Anisotropic-scaling localization in higher-dimensional non-Hermitian systems

TL;DR

This work identifies anisotropic-scaling localization (ASL) as a distinct higher-dimensional non-Hermitian localization phenomenon, where corner or edge states exhibit size-dependent localization lengths that scale anisotropically with system dimensions. By solving minimal 2D models and analyzing BBH-type higher-order boundaries, the authors show ASL can arise from two mechanisms: effective bulk-edge couplings and edge junctions, and they establish a scaling law linking ASL length to system size. They further demonstrate that ASL can coexist with non-Hermitian skin states, produce complex-energy spectra forming loops, and persist under disorder and long-range couplings, extending the framework into 3D and experimental platforms like cold atoms. Overall, ASL provides a unifying, analytically tractable pathway to classify and understand complex, size-dependent localization in finite non-Hermitian lattices, with clear implications for detecting weak non-Hermiticity and guiding experimental realizations.

Abstract

Spatial localization of quantum states is one of the focal points in condensed matter physics and quantum simulations, as it signatures profound physics such as nontrivial band topology and non-reciprocal non-Hermiticity. Yet, in higher dimensions, characterizing state localization becomes elusive due to the sophisticated interplay between different localization mechanisms and spacial geometries. In this work, we unveil an exotic type of localization phenomenon in higher-dimensional non-Hermitian systems, termed anisotropic-scaling localization (ASL), where localization lengths follow distinct size-dependent scaling rules in an anisotropic manner. Assisted with both analytical solution and numerical simulation, we find that ASL can emerge from two different mechanisms of effective bulk couplings or one-dimensional junction between different 1D edges, depending on how non-reciprocity is introduced to the system. The competition between ASL states and edge non-Hermitian skin states are further identified by their complex and real eigenenergies, respectively. Our results resolve the subtle co-existence of loop-like spectrum and skin-like localization of boundary states in contemporary literature, and provide a framework to classify the intricate higher-order non- Hermitian localization regarding their localization profiles.

Figures (13)

