Supermetal-insulator transition in a non-Hermitian network model

We study a non-Hermitian and nonunitary version of the two-dimensional Chalker-Coddington network model with balanced gain and loss. This model belongs to the class ${\mathrm{D}}^{†}$ with particle-hole ${\mathrm{symmetry}}^{†}$ and hosts both the non-Hermitian skin effect as well as exceptional points. By calculating its two-terminal transmission, we find a contact effect induced by the skin effect, which results in a nonquantized transmission for chiral edge states. In addition, the model exhibits an insulator to “supermetal” transition, across which the transmission changes from exponentially decaying with system size to exponentially growing with system size. In the clean system, the critical point separating insulator from supermetal is characterized by a non-Hermitian Dirac point that produces a quantized critical transmission of 4, instead of the value of 1 expected in Hermitian systems. This change in critical transmission is a consequence of the balanced gain and loss. When adding disorder to the system, we find a critical exponent for the divergence of the localization length $\nu \approx 1$, which is the same as that characterizing the universality class of two-dimensional Hermitian systems in class D. Our work provides a way of exploring the localization behavior of non-Hermitian systems, by using network models, which in the past proved versatile tools to describe Hermitian physics.

Figure 1

(a) Illustration of the nonunitary network model. ${\psi }_i$ ($i=1$, 2, 3, 4) represents a propagating mode inside the unit cell (blue dashed square). Red and green nodes correspond to two types of $2\times 2$ scattering matrices $S_1$ and $S_2$ with $s\equiv sin\alpha$ and $c\equiv cos\alpha$, respectively. (b), (c) The real and imaginary eigenphase spectra in momentum space, respectively. The parameters are $\alpha =\pi /4$ and $\gamma =0.5$.

Figure 2

Topological phase limit with $\alpha =\pi /2$ (a) and trivial phase limit with $\alpha =0$ (b). (c), (d) Their corresponding ribbon geometry eigenphase spectra. The ribbon consists of three unit cells in the vertical direction and is infinite in the horizontal direction. The green, blue, and red colors indicate eigenstates localized at the bulk, the top edge, and the bottom edge, respectively.

Figure 3

(a) The eigenphases obtained in a ribbon geometry with a width of of 50 unit cells (purple color) are circled by the eigenphases of the infinite system. Outside of these winding contours, there appear chiral edge modes located on the ribbon boundaries (black points, shown by arrows). (b) The probability distribution of all the eigenvectors with open boundary conditions. All bulk states are pushed to the bottom boundary, but the top edge mode remains unaffected. The inset shows the sum of probability densities for all states on the bottom three unit cells, which we label “LDOS” by analogy to Hermitian systems. The summed probability density increases linearly with the width of the ribbon, indicating that an extensive number of states are present on the bottom boundary. The plots are with $\alpha =0.35\pi , \gamma =0.5$, and for panel (b) we chose $k_x=0.05$.

Figure 4

(a) The horizontal transmission $G_{\text{h}}$ with OBC as a function of $\alpha$ and $\gamma$. The system size is $L_x=3000$ and $L_y=60$. (b) $G_{\text{h}}$ for different $L_y$ with open boundary condition as a function of $L_x$. This plot is with $\alpha =0.3\pi$ and $\gamma =0.5$.

Figure 5

The transmission between top two leads as a function of $\alpha$. $G_{ij}$ corresponds to the transmission from $j$ to $i$. All plots are with $\gamma =0.3$ and a total system size $76\times 76$ unit cells. The inset is a sketch of the three-terminal geometry for OBC (left) and PBC (right) along the $x$ direction. The arrows indicate the chiral edge mode. Here, leads 1 and 2 have a width of 19 unit cells, and the separation between them is 20 unit cells. Lead 3 extends over the entire bottom boundary (or circumference of the cylinder, with PBC). Thus, $G_{12}$ (or $G_{21}$) between lead 1 and lead 2 provides the transmission of the chiral edge state along the top boundary of the system.

Figure 6

$\text{ln}\left[G_{\text{h}}\right]$ and $\text{ln}\left[r\left({\mathfrak{r}}_R{\mathfrak{r}}_L^{\prime }\right)\right]$ as a function of system size $L_x$. All plots are for $L_y=50, \alpha =\pi /4$, and $\gamma =0.05$.

Figure 7

The vertical transmission as a function of $\alpha$ and $\gamma$ both with periodic boundary conditions (a), (c) and with open boundary conditions (b), (d). (a), (b) Represent the transmission from top to bottom, (c), (d) Represent the transmission from bottom to top. All plots are obtained for a system size $60\times 60$ unit cells.

Figure 8

(a) The logarithm of transmission $G_{\text{v}}^{\text{t}\to \text{b}}$ of different system sizes with periodic boundary conditions as a function of $\gamma$ for $\alpha =0.3\pi$ and system size $10L\times L$ ($L_x\times L_y$). The solid lines are the fit using Eq. (18). (b) The critical transmission for different $\alpha$ as a function of system size $L_y$. This plot is obtained for $L_x=20$. When $\alpha =\pi /4$, the Dirac point occurs for $\gamma =0$, and thus shows the critical transmission characteristic of Hermitian Dirac cones $G=1$. For all other values of $\alpha$, the transmission is 4.

Figure 9

(a) The average transmission $\langle \text{ln}{G}_{\text{v}}^{\text{t}\to \text{b}}\rangle$ as a function of disorder strength $W$ and $\alpha$. The system size is $30\times 30$ and $\gamma =0.5$. (b), (c) Finite-size scaling fit of $\langle \text{ln}{G}_{\text{v}}^{\text{t}\to \text{b}}\rangle$ for $(\gamma ,\alpha )=(0.5,\pi /2)$ and $(0.2,0.35\pi )$, respectively. (d) Finite-size scaling fit of $\langle \text{ln}{G}_{\text{h}}\rangle$ with open boundary condition and $(\gamma ,\alpha )=(0.5,0)$.

Figure 10

$\text{arg}\left(\text{det}\right[\mathcal{S}\left(\theta \right)-{\mathbb{I}}_{4\times 4}\left]\right)$ as a function of $\theta$ (the parameter of the closed loop $s^1$), when encircling an EP (a) or not (b). Blue dots and orange triangles correspond to two different closed loops. The latter enclose one or the other EP in (a). All the plots are with $\alpha =\pi /4$ and $\gamma =0.5$.

Figure 11

(a) The probability distribution of the top edge state as a function of $\gamma$ with $\alpha =0.35\pi$ (${\gamma }_{\text{EP}}\approx 0.937$) and a system size $L_y=20$ unit cells. (b) The edge-mode probability distribution is summed over half of the system, corresponding to unit cells with position $n\le L_y/2$. This sum is plotted as a function of $\alpha$ and $\gamma$, using $L_y=80$. The blue dashed-dotted line indicates the critical $\gamma$ for EPs, showing that the top edge mode is pushed to the bottom boundary as soon as EPs appear in the spectrum of the infinite system. All plots are with $k_x=0.05$.

Figure 12

(a) Illustration of the nonunitary network model with the second-order skin effect. (b), (c) The real and imaginary eigenphase spectra. (d), (e) The ribbon geometry eigenphase spectra encircled by the bulk dispersion along the $k_x$ and $k_y$ directions, respectively. (f) The probability distribution of all the eigenstates with a system size $20\times 20$. All of the plots are obtained using $\alpha =\pi /4$ and $\gamma =0.5$.