Eigenenergies and quantum transport properties in a non-Hermitian quantum-dot chain with side-coupled dots

We study the eigenenergies of a one-dimensional quantum-dot chain with its each dot side coupling to two additional dots. It is found that when the $\mathcal{P}T$-symmetric complex potentials are introduced to the side-coupled dots, the eigenlevel degeneracy is broken. However, further increasing the complex potentials induces a kind of two-degree degeneracy of the eigenlevels. This is accompanied by the $\mathcal{P}T$-symmetry breaking, with the appearance of the complex part of the eigenlevels. These changes exactly affect the quantum transport properties of the chain. First, in the case of weak $\mathcal{P}T$-symmetric complex potentials, a group of transmission function peaks arise at the center of the transmission function spectrum. When the eigenlevel degeneracy takes place, the degenerated eigenlevels decouple from the leads and the corresponding peaks disappear in the transmission function spectra. We believe that this work provides helpful information for a better understanding of the eigenenergies and quantum transport properties in non-Hermitian systems.

Figure 1

Schematic of a non-Hermitian QD chain, whose terminal QDs of the chain couple to lead-$L$ and lead-$R$, respectively. In addition, each QD in the chain couples to two QDs laterally, which are affected by the $\mathcal{P}T$-symmetric complex on-site chemical potentials.

Figure 2

Spectra of the eigenenergies of the $\mathcal{P}T$-symmetric chain with $N=1$, affected by the presence of $\mathcal{P}T$-symmetric complex potentials. Relevant parameters are taken to be ${\varepsilon }_l={\xi }_{lj}=0$ and $v_c=0.5$. (a) and (b) Description of the real and imaginary parts of the eigenenergies.

Figure 3

Real and imaginary parts of the eigenenergies of the QD chain with its cell number from 2 to 5, affected by the presence of $\mathcal{P}T$-symmetric complex potentials. Relevant parameters are taken to be ${\varepsilon }_l={\xi }_{lj}=0$ and $t_c=v_c=0.5$. (a) and (b) Real and imaginary parts of $N=2$; (c) and (d) results of $N=3$; (e) and (f) $N=4$; (g) and (h) $N=5$.

Figure 4

Real and imaginary parts of the eigenenergies of the $N=3$ and $N=4$ chains due to $\mathcal{P}T$-symmetric complex potentials applied on the side-coupled QDs partially. (a)–(d) Results of $N=3$ structure when the potentials applied on the ending and nonending cells. (e)–(h) Corresponding results of the structure of $N=4$.

Figure 5

(a) Spectrum of the real eigenenergy (i.e., $E_1$) of the infinitely long QD chain. (b) and (c) Real and imaginary parts of $E_2$.

Figure 6

Spectra of the TF of the triple-cell system, following the increase of $\mathcal{P}T$-symmetric complex potentials. The chain-lead coupling is taken to be $t_{\alpha }=0.1$.

Figure 7

(a)–(c) TF spectra of the systems of $N=2$, 3, and 5, due to the increase of $\mathcal{P}T$-symmetric complex potentials. The chain-lead coupling is taken to be $t_{\alpha }=0.1$. (d) Results of the triple-cell structure when the complex potentials are applied on the ending cells.

Figure 8

Spectra of the TF of our considered system, in the absence and presence of $\mathcal{P}T$-symmetric complex potentials. Relevant parameters are taken to be ${\varepsilon }_l={\xi }_{lj}=0$ and $t_c=v_c=t_{\alpha }=0.5$.

Figure 9

TF spectra of the $N=3$ and $N=4$ systems, influenced by the increase of $\mathcal{P}T$-symmetric complex potentials. The other parameters are the same as those in Fig. 2.

Figure 10

Spectra of the TF of our considered system, influenced by the increase of $\mathcal{P}T$-symmetric complex potentials. (a) and (b) $N=3$ results of the potentials applied on the ending and nonending cells, respectively. (c) and (d) Corresponding results of the $N=4$ structure.