Spin-orbit interactions may relax the rigid conditions leading to flat bands

Flat bands are of extreme interest in a broad spectrum of fields since, given their high degeneracy, a small perturbation introduced in the system is able to push the ground state in the direction of an ordered phase of interest. Hence, the flat-band engineering in real materials attracts huge attention. However, manufacturing a flat band represents a difficult task because its appearance in a real system is connected to rigid mathematical conditions relating a part of Hamiltonian parameters. Consequently, whenever a flat band is to be manufactured, these Hamiltonian parameters must be tuned exactly to the values fixed by these rigid mathematical conditions. Here we demonstrate that taking the many-body spin-orbit interaction into account, which can be continuously tuned, e.g., by external electric fields, these rigid mathematical conditions can be substantially relaxed. Consequently, we show that a $\sim 20\%–30\%$ variation in the Hamiltonian parameters rigidly fixed by the flat-band conditions can also lead to flat bands in the same, or in a bit displaced, position on the energy axis. This percentage can even increase to $\sim 80\%$ in the presence of an external magnetic field. This study is made for the case of conducting polymers. These systems are relevant not only because they have broad application possibilities, but also because they can be used to present the mathematical background of the flat-band conditions in full generality, in a concise, clear, and understandable manner applicable everywhere in itinerant systems.

Figure 1

The pentagonal unit cell, with the nearest-neighbor hopping matrix elements ($t,t_h,t_c,t_f$), the Rashba couplings ($\lambda ,{\lambda }_c$), the onsite one-particle potentials (${\epsilon }_1,{\epsilon }_2,{\epsilon }_3,{\epsilon }_4$), and the external magnetic field ($B$).

Figure 2

Schematic plot of the polyaminotriazole cell.

Figure 3

Notations used for the pentagonal unit cell.

Figure 4

(a) The $\lambda$ values necessary to achieve a flat band with different fixed ${\lambda }_c$ values, after changing the $t_c=t_c^{\text{rfbc}}$ value defined by the rigid flat-band conditions by $\mathrm{\Delta }{t}_c$. (b) The ${\lambda }_c$ values necessary to achieve a flat band with different fixed $\lambda$ values, after changing the $t_c=t_c^{\mathrm{rfbc}}$ value defined by the rigid flat-band conditions by $\mathrm{\Delta }{t}_c$. For exemplification, we have used the Set 1 of Hamiltonian parameter data from Appendix pp4.

Figure 5

The $\lambda , {\lambda }_c$ values necessary to achieve a flat band after changing the $t_c=t_c^{\text{rfbc}}$ value defined by the rigid flat-band conditions by $\mathrm{\Delta }{t}_c$. The changed $\mathrm{\Delta }{t}_f$ values were obtained by changing the $t_h$ hopping magnitude, therefore, the value of $t_h$ is continuously changing along the solid lines, from the original $t_h$ to $1.5t_h$. In case (a) one has $\lambda =\text{constant}$, while in the case (b) ${\lambda }_c=\text{constant}$ holds. For exemplification, we have used the Set 1 of Hamiltonian parameter data from Appendix pp4.

Figure 6

The $\overline{\lambda }=\lambda ={\lambda }_c$ value necessary to compensate the deviation $\mathrm{\Delta }{t}_c\ne 0$ from $t_c^{\text{rfbc}}$ in order to recreate the flat band at the origin of the energy axis. For exemplification, we have used the Set 1 of Hamiltonian parameter data from Appendix pp4.

Figure 7

(a) $\lambda$ and (b) ${\lambda }_c$ spin-orbit coupling values deduced at $B\ne 0$ (upper) and $B=0$ (lower) necessary to compensate the deviation $\mathrm{\Delta }{t}_c\ne 0$ from $t_c^{\text{rfbc}}$ in order to recreate the flat band at the origin of the energy axis. For exemplification, we have used the Set 2 of Hamiltonian parameter data from Appendix pp4.

