Figure 1

Evolution of the cross sections of the global phase diagram under $t$ increase for $\left|U\right|=1$, in terms of the chemical potentials (upper panels) and densities (lower panels) of the localized and conduction electrons. In phase subscripts throughout this paper, the first and second subscripts respectively describe localized and conduction electron densities, as dilute (d) or dense (D). The dotted and thick full lines are respectively first- and second-order phase transitions. Phase separation (i.e., phase coexistence) occurs inside the dotted boundaries, as identified appropriately but not repeatedly in the figure. The dashed lines are not phase transitions, but smooth changes between the different density regions of the disordered ($\delta$) phase. The charge-ordered phases are denoted by CO. The charge-ordered phases occur as strips near the half filling of the conduction electrons. Phase separation occurs near the simultaneous half filling of both the localized and conduction electrons. The four rounded coexistence regions, two of which disappear as $t$ is increased, are two-phase coexistence regions between the disordered $\delta$ phases that are distinguished by electron densities. The narrow triangular regions are three-phase coexistence regions between the $\delta$ phases. The second-order transition lines bounding the charge-ordered CO phases terminate at critical endpoints on the coexistence regions. These endpoints, as $t$ is increased, move past each other, as detailed in Sec. 4 below, leading to coexistence regions between the different charge-ordered phases and between the charge-ordered and disordered phases.Reuse & Permissions

Figure 2

Renormalization-group flow basins of the $t=0$ classical submodel, in the chemical potentials (upper panel) and densities (lower panel) of the localized and conduction electrons. In phase subscripts throughout this paper, the first and second subscripts respectively describe localized and conduction electron densities, as dilute (d) or dense (D). The dashed lines are not phase transitions, but smooth changes between the four different density regions of the disordered ($\delta$) phase.Reuse & Permissions

Figure 3

Constant $t/\left|U\right|$ cross sections of the phase diagram for interaction $\left|U\right|=0.1$, in terms of the chemical potentials (upper panels) and densities (lower panels) of the localized and conduction electrons. In phase subscripts throughout this paper, the first and second subscripts respectively describe localized and conduction electron densities, as dilute (d) or dense (D). The full lines are second-order phase transitions. The dashed lines are not phase transitions, but smooth changes between the different density regions of the disordered ($\delta$) phase. The charge-ordered phases are denoted by CO. Details are shown in Fig. 4. Thus, for low values of the interaction, all phase boundaries are second order and there is no phase coexistence.Reuse & Permissions

Figure 4

Zoomed portion of Fig. 3, for the $\left|U\right|=0.1$, $t/\left|U\right|=10$ phase diagram.Reuse & Permissions

Figure 5

Constant $t/\left|U\right|$ cross sections of the phase diagram for interaction $\left|U\right|=1$, in terms of the chemical potentials (upper panels) and densities (lower panels) of the localized and conduction electrons. The dotted and thick full lines are respectively first- and second-order phase transitions. Phase separation (i.e., phase coexistence) occurs inside the dotted boundaries, as identified in the figure. The details of the coexistence region in the lower-right panel are given in Fig. 6. The quadruple point Q tie line is shown as the thin straight line. The dashed lines are not phase transitions, but smooth changes between the different density regions of the disordered ($\delta$) phase.Reuse & Permissions

Figure 6

Zoomed portion of Fig. 5, for the $\left|U\right|=1$, $t/\left|U\right|=10$ phase diagram. The coexistence tie lines of the critical endpoints E and of the double critical endpoints E${}_2$ are shown. Inside each region delimited by dotted lines and the endpoint tie lines, phase separation (i.e., phase coexistence) occurs between phases as identified on this figure.Reuse & Permissions

Figure 7

Constant $t/\left|U\right|$ cross sections of the phase diagram for interactions $\left|U\right|=1.5,1.845628,10$ (left to right), in terms of the chemical potentials (upper panels) and densities (lower panels) of the localized and conduction electrons. The dotted and thick full lines are respectively first- and second-order phase transitions. The tie lines of the critical endpoints E and of the double critical endpoints E${}_2$ are shown by thin straight lines. Phase separation, (i.e., phase coexistence) occurs within the regions bounded by the dotted lines and these endpoint tie lines, this coexistence being between the phases seen on each side of the dotted lines. In the upper-right chemical-potential panel, the first-order phase boundary exhibits, from the left to right, a sequence of the maximum, minimum, maximum, minimum points; the four corresponding tie lines are also shown in the lower-right density panel. These tie lines abut, on one end, very near maxima and minima of the lower and upper branches of the coexistence boundaries. The dashed lines are not phase transitions, but smooth changes between the different density regions of the $\delta$ phase.Reuse & Permissions

Figure 8

Zoomed portions of Fig. 7, for the $\left|U\right|=1.845628,t/\left|U\right|=5.41821$ phase diagram.Reuse & Permissions

Figure 9

Two zoomed portions of Fig. 7, for the $\left|U\right|=10,t/\left|U\right|=1$ phase diagram.Reuse & Permissions