• Figure 1: (a) A sketch of the model with $L_x=L_y=6$. Blue arrows indicate the non-Hermitian non-reciprocal pumping direction. (b1)-(b3) Energy spectra under OBCs with $L_x=30$ and different $L_y$. An extra term of pseudospin-dependent imaginary energies $ig\sigma_z$ with $g=0.15$ are added to the Hamiltonian for (b3). Eigenenergies are marked by different colors according to the fractal dimension $D[\psi_n]$. (c) Summed distribution of bulk states (yellow dots) and edge states (blue dots) in (b1). Size of each dot indicates the value of the total density for bulk and edge states at each lattice site, defined as $\rho_\text{bulk/edge}=\sum_n\vert\psi_{n}(x,y)\vert^2$, where the summation of $n$ runs over all states with $D[\psi_n]\approx 2$ and $D[\psi_n]\approx 0$, respectively. $\rho_{\rm edge}$ along the bottom and top edges ($y=1$ and $y=L_y$) is further demonstrated in (d). Other parameters are $t_1=0.25,t_2=1.0,t_x^+=0.35,t_x^-=0.05$.
• Figure 2: Average localization length (along $x$ direction) of corner states, $\bar{\xi}_x$, versus different parameters under OBCs. (a) $\bar{\xi}_x$ (dots) versus $L_y$ with $L_x=24$ and $36$, respectively, in comparison with the analytical localization lengths of ASL ($\xi_x^{\rm AS}$, dash lines) and NHSE states ($\xi_x^{\rm skin}$, gray line). (b) Numerical (dots) and analytical (dashed line) results of the maximal imaginary eigenenergies, $\text{Max}[\text{Im}(E)]$, versus $L_y$ corresponding to (a). It is seen that $\bar{\xi}_x\approx{\rm Max}\{ {\xi}_x^{\rm skin},{\xi}_x^{\rm AS}\}$, and the transition point of ${\xi}_x^{\rm skin}={\xi}_x^{\rm AS}$ matches the complex-real transition of eigenenergies. (c) $\Delta\xi_x\equiv\bar{\xi}_x-\xi_x^{\rm skin}$ versus $L_x$ and $L_y$. The analytical result of $\xi_x^{\rm AS}=\xi_x^{\rm skin}$ (red line) consists with the transition between $\Delta\xi_x=0$ and $\Delta\xi_x>0$. Negative $\Delta\xi_x$ (blue) is observed around the transition line between ASL and NHSE. (d) $\xi_x^{\rm AS}$ of Eq. \ref{['eq:infinite_L']} (dashed line) and $\bar{\xi}_x$ (dots) versus $t_1$ for our model at different sizes. With the size increased, numerical results are seen to approach the analytical prediction for the thermodynamic limit. (e) $\Delta\xi_x$ versus $t_1$ and $t_x^+$ for $L_x=L_y=30$, with analytical results of $\xi_x^{\rm AS}=\xi_x^{\rm skin}$ at this size (red line) and the thermodynamic limit (cyan dash line). Parameters are $t_1=0.25,t_2=1,t_x^+=0.35$, $t_x^-=0.05$, unless specified otherwise in the figures. In (c) and (e), $10^{-3}$ is chosen as the threshold of negative $\Delta\xi_x$ to distinguish from the black region with small negative $\Delta\xi_x$ due to numerical inaccuracy.
• Figure 3: NHSE and ASL of the non-Hermitian BBH model. (a1) Energy spectrum of the non-Hermitian BBH model with non-Hermicity only along $x$ direction. Insets display the distribution of the eigenstates marked by arrows. (a2) Comparison between the eigenenergies of the edge states (colored solid dots) in (a1) and that of the effective 1D junction model formed by the edges of the 2D system (gray circles). Colors indicate the edge distribution ratio $r_{\rm LR}$ of each edge states. Parameters are $t_x^+=2,t_x^-=0.25,t_y^+=t_y^-=1,t'=0.25,N_x=30$ and $N_y=20$. (b1) and (b2) the same as (a1) and (a2), but with non-Hermicity along both directions. Parameters are $t_x^+=1.5,t_y^+=2.5,t_x^-=t_y^-=0.5,t'=0.25$ and $N_x=N_y=20$. (c) Edge spectra with positive real energy (marked by $r_{\rm LR}$), with $N_x=20$, ${\rm Re}[E]>0$, and $N_y=20,30,35$ for (c1)-(c3) respectively. (d) Real part of edge spectra with $N_x=20$ and ${\rm Re}[E]>0$ versus $N_y$. Colors indicate the value of $\ln\vert{\rm Im}[E]\vert$. The three dash lines mark the cases for (c1) to (c3), respectively. (e) Localization length along $x$ ($y$) direction of corner state with the minimal (maximal) absolute value of real energy, $\xi_x(E_{\rm min})$ [$\xi_y(E_{\rm max})$], versus the system's size $N_x$ and $N_y$. Here $\xi_{x/y}$ is defined regarding the sublattice strucutre of the BBH model; see Supplemental Materials suppmat for more details. Other parameters in (c) to (e) are same as that in (b).
• Figure S 1: (a) Energy spectrum of the non-Hermitian BBH model with non-Hermiticity added along $x$ direction. The system's size is $N_x=30$ and $N_y=10,20,30$ for (a1) to (a3), respectively. (b) Comparison between the eigenenergies of the edge states (colored solid dots) in (a) and that of $H_\text{eff}^\text{y-edge}$, the effective 1D ladder model formed by the top and bottom edges of the 2D systems with size-dependent couplings induced by the bulk (black squares). Colors indicate the edge distribution ratio $r_{\rm LR}$ of each edge state. $r_{\rm LR}\approx 1$ and $0$ correspond to eigenstates distribute mostly on the left/right and top/bottom edges, respectively. It is seen that the eigenenergies of $H_\text{eff}^\text{y-edge}$ match well with those of the top/bottom edge states. (c) and (d) the same plots as in (a) and (b), but with non-Hermiticity added along both directions. The eigenenergies of the edge states and that of $H_\text{eff}^\text{y-edge}$ do not coincide with each other. Parameters are $t_x^+=2,t_x^-=0.25,t_y^+=t_y^-=1,t'=0.25$ in (a) and (b), and $t_x^+=1.5,t_x^-=0.5,t_y^+=2.5,t_y^-=0.5,t'=0.25,N_x=N_y=20$ in (c) and (d).
• Figure S 2: ASL of the 3D BBH model with non-Hermiticity added along $x$ direction. (a) Energy spectrum of the 3D non-Hermitian BBH model under OBCs with $N_x=N_y=N_z=8$. Eigenenergies are marked by different colors according to the 3D fractal dimension, defined as $D[\psi_n]=-\ln I/\ln\sqrt[3]{8N_x N_y N_z}$ and $I=\sum_{x,y,z}|\psi_{x,y,z}|^4$. Corner states correspond to $0\lesssim D[\psi_n]<1$, and form two loops in the complex energy plane. (b) Summed distribution of corner states with loop-like spectrum in (a). Size of each red sphere indicates the value of the total density for corner states at each site, defined as $\rho_{\rm loop}=\sum_n\vert\psi_n(x,y,z)\vert^2$, where the summation of $n$ runs over all states with loop-like spectra in (a). (c) Average localization length along $x$ direction of corner states with loop-like spectra, $\bar{\xi}_x$, versus $N_x$ with different $N_y=N_z$. (d) $\bar{\xi}_x$ versus $N_y$ with $N_x=N_z$. (e) $\bar{\xi}_x$ versus $N_z$ with $N_x=N_y$. Gray solid lines in (c)-(e) indicate $\xi_x^{\rm skin}=-2/\ln(t_x^-/t_x^+)$. (f) Inverse of the average localization length along $x$ direction of corner states with loop-like spectra, $1/\bar{\xi}_x$, versus $N_z$ with $N_x=N_y$. (g) $\bar{\xi}_x$ versus $N_y$ and $N_z$ with $N_x=8$. (h) $1/\bar{\xi}_x$ versus $N_z$ with $N_x=8$ and $N_y=6,8,10,12$ respectively. Gray solid lines in (f) and (h) indicate $1/\xi_x^{\rm skin}$. Other parameters are $t'=0.5,t_y=t_z=1,t_x^-=0.75,t_x^+=4.25$.