Pair states in one-dimensional Dirac systems

Analytic solutions of the quantum relativistic two-body problem are obtained for an interaction potential modeled as a one-dimensional smooth square well. Both stationary and moving pairs are considered and the limit of the $\delta$-function interaction is studied in depth. Our result can be utilized for understanding excitonic states in narrow-gap carbon nanotubes. We also show the existence of bound states within the gap for a pair of particles of the same charge.

Figure 1

Smooth Klein step for $L=0.1$ (solid line) and $L=0.001$ (dashed line).

Figure 2

Energy spectrum of an electron-hole pair interacting via a one-dimensional smooth square well as a function of $\mathrm{\Delta }W/L$ for $2W{V}_0/L=10$. The gray dashed (topmost) and black dashed (bottommost) lines represent $E=2\mathrm{\Delta }$ and $E-2V_0=2\mathrm{\Delta }$, respectively. Lines 2 and 4 (in red) correspond to the even modes, while lines 1 and 3 (in blue) correspond to the odd modes.

Figure 3

Dependence of the exciton energy $E_b$ on the interaction strength $2V_0$: The left-hand side is for a semiconductor nanotube with $\mathrm{\Delta }W/L=1$; the right-hand side is for a narrow-gap tube with $\mathrm{\Delta }W/L=0.01$. The different lines correspond to different excitonic states.

Figure 4

Thick gray dotted, dashed, and dot-dashed lines denote $E=\sqrt{{\tilde{K}}^2+4{\mathrm{\Delta }}^2}$ for $\tilde{K}=0$, 1, and 2, respectively, while their black counterparts denote $E=\sqrt{{\tilde{K}}^2+4{\mathrm{\Delta }}^2}-V_0$. The solid green, blue, and red lines (the lowest, second from the bottom, and top lines in each set of lines, respectively) correspond to the eigenvalues of $\tilde{K}=0$, 1, and 2, respectively, and their dashed counterparts mark $E=\tilde{K}$.

Figure 5

Dependence of the exciton binding energy $E_b$ on the interaction strength $2V_0$ with $\mathrm{\Delta }W/L=0.01$ for $K=0$ (black topmost line in each set), $K=\mathrm{\Delta }$ (blue line, second from top in each set), $K=2\mathrm{\Delta }$ (green line, third from top in each set), and $K=3\mathrm{\Delta }$ (red bottommost line in each set).

Figure 6

Dependence of the exciton binding energy $E_b$ on the interaction strength $\alpha$, for an electron-hole pair interacting via the interaction potential (9) in the $\delta$-function limit. The different lines correspond to different excitonic states. The crosses indicate when $E_b=2\mathrm{\Delta }$, which occurs at $\alpha =2(1+n)\pi$, where $n=0,1,2,3,...$.