Diversity of charge orderings in correlated systems

The phenomenon associated with inhomogeneous distribution of electron density is known as a charge ordering. In this work, we study the zero-bandwidth limit of the extended Hubbard model, which can be considered as a simple effective model of charge ordered insulators. It consists of the on-site interaction $U$ and the intersite density-density interactions $W_1$ and $W_2$ between nearest neighbors and next-nearest neighbors, respectively. We derived the exact ground state diagrams for different lattice dimensionalities and discuss effects of small finite temperatures in the limit of high dimensions. In particular, we estimated the critical interactions for which new ordered phases emerge (laminar or stripe and four-sublattice-type). Our analysis show that the ground state of the model is highly degenerated. One of the most intriguing finding is that the nonzero temperature removes these degenerations.

Figure 1

Schematic representation of all possible homogeneous solutions for four-sublattice orderings at the ground state. Different sizes of dots correspond to different concentrations $n_{\alpha }$ ($\alpha =A,B,C,D$) in the sublattices. Note that each pattern can be realized in a few distinct forms (Tables 1 and 2) due to specific concentrations of each site.

Figure 2

Ground state phase diagrams as a function of chemical potential for $W_1>0$ and (a) $U/|W_1|\le 0.00$, (b) $U/|W_1|=0.25$, (c) $U/|W_1|=0.50$, (d) $U/|W_1|=0.75$, (e) $U/|W_1|=1.00$, and (f) $U/|W_1|=2.00$ ($\overline{\mu }=\mu -\frac{1}{2}U-W_1-W_2$). Each region is labeled by electron concentrations in each sublattice, $\left(\begin{array}{cc}{n}_A& n_B\\ n_D& n_C\end{array}\right)$ (cf. Table 1). Regions filled by a slantwise pattern: “reduced long-range order” in $d=1,2$. At the boundaries for $d=2$: solid lines: macroscopic degeneration; dashed lines (and inside the region filled by horizontal pattern): infinite degeneration but not macroscopic; dotted lines: finite degeneration (modulo spin).

Figure 3

Ground state phase diagrams as a function of electron concentration for $W_1>0$ and (a) $U/|W_1|\le 0.00$, (b) $U/|W_1|=0.25$, (c) $U/|W_1|=0.50$, (d) $U/|W_1|=0.75$, (e) $U/|W_1|=1.00$, (f) $U/|W_1|=2.00$. Each rectangular region of the diagrams is labeled by the name of homogeneous phase with the lowest free energy (cf. Table 2). In regions filled by slantwise pattern the corresponding PS state is stable at $T=0$ and at infinitesimally small $T>0$. In regions filled by grating pattern the homogeneous phase and PS state have equal free energies at $T=0$ whereas the PS state is stable at infinitesimally small $T>0$. In nonfilled regions the homogeneous phase and PS state have equal free energies at $T=0$ whereas the homogeneous phase is stable at infinitesimally small $T>0$. At vertical boundaries only the homogeneous phases occur (cf. Table 1 and Fig. 2). The rectangular points indicate discontinuous transitions at commensurate fillings.

Figure 4

Ground state phase diagrams as a function of chemical potential for $W_1<0$ and (a) $U/|W_1|\le 0$, (b) $U/|W_1|>0$ ($\overline{\mu }=\mu -\frac{1}{2}U-W_1-W_2$). Each region is labeled by electron concentrations in each sublattice: $\left(\begin{array}{cc}{n}_A& n_B\\ n_D& n_C\end{array}\right)$ (cf. Table 1). Dashed lines denote infinite degeneration but not macroscopic, whereas dotted lines indicate finite degeneration (at the boundaries for $d=2$).

Figure 5

Ground state phase diagrams as a function of electron concentration for $W_1<0$ and (a) $U/|W_1|\le 0$, (b) $U/|W_1|>0$. Each rectangular region of the diagrams is labeled by the name of the homogeneous phase with the lowest free energy (cf. Table 2). In all regions the corresponding PS states are stable at $T=0$ and at infinitesimally small $T>0$ (excluding vertical boundaries). The rectangular points indicate discontinuous transitions at commensurate fillings (cf. Fig. 2).

Figure 6

Schematic illustration of nonmixing block alignment in $d=2$. Note that the FNO block at the interface between the NO and CBO phases emerges (shadow, dotted line). For $W_2/\left|W_1\right|<0$ the FNO block does not yield the minimal energy.

Figure 7

Schematic illustration of partial-mixing block alignments in $d=2$ at the STO–CBO interface. In the left panel an allowed block orientation is shown (at the interfaces the STO or CBO blocks emerge). In the right panel a forbidden configuration is present, where in addition the FNO blocks emerge at the interface.

Figure 8

Schematic illustration of full-mixing block alignment in $d=2$. All building blocks at the interface between the CBO and FCBO phases are blocks of one of these mixing phases. An exemplary block at the interface is denoted by the shadow (dashed line).