Effective Ising model for correlated systems with charge ordering

Collective electronic fluctuations in correlated materials give rise to various important phenomena, such as charge ordering, superconductivity, Mott insulating and magnetic phases, and plasmon and magnon modes. Unfortunately, the description of these correlation effects requires significant effort, since they almost entirely rely on strong local and nonlocal electron-electron interactions. Some collective phenomena, such as magnetism, can be sufficiently described by simple Heisenberg-like models that are formulated in terms of bosonic variables. This fact suggests that other many-body excitations can also be described by simple bosonic models in the spirit of Heisenberg theory. Here we derive an effective bosonic action for charge degrees of freedom for the extended Hubbard model and define a physical regime where the obtained action reduces to a classical Hamiltonian of an effective Ising model.

Figure 1

Double occupancy of the extended Hubbard model shown on the $U-V$ phase diagram. Calculations are performed in the normal phase where the value of the double occupancy $d$ is depicted by color. The gray part corresponds to the charge ordered phase. Values of the double occupancy at the phase boundary are explicitly mentioned. The area depicted by the black dashed line corresponds to the case of large value of the double occupancy $d\gtrsim 70\%d_{\max }$ and shows the regime where charge excitations can be described by an effective Ising model. Values of Coulomb interactions $U$ and $V$ are given in units of half of the bandwidth ($W/2=4t=1$). Therefore, the effective Ising model can be used for a broad range of values of the Coulomb interaction, which may even exceed half of the bandwidth. The inverse temperature is $\beta =50$.

Figure 2

Frequency dependence of the effective local Coulomb interaction $U_{\nu {\nu }^{\prime }\omega }^{\mathrm{eff}}$ obtained for different values of $U$ at the phase boundary between the normal and CO phases at the ${\nu }^{\prime }=0$ and $\omega =0$ point for $\beta =50$. As the double occupancy $d$ is decreased, the dependence of the effective interaction $U_{\nu {\nu }^{\prime }\omega }^{\mathrm{eff}}$ on fermionic frequency becomes larger.

Figure 3

Frequency dependence of the effective local Coulomb interaction $U_{\omega }^{\mathrm{eff}}$ obtained for $\beta =50$ close to the phase boundary between the normal and charge ordered phases for different values of the actual Coulomb interaction $U$. When the double occupancy is decreased, the difference between the effective and actual local Coulomb interactions becomes more notable.

Figure 4

Phase boundary between the normal (N) and charge-ordered (CO) phases of the hole-doped extended Hubbard model in the space of nearest-neighbor interaction $V$ and chemical potential $\mu$. Calculations are performed for the Ladder DB and RPA+EDMFT approaches in the regime of large double occupancy where the use of the simplified approximation is justified. The DCA data is kindly provided by authors of the Ref. [50]. The local interaction is $U=0$ (left panel), $U=0.5$ (middle panel), and $U=1$ (right panel). The hopping amplitude is $t=0.25$. $\mu =0$ corresponds to the half filling. All data are obtained for $\beta =12.5$ ($T=0.08$).

Figure 5

Frequency dependence of the effective local Coulomb interaction $U_{\nu {\nu }^{\prime }\omega }^{\mathrm{eff}}$ obtained for different values of the actual Coulomb interaction $U=0.1,0.5,1.0,1.5$ (from left to right) in the normal phase close to the charge ordering for different values of fermionic ${\nu }^{\prime }$ and bosonic $\omega$ frequencies for $\beta =50$. The dependence of the effective interaction on fermionic frequency becomes larger at larger Coulomb interaction.