Effective momentum-momentum coupling in a correlated electronic system

We present a way to partly reincorporate the effects of the localized bonding electrons on the dynamics of their itinerant counterparts in Hubbard-like Hamiltonians. This is done by relaxing the constraint that the former should be entirely frozen in the chemical bonds between the underlying lattice sites through the employment of a Born-Oppenheimer-like ansatz for the wave function of the whole electronic system. Accordingly, the latter includes itinerant as well as bonding electron coordinates. Going beyond the adiabatic approximation, we show that the net effect of virtual transitions of bonding electrons between their ground and excited states is to furnish the itinerant electrons with an effective interelectronic momentum-momentum interaction. Although we have applied these ideas to the specific case of rings, our assumptions can be generalized to higher-dimensional systems sharing the required properties of which we have made use herein.

Figure 1

Orbital structure of the rings we consider in this work. Here, we illustrate a ring with six sites, which consists of a prototype of the benzene molecule. (a) Single-band model considered in Sec. 2a, with one $p_z$ orbital per site. In this case, only the degrees of freedom of the $\pi$ electrons are taken into account. The bonding $\sigma$ electrons are frozen and incorporated into the ring's sites, as illustrated in the inset, and they only contribute by generating the static crystal potential. (b) Three-band model, with one $p_z$ orbital and two $s{p}^2$ orbitals, which we denominate the left $s{p}^2$ orbital ($L$) and right $s{p}^2$ orbital ($R$) according to the right-hand rule. As shown in the inset of panel (b), the degrees of freedom of the third $s{p}^2$ orbital of each site, as well as those of the valence orbitals of another atom that might bind to it (hydrogen atom, in the case of a benzene molecule), are frozen and incorporated to the ring's sites. The ring's sites are always enumerated in ascending order in the counterclockwise direction.

Figure 2

Energy spectrum of the Hubbard Hamiltonian [Eq. (2)]. The panels show the energy levels, as a function of $U/t$ for rings with (a) $N=3$ sites, (b) $N=4$ sites, (c) $N=5$ sites, and (d) $N=6$ sites at the half-filling regime ($N=N_e$). In panels (b) to (d), only a few of the low-lying energy levels are shown.

Figure 3

Energy “surfaces” of the $\sigma$-electrons. Illustration of the energy levels ${\lambda }_{\nu }\left(\mathbf{R}\right)$ of the $\sigma$-electrons as a function of the $\pi$-electron configurations as if they were a function of a scalar variable, in analogy to the simpler standard Born-Oppenheimer approximation. Panel (a) represents the first three low-lying ${\lambda }_{\nu }\left(\mathbf{R}\right)$. Panel (b) focuses only on the first two $\sigma$-electron energy levels. In each one of them, the $\pi$-electron Hubbard spectrum is represented by the horizontal black lines. The blue arrows indicate virtual excitations that may occur in the system if the energy separation ($\mathrm{\Lambda }$) between the two $\sigma$-electron surfaces is comparable with the $\pi$-electron bandwidth, which is set by the $\pi$-electron hopping amplitude $t_0$.

Figure 4

Effective interaction. Illustration of two types of two-body processes that appear in the effective interaction Eq. (47). (a) is the “Bubble term,” while (b) is the extended term that favors the electron delocalization. The Hermitian conjugates of (a) and (b) just reverse the direction of the arrows.

Figure 5

Energy spectrum of the extended Hubbard Hamiltonian. The panels show the energy levels of Eq. (48), as a function of $U/t$ for rings with (a) $N=3$ sites, (b) $N=4$ sites, (c) $N=5$ sites, and (d) $N=6$ sites at the half-filling regime ($N=N_e$) and with $\lambda /t=0.04$. In the panels (b) to (d), only a few of the low-lying energy levels are plotted. The dashed lines correspond to $\lambda =0$, i.e., the spectrum of the standard Hubbard model defined in Eq. (2).