Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree $d$. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around $W\sim 20$ for a random regular graph $d=4$. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around $W\sim 5$, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

Figure 1

Numerical study of the single-particle Anderson transition for the RRG using the fractal dimension $D_4$ as a localization measure. The results of calculations for networks with $N=1024$ nodes are shown. (a)–(f) Adjacency matrix spectra for different values of reciprocity $r$ and hopping asymmetry $p$. Eigenvalues are colored according to the values of the fractal dimension $D_4$; the color of the frame reflects the parameters set at the corresponding point in (g)–(j). (a)–(c) Same spectrum, but colored by the left, biorthogonal, and right $D_4$ for $d=4, r=0.2, p=0.5$; (d) the right $D_4$ for $d=4, r=0.2, p=0.05$; (e) the right $D_4$ for $d=4, r=0.9, p=0.05$; (f) the right $D_4$ for $d=4, r=0.9, p=0.5$. The left and right localized states cluster in the center of the adjacency matrix spectrum with the formation of a characteristic mobility edge. Dependence of the (g),(i) left and right and (h),(j) biorthogonal fractal dimension $D_4$ on (g),(h) reciprocity $r$ and node degree $d$ for $p=0.5$, and (i),(j) $r$ and hopping asymmetry $p$ for $d=4$. Fractal dimensions are averaged over 16 realizations, averaging over the left and right fractal dimensions calculated as independent. The black lines on (g),(h) and the black dots on (i),(j) denote the critical value of reciprocity calculated by (9) for the studied system size.

Figure 2

Fractal dimension for states sorted in ascending order. (a) Left and right fractal dimension $D_2$. (b) Biorthogonal fractal dimension $D_2$. The green (light-gray) dash-dotted and solid lines show states for the adjacency matrix without diagonal disorder ($W=0$); the red (dark-gray) dash-dotted and solid lines show states with diagonal disorder $W=1$. The solid lines represent numerical data for $N=1024$ and 64 realizations; the dash-dotted lines correspond to $N=4096$ and 16 realizations. The blue dotted and purple dashed lines show states with ($W_{\mathrm{EP}}={10}^{-3}$) and without ($W_{\mathrm{EP}}=0$) small diagonal disorder, for the case without $\epsilon$-perturbation of the adjacency matrix (see Sec. 4a for details; the adjacency matrix is defective without $\epsilon$-perturbation and without diagonal disorder at the studied parameters). (c) Logarithmic scale of unnormalized spectral density with $\epsilon$-perturbation and $W=0$. The results of the calculations for the modified RRG $d=4$ ($N=1024$ nodes, 64 realizations) with reciprocity $r=0.1$ and hopping asymmetry $p=0.5$.

Figure 3

Numerical study of the single-particle Anderson transition for the RRG using the fractal dimension $D_6$ as a localization measure calculated from the average ${\mathrm{IPR}}_q$ over the spectrum and all realizations. The results of the calculations for networks with $N=1024$ nodes are shown. Dependence of the (a) left and right and (b) biorthogonal fractal dimension $D_6$ on reciprocity $r$ and hopping asymmetry $p$ for $d=4$. Fractal dimensions are averaged over 16 realizations, averaging over the left and right fractal dimensions calculated as independent.

Figure 4

The emergence of localization in various models of random graphs ($N=1024$) with varying hopping asymmetry $p$, reciprocity $r$, and specific parameters of models, considering left and right eigenvectors separately (see Sec. 3 for a description of the specific parameters of each model). The fractal dimension $D_4$ is averaged for $8\times 2$ samples per color cell. (a),(b) BA model with different hopping asymmetry $p$, reciprocity $r$, and minimal node degree $m$. (c),(d) ER model with different $p, r$ and the probability of the edge appearance, $p_{\mathrm{ER}}$. (e)–(g) WS model with different $p, r$, number of connected neighbors in a ring topology $k$ for a node, and the probability of the edge rewiring, $p_{WS}$. (h) RL model with different $p$ and $r$.

Figure 5

Examples of structural patterns that provide the emergence of localization transition in the RL model with periodic boundary conditions ($8\times 8$ nodes and $r=0$): (a) drain nodes, (b) source nodes, (c) TEN out, (d) TEN in. Color plots show the square module of the eigenvector with normalization, ${\sum }_n{\left|{\varphi }^{L\left(R\right)}\left(n\right)\right|}^2=1$.

Figure 6

(a) Left and right fractal dimensions $D_2$ for the RRG for different $r$ and $W, p=0.5$; cuts by $r$ (inset). (b) Biorthogonal fractal dimensions $D_2$ for the RRG for different $r$ and $W, p=0.5$; cuts by $r$ (inset). Spectra are colored by fractal dimension $D_2$: (c) left, (d) biorthogonal, and (e) right fractal dimensions for the same realization, $W=4, r=0.375$; (f)–(h) right and (i)–(k) biorthogonal fractal dimensions with different absolute values of diagonal disorder, $W=5,10,15, r=0.25$. The system size is $N=1024$.

Figure 7

Examples of spectra of the RRG $d=4$ with $N=1024$ adjacency matrices depending on $r$ and $W, p=0.5$, colored by right fractal dimensions. For all subfigures, the diagonal disorder distribution has the same realization from $[-1/2;+1/2]$ but multiplied by $W$.

Figure 8

Examples of spectra of the RRG $d=4$ with $N=1024$ adjacency matrices depending on $r$ and $W, p=0.5$, colored by biorthogonal fractal dimensions. For all subfigures, the diagonal disorder distribution has the same realization from $[-1/2;+1/2]$ but multiplied by $W$.