Electron self-energy from quantum charge fluctuations in the layered $t-J$ model with long-range Coulomb interaction

Employing a large-$N$ scheme of the layered $t-J$ model with the long-range Coulomb interaction, which captures the fine details of the charge excitation spectra recently observed in cuprate superconductors, we explore the role of charge fluctuations on the electron self-energy. We fix the temperature at zero and focus on quantum charge fluctuations. We find a pronounced asymmetry of the imaginary part of the self-energy $\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},\omega )$ with respect to $\omega =0$, which is driven by strong electron correlation effects. The quasiparticle weight is reduced dramatically, which occurs almost isotropically along the Fermi surface. Concomitantly, an incoherent band and a sharp side band are generated and acquire sizable spectral weight. All these features are driven by the usual on-site charge fluctuations, which are realized in a rather high-energy region and yield plasmon excitations. On the other hand, the low-energy region with the scale of the superexchange interaction $J$ is dominated by bond-charge fluctuations. Surprisingly, compared with the effect of on-site charge fluctuations, their effect on the electron self-energy is much weaker, even if the system approaches close to bond-charge instabilities. Furthermore, quantum charge dynamics does not produce a clear kink nor a pseudogap in the electron dispersion.

Figure 1

Typical charge excitation spectrum [intensity map of $\mathrm{Im}{D}_{11}(\mathbf{q},\nu )$] in the plane of momentum $\mathbf{q}$ and energy $\nu$ for $q_z=\pi$ (a) and $q_z=0$ (b). The result for $q_z=\pi$ is a representative one of results for other $q_z(\ne 0)$. The dotted curve is the upper limit of the continuum excitations.

Figure 2

Intensity map of $A(\mathbf{k},\omega )$ along the direction $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$ for $k_z=\pi$. The quasiparticle dispersion in the presence of the self-energy is in yellow. The white curve is the quasiparticle dispersion obtained in leading order theory without the self-energy [Eq. (2)]. See also Fig. 12 (a) obtained after a particle-hole transformation.

Figure 3

Imaginary part of the electron self-energy, $-\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},\omega )$, as a function of $\omega$ for several choices of $\mathbf{k}$. The positive energy region is magnified in the right panel. Results do not depend much on a value of $k_z$ and $k_z=\pi$ is taken as a representative one. See also Fig. 15 obtained after a particle-hole transformation.

Figure 4

Energy dependence of $-\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},\omega ), \mathrm{Re}\mathrm{\Sigma }(\mathbf{k},\omega )$, and $A(\mathbf{k},\omega )$ for $\mathbf{k}=(0,0,\pi )$ (a), $(\pi ,\pi ,\pi )$ (b), and $(\pi ,0,\pi )$ (c). The line of $\omega -\varepsilon \left(\mathbf{k}\right)$ is also drawn. The scales of $-\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},\omega )$ and $A(\mathbf{k},\omega )$ correspond to the right vertical axis and their units are taken differently to clarify their peak structure in the same panel. See also Fig. 16 obtained after a particle-hole transformation.

Figure 5

Intensity map of $A(\mathbf{k},\omega )$ along the direction $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$; $k_z$ dependence is weak and $k_z=\pi$ is taken as a representative value. See also Fig. 12 obtained after a particle-hole transformation.

Figure 6

The density of states for three difference cases: one is at leading order in the large-$N$ theory [Eq. (2)] and the other two are $2\times 2$ and $6\times 6$, where the self-energy effect is taken into account from on-site charge fluctuations and both on-site charge and bond-charge fluctuations, respectively. The results for $2\times 2$ and $6\times 6$ almost overlap with each other around $\omega /t=0$ and 1. See also Fig. 12 obtained after a particle-hole transformation.

Figure 7

Intensity map of $A(\mathbf{k},\omega )$ along the direction $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$ in the presence of both on-site charge and bond-charge fluctuations. $k_z$ dependence is weak and $k_z=\pi$ is taken as a representative value. See also Fig. 12 obtained after a particle-hole transformation.

Figure 8

(a) Quasiparticle weight $Z\left(\phi \right)$ along the Fermi surface for $k_z=0$ and $\pi$ at $\delta =0.15$. The momentum is measured by the angle $\phi$ defined in the inset. Four different cases are plotted: (i) $6\times 6$, where full charge fluctuations are included, (ii) $2\times 2$, where only on-site charge fluctuations are considered, (iii) $d\mathrm{bond}$ and (iv) $d\mathrm{CDW}$, where fluctuations associated with $d\mathrm{bond}$ and $d\mathrm{CDW}$ are considered, respectively. In the inset, Fermi surfaces shown by solid and dotted lines correspond to $k_z=\pi$ and 0, respectively. (b) Doping dependence of $Z\left(\phi \right)$ at $\phi =0$ and $\pi /4$ for four different cases. The $d\mathrm{CDW}$ and $d\mathrm{bond}$ are stabilized below ${\delta }_c\approx 0.07$ and 0.06, respectively. After the particle-hole transformation discussed in Sec. 3f, the results stay intact except that the momentum in the inset in (a) should be transformed as $(0,0)\to (\pi ,\pi )$ and $(\pi ,\pi )\to (0,0)$.

