Arbitrary Control of Non-Hermitian Skin Modes via Disorder and An Electric Field

TL;DR

The work addresses arbitrary control of the non-Hermitian skin effect (NHSE) in two-dimensional lattices by combining disorder with a static electric field. Analytically, the disorder-free case is solved via a similarity transformation, revealing Stark localization and field-driven Bloch oscillations that suppress NHSE; with disorder, biorthogonal Wannier–Stark states unlock transverse transport, allowing boundary localization at programmable positions by tuning the field orientation and nonreciprocal direction. The study further shows geometry-dependent skin modes in reciprocal lattices, and demonstrates robustness through ultra-long-time dynamics and open-system (Liouvillian) mappings that reproduce the same transport-localization behavior. Together, these results establish a tunable mechanism for boundary accumulation and directed transport with potential applications in classical metamaterials and quantum materials, including realizations in open quantum settings.

Abstract

The non-Hermitian skin effect (NHSE), characterized by the accumulation of a macroscopic number of bulk states at system boundaries, is a hallmark of non-Hermitian physics. However, effective control of skin-mode localization in higher-dimensional systems remains a significant challenging. Here, we propose a versatile approach to manipulate the localization of skin modes in two-dimensional non-Hermitian lattices by combining disorder with a static electric field. While the electric field alone suppresses the NHSE in a clean system, the introduction of disorder induces transverse wave-packet transport perpendicular to the field. In nonreciprocal lattices, when the nonreciprocal hopping is misaligned with the electric field, the hopping component perpendicular to the field guides wave-packet propagation and produces boundary localization. By tuning the relative orientation between the electric field and the nonreciprocal hopping direction, the boundary localization position can be continuously and arbitrarily controlled. We further demonstrate distinct geometry-dependent manipulation of skin modes in reciprocal lattices, where controllable boundary localization emerges solely from the lattice geometry. These results establish a robust and tunable mechanism for engineering boundary accumulation and directed transport in non-Hermitian systems, offering new opportunities for applications in classical platforms and quantum materials.

Figures (7)

• Figure 1: (a) Schematic of the 2D Hatano–Nelson model with nonreciprocal hopping, subject to a static electric field $\mathbf{F}$, oriented at angle $\phi$ (magenta arrow), and random on-site disorder. The nonreciprocal hopping direction (green arrow) is encoded in the vector $\mathbf{g} = (g\cos\theta,~g\sin\theta)$. (b) Center-of-mass $\langle x(t_\textrm{f}) \rangle$ and $\langle y(t_\textrm{f}) \rangle$, in the clean limit ($\xi=0$), at different $\theta \in (-\pi,\pi]$ (red arrows indicate increasing $\theta$) and grouped into quadrants $\textrm{I}$--$\textrm{IV}$. The initial Gaussian wave packet is centered at $\mathbf{r}=(0,0)$ on a $61\times61$ lattice with $g/J=1$ and $F/J=0$, evolved up to $t_\textrm{f}=20T_J$ with $T_J=2\pi/J$. (c) Time evolution of the center-of-mass $\langle \mathbf{r}(t) \rangle$ [red solid (blue dashed) line denotes $x$ ($y$) component] in the clean limit ($\xi=0$), for $(g/J,\theta,F/J,\phi) = (1,\pi/4,0,0)$, with time in units of $T_J$. (d) $\langle \mathbf{r}(t) \rangle$ under a finite electric field $F/J=0.8$ oriented at $\phi=\pi/4$, with time rescaled by $T_x=2\pi/F_x$.
• Figure 2: Time-evolution trajectory of the center of mass, $\langle x(t) \rangle$ and $\langle y(t) \rangle$, for an initially Gaussian wave packet centered at the origin, under an electric field with $F/J = 1.5$ and random on-site disorder $\xi/J = 1.0$. (a) Hermitian case with $g/J = 0$ and $\phi = \pi/4$. (b–f) Non-Hermitian cases with $g/J = 1$, for $(\theta,\phi) = (\pi/4,\pi/4)$ (b), $(\pi/4,-3\pi/4)$ (c), $(\pi/4,\pi/12)$ (d), $(\pi/4,5\pi/12)$ (e), and $(\pi,\pi/4)$ (f). Circle color encodes time in units of the Bloch period $T_B = 2\pi/F$. Insets show the corresponding probability density distributions after long-time evolution. (g–l) Corresponding time-resolved second moment $\langle \mathbf{r}^2(t) \rangle$, where red solid and blue dashed lines denote components parallel and perpendicular to the electric field, respectively. All results are averaged over 1000 disorder realizations.
• Figure 3: Trajectories of the center of mass, $\langle x(t_\mathrm{f})\rangle$ and $\langle y(t_\mathrm{f})\rangle$, of an origin-centered Gaussian wave packet after long-time evolution under nonreciprocal hopping, electric field, and random on-site disorder, versus field orientation $\phi$ (a), nonreciprocity strength $g$ (b), field strength $F$ (c), and disorder strength $\xi$ (d). Insets mark the NHSE (green) and field (magenta) orientations. Colors denote the varied parameter (increasing along red arrows). Except for the varied parameters, others are fixed at $g/J=1.0$, $F/J=1.5$, $\xi/J=1.0$ and $t_\textrm{f} = 20 T_B$, with $(\theta, \phi)=(\pi/4,0)$ in (a), and $(\pi/4,\pi/3)$ in (b-d). (e-h) IPR of the evolved states at time $t_\textrm{f}$. All results are averaged over 1000 disorder realizations.
• Figure 4: (a,b) Spatial distributions of eigenstates $W(\mathbf{r})$ for the square geometry (a) and the up- and down-triangle geometries (b), with color bars indicating intensity. (c–f) Time-evolution trajectories of the center of mass, $\langle x(t)\rangle$ and $\langle y(t)\rangle$, for an initial wave packet located at the site marked by the red arrow. Results are shown for (c) the square geometry with $(F/J,\phi) = (2.5,\pi/3)$ and for (d–f) triangle geometries with (d) $(2.0,\pi/24)$, (e) $(2.5,\pi/3)$, and (f) $(4.0,7\pi/12)$. Circle color denotes time in units of $T_B = 2\pi/J$. Insets display the probability density distributions after a long-time evolution. Parameters: $J=1$, $J_x = 1.0J$, $J_y = 0.5iJ$, $\xi/J = 0.5$, and $L_x \times L_y = 61 \times 61$. All results are averaged over 1000 disorder realizations.
• Figure S1: Time-resolved center-of-mass $\langle \mathbf{r}(t) \rangle$ obtained numerically (top panels), and analytically (bottom panels), for an initial state $\psi_{\mathbf{r}^\prime}(0) = \delta_{\mathbf{r}^\prime,0}$ centered at the origin on $61\times61$ square lattice. Left: $F/J=0.8$ with $\theta=\pi/4$ for different values of $\phi$. Right: $F/J=1.5$ with $\phi=\pi/4$ for different values of $\theta$. Time is measured in unit of $T_x = 2\pi/F_x$. Other parameter used is $g/J=1.0$.