi }\left(\omega \right)$ at the same temperature. Bottom right panel: temperature $T=150$ K, ${\Delta }_0=0.0642,$ and the identical data as in the bottom left panel.

Figure 9

With $n=0.85$ and $T=300\text{K}$, and the other parameters from set I in Eq. (54). The spectral function for the inferred Dyson self-energy ${\rho }_{\Sigma }(\xi =0,x)$ using Eq. (23) and the the input Fermi-liquid spectral function ${\rho }_{\Phi }\left(x\right)$ over a larger energy range. Note the distinctive asymmetry in shape of ${\rho }_{\Sigma }$ below and above the Fermi energy.

Figure 10

The spectral shapes possible are seen by plotting the shape function at different values of the parameter $Q$. In this curve, $\overline{\gamma }$ is the $\gamma (Q,\overline{\epsilon })$ of Eq. (60) normalized to unit area in the natural interval $[-1,1]$ for the variable $\overline{\epsilon }$.

Figure 11

The same shapes as in Fig. 10 but now $\gamma$ is normalized to unity at the peak as in Eq. (60). The sharp peaks for small $Q\le 0.25$ flatten out as $Q$ increases with a left skew asymmetry that is characteristic of this functional dependence.