Figure 9

(a), (b) Spectral function $A({\mathbf{k}}_F,\omega )$ as a function of $\omega$ at the Fermi momentum; $k_z=\pi$ is taken as a representative one. ${\mathbf{k}}_F^{\mathrm{N}}$ and ${\mathbf{k}}_F^{\mathrm{AN}}$ are the Fermi momenta at the nodal $(\phi =\pi /4)$ and antinodal $(\phi =0)$ points, respectively. In (a), only on-site charge fluctuations (denoted by $2\times 2$) are considered whereas the effect of both on-site charge and bond-charge fluctuations ($6\times 6$) are taken into account in (b). (c) Intensity map of $A(\mathbf{k},0)$ at the Fermi energy in the first quadrant of the Brillouin zone for $k_z=0$ as a representative one. See also Figs. 12 and 12 obtained after a particle-hole transformation.

Figure 10

Results for parameters appropriate for h-cuprates at $\delta =0.15$. (a) Intensity map of $A(\mathbf{k},\omega )$ in the presence of the on-site charge fluctuations along the direction $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$; $k_z$ dependence is weak and $k_z=\pi$ is taken as a representative value. (b) Intensity map of $A(\mathbf{k},\omega )$ in the presence of both on-site charge and bond-charge fluctuations. (c) Quasiparticle weight $Z\left(\phi \right)$ along the Fermi surface for four cases: $6\times 6$ where full charge fluctuations are included, $2\times 2$ where only on-site charge fluctuations are considered, and $d\mathrm{bond}$ and $d\mathrm{CDW}$ where fluctuations associated with $d\mathrm{bond}$ and $d\mathrm{CDW}$ are considered, respectively. The results for e-cuprates corresponding to (a)–(c) are shown in Figs. 5, 7, and 8, respectively.

Figure 11

Results for h-cuprates at $\delta =0.15$. (a) Spectral function $A({\mathbf{k}}_F,\omega )$ as a function of $\omega$ at the Fermi momentum. ${\mathbf{k}}_F^{\mathrm{N}}$ and ${\mathbf{k}}_F^{\mathrm{AN}}$ are the Fermi momenta at the nodal $(\phi =\pi /4)$ and antinodal $(\phi =0)$ points. (b) Intensity map of $A(\mathbf{k},0)$ at the Fermi energy in the first quadrant of the Brillouin zone. Corresponding results to e-cuprates are given in Figs. 9 and 9.

Figure 12

Intensity map of $A(\mathbf{k},\omega )$ along the direction $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$ after the particle-hole transformation (a) around the Fermi energy corresponding to Fig. 2, and (b) and (c) in a larger energy region corresponding to Fig. 5 and 7, respectively. (d) The density of states after the particle-hole transformation of Fig. 6. (e) The spectral function $A({\mathbf{k}}_F,\omega )$ at the nodal and antinodal Fermi momenta. (f) Intensity map of $A(\mathbf{k},0)$ at the Fermi energy in the first quadrant of the Brillouin zone. (e) and (f) are obtained after the particle-hole transformation of Figs. 9 and 9, respectively.

Figure 13

Intensity map of $A(\mathbf{k},\omega )$ in the absence of the long-range Coulomb interaction along the direction $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$ in the hole picture; both on-site charge and bond-charge fluctuations are taken into account. $k_z$ dependence is weak and $k_z=\pi$ is taken as a representative value. The corresponding result in the particle picture is presented in Appendix pp2.

Figure 14

(a) Propagators and vertices in a large-$N$ theory up to $O(1/N)$. Solid line represents the fermion propagator $G_{p{p}^{\prime }}^{\left(0\right)}$; dashed line the $6\times 6$ boson propagator $D_{ab}^{\left(0\right)}$ for the six-component fields $\delta X^a$; wavy line the ghost propagator ${\mathcal{D}}_{p{p}^{\prime }}^{\left(0\right)}$. ${\mathrm{\Lambda }}_a^{p{p}^{\prime }}$ and ${\mathrm{\Gamma }}_a^{p{p}^{\prime }}$ describe the three-point interaction between two fermions with spin indices $p$ and $p^{\prime }$ and one boson with the component $a$, and two ghost fields with one boson, respectively. ${\mathrm{\Lambda }}_{ab}^{p{p}^{\prime }}$ and ${\mathrm{\Gamma }}_{ab}^{p{p}^{\prime }}$ are the four-point interactions among the two fermions and two bosons, and two ghost fields and two bosons, respectively. (b) The irreducible boson self-energy ${\mathrm{\Pi }}_{ab}$ and the renormalized boson propagator $D_{ab}$. (c) The electron self-energy $\mathrm{\Sigma }(\mathbf{k},\omega )$ at the order of $1/N$.

Figure 15

Imaginary part of the electron self-energy, $-\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},\omega )$, as a function of $\omega$ for several choices of $\mathbf{k}$ after the particle-hole transformation of Fig. 3.

Figure 16

Energy dependence of $-\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},\omega ), \mathrm{Re}\mathrm{\Sigma }(\mathbf{k},\omega )$, and $A(\mathbf{k},\omega )$ at $\mathbf{k}=(0,0,\pi )$ (a), $(\pi ,\pi ,\pi )$ (b), and $(\pi ,0,\pi )$ (c) after the particle-hole transformation of Fig. 4.

Figure 17

Intensity map of $A(\mathbf{k},\omega )$ in the absence of the long-range Coulomb interaction along the direction $(\pi ,\pi )-(0,0)-(\pi ,0)-(\pi ,\pi )$ in the particle picture; both on-site charge and bond-charge fluctuations are taken into account. $k_z$ dependence is weak and $k_z=\pi$ is taken as a representative value. The corresponding result in the hole picture is presented in Fig. 13.