Non-Hermitian Aharonov-Bohm effect in the quantum ring

We investigate the topological properties, energy spectrum, and persistent currents of a non-Hermitian ring with anti-Hermitian hopping terms. It is demonstrated that the anti-Hermitian hopping can effectively induce a synthetic gauge field. As the magnetic flux of the synthetic gauge field threads through the ring, the non-Hermitian system exhibits an Aharonov-Bohm effect. For the case of a non-Hermitian ring in the topological phase, the system, having an energy spectrum structure with a real gap, supports an imaginary persistent current. For the trivial case, a non-Hermitian system with an imaginary gap supports a real persistent current. Furthermore, we also investigate the transport property of a non-Hermitian Aharonov-Bohm ring connected by two semi-infinite leads. We find that the transmission coefficient shows the Aharonov-Bohm quantum oscillation as a function of the synthetic gauge field.

Figure 1

(a) Schematic illustration of a non-Hermitian AB effect system. An array of dimeric lattices form a closed circle in the central scattering region. The nonzero magnetic flux threading the ring is induced by the anti-Hermitian coupling processes. The left and right terminals are exactly extended from unified lattices. (b) Detail of the tight-binding model of the asymmetric coupling processes. $t_L=t_1$ and $t_R=-t_1^{*}$ are intracell anti-Hermitian hopping terms which induce the Peierls phase, and $t_2$ is the intercell hopping term. The yellow box indicates the unit cell. (c) The interaction between two adjacent lattices in the unit cell has an angle in the complex plane.

Figure 2

Distribution of the energy spectrum in (a) the topological phase and (b) the trivial phase. The solid blue circles and the open red circles represent the eigenvalues of the non-Hermitian system under OBC and PBC, respectively. (c) Distribution of the eigenvectors on one-dimensional open chains. The blue and red lines show the wave functions of edge eigenvectors, and the remaining lines show the wave function of the bulk eigenvectors. (d) Geometric characterization of the non-Hermitian system. The parameters of the blue loop and the red loop correspond to the parameters in (a) and (b), respectively. The parameters are $N=56, t=0.5, t_2=1$, and (a) $\gamma =0.5$ and (b) $\gamma =1$.

Figure 3

Complex-energy spectra for the AB ring as functions of the real magnetic flux $\mathrm{\Phi }$. The projections of the energy spectra onto the complex plane are also presented with red lines for a better view. The system parameters are $N=6, t_2=1$, and (a) $|t_1|=1$, (b) $|t_1|=0.5$, (c) $|t_1|=1.5$, and (d) $t_1=1$.

Figure 4

Dimensionless persistent currents for the AB ring as functions of the magnetic flux $\mathrm{\Phi }$. The current-flux relation $I/I_0\left(\mathrm{\Phi }\right)$ is exhibited in (a) the $(I_{\mathrm{Re}\left(I\right)=\mathrm{Im}\left(I\right)},\mathrm{\Phi })$ plane, (b) the $(I_{\mathrm{Re}\left(I\right)=0},\mathrm{\Phi })$ plane, and (c) the $(I_{\mathrm{Im}\left(I\right)=0},\mathrm{\Phi })$ plane. The system parameters are $N=60, t_2=1$, and (a) $|t_1|=1$, (b) $|t_1|=0.5$, and (c) $|t_1|=1.5$.

Figure 5

(a), (c), (e), and (g) The transmission coefficient $T$ as a function of the Fermi energy $E$ and the magnetic flux $\mathrm{\Phi }$. (b), (d), (f), and (h) The transmission coefficient $T$ as a function of the magnetic flux $\mathrm{\Phi }$ with the fixed Fermi energy $E=0$ for (b) and (h), $E=0.86$ for (d), and $E=1.15i$ for (f). The coupling amplitude of intracell anti-Hermitian hopping is $|t_1|=1$ for (a) and (b), $|t_1|=0.5$ for (c) and (d), and $|t_1|=1.5$ for (e) and (f). The coupling amplitude of intracell Hermitian hopping is $|t_1|=1$ for (g) and (h). The other system parameters are $N=6$ and $t_2=1$. The coupling term between the left (right) terminal and the central scattering region is set to 0.4.