Figure 8

The $\overline{\lambda }=\lambda ={\lambda }_c$ spin-orbit coupling values deduced at $B\ne 0$ (upper) and $B=0$ (lower) necessary to compensate the deviation $\mathrm{\Delta }{t}_c\ne 0$ from $t_c^{\text{rfbc}}$ in order to recreate the flat band at the origin of the energy axis. For exemplification, we have used the Set 3 of Hamiltonian parameter data from Appendix pp4.

Figure 9

The ${\lambda }_c$ spin-orbit coupling values necessary to compensate at $\lambda =0$ the common deviations $\mathrm{\Delta }{t}_c\ne 0$ and $\mathrm{\Delta }{t}_f\ne 0$ at (a) $B=0$ and (b) $B\ne 0$, in order to recreate a flat band placed originally (at $\lambda ={\lambda }_c=B=0$) in the origin of the energy axis $\epsilon =0$. The new position of the flat band is at $\epsilon =\mathrm{\Delta }E$. In this figure $\mathrm{\Delta }{t}_c/t_c<0$ holds. For exemplification, we have used the Set 4 of Hamiltonian parameter data from Appendix pp4.

Figure 10

The ${\lambda }_c$ spin-orbit coupling values necessary to compensate at $\lambda =0$ the common deviations $\mathrm{\Delta }{t}_c\ne 0$ and $\mathrm{\Delta }{t}_f\ne 0$ at (a) $B=0$ and (b) $B\ne 0$ in order to recreate a flat band placed originally (at $\lambda ={\lambda }_c=B=0$) in the origin of the energy axis $\epsilon =0$. The new position of the flat band is at $\epsilon =\mathrm{\Delta }E$. In this figure $\mathrm{\Delta }{t}_c/t_c>0$ holds. For exemplification, we have used the Set 4 of Hamiltonian parameter data from Appendix pp4.

Figure 11

The ${\lambda }_c=\lambda =\overline{\lambda }$ spin-orbit coupling values necessary to compensate the common deviations $\mathrm{\Delta }{t}_c\ne 0$ and $\mathrm{\Delta }{t}_f\ne 0$ at $B\ne 0$ and (a) $\mathrm{\Delta }{t}_c/t_c<0$ and (b) $\mathrm{\Delta }{t}_c/t_c>0$, in order to recreate a flat band placed originally (at $\overline{\lambda }=B=0$) at the origin of the energy axis $\epsilon =0$. The new position of the flat band is at $\epsilon =\mathrm{\Delta }E$. For exemplification, we have used the Set 4 of Hamiltonian parameter data from Appendix pp4.

Figure 12

The many-body spin-orbit interaction effect on the whole band structure: (a) the $\lambda ={\lambda }_c=B=0$ case providing dispersive bands, and (b) at $B=0$ the $\lambda /t=0.178$ providing a double-degenerate flat band (see Appendix pp3). For the Hamiltonian parameters, see text.

Figure 13

The flat band originally placed at the origin of the energy axis becomes dispersive under the action of $\mathrm{\Delta }{t}_c$ at zero SOI couplings. The line in the middle of the dispersive band (at position $\mathrm{\Delta }{\overline{E}}_{\text{median}}$) shows the median of the band, while $\mathrm{\Delta }{\overline{E}}_{\text{min}}$ and $\mathrm{\Delta }{\overline{E}}_{\text{max}}$ denote the minimum and maximum positions in the dispersive band relative to the median. The $\mathrm{\Delta }{\overline{E}}_{\alpha }, \alpha =\text{median,}\text{min,}\text{max}$, values are indicated in $t$ units. For exemplification, we have used the Set 1 of Hamiltonian parameter data from Appendix pp4.

Figure 14

The different $\mathrm{\Delta }{\overline{E}}_{\alpha }$ quantities from Fig. 12 as function of $\mathrm{\Delta }{t}_c$ normalized to $t_c=t_c^{\text{rfbc}}$. The presented cases: (a) the median ($\mathrm{\Delta }{\overline{E}}_{\text{median}}$), (b) the maximum ($\mathrm{\Delta }{\overline{E}}_{\text{max}}$), and (c) the minimum ($\mathrm{\Delta }{\overline{E}}_{\text{min}}$). For exemplification, we have used the Set 1 of Hamiltonian parameter data from Appendix pp4.