Magnetic Control of the Non-Hermitian Skin Effect in Two-Dimensional Lattices

TL;DR

This work addresses how magnetic fields influence the non-Hermitian skin effect (NHSE) in two-dimensional single-band lattices and clarifies how reciprocity and boundary geometry mediate this control. A theoretical framework is developed and applied to a non-Hermitian Harper–Hofstadter extension, deriving an effective 1D model along the strip and using spectral winding and real-space diagnostics to assess NHSE under flux. The results show that magnetic fields suppress the geometry-dependent NHSE in reciprocal lattices and can mitigate or alter NHSE in nonreciprocal lattices via Landau- or Anderson-type bulk localization, or by partial reciprocity restoration. These insights advance the understanding of gauge-field control of boundary phenomena in non-Hermitian systems and point to future directions including multiband, higher-order, and many-body/nonlinear extensions.

Abstract

The non-Hermitian skin effect (NHSE) -- the anomalous boundary accumulation of an extensive number of bulk modes -- has emerged as a hallmark of non-Hermitian physics, with broad implications for transport, sensing, and topological classification. A central open question is how magnetic or synthetic gauge fields influence this boundary phenomenon. Here, we develop a theoretical framework for magnetic control of the NHSE along line boundaries in two-dimensional single-band lattices. Using a non-Hermitian extension of the anisotropic Harper--Hofstadter model as a representative example, we show that magnetic fields suppress the geometric skin effect in reciprocal models, whereas skin localization can persist in nonreciprocal systems. The analysis disentangles the interplay of flux, nonreciprocity, and boundary geometry, revealing that magnetic fields mitigate or suppress the NHSE through distinct physical mechanisms -- such as bulk localization via Landau or Anderson physics, or the restoration of effective reciprocity. In particular, the geometry-dependent skin effect in reciprocal systems is found to be fragile against even weak magnetic fields.

Figures (8)

• Figure 1: (a) Schematic of a 2D crystal with a magnetic field $B$ applied orthogonal to the crystal plane $(X,Y)$. $\mathbf{a}_X = a_X \mathbf{u}_X$ and $\mathbf{a}_Y = a_Y \mathbf{u}_Y$ are the primitive vectors of the Bravais lattice $\mathcal{B}$, forming an angle $\alpha$ with each other. We consider a strip geometry (shaded area) defined along the $x$ and $y$ directions, where the system is infinite along $x$ (equivalently, periodic boundary conditions are imposed, $x$-PBC) and finite along $y$ with open boundary conditions ($y$-OBC). The $y$ axis is rotated relative to $x$ by the same angle $\alpha$ between the primitive vectors, while the $x$ axis is rotated by $\theta$ with respect to the principal axis $X$ of the crystal. (b) Schematic of the non-Hermitian anisotropic Harper-Hofstadter model (rectangular lattice with nearest-neighbor hopping).
• Figure 2: Effect of a rational magnetic flux on the energy spectrum and skin localization for the non-reciprocal Harper-Hofstadter model in a slab geometry [Eq.(32)]. Parameter values are $h_X=0$, $h_Y=0.2$, $J_X=J_Y=1$ and magnetic flux $\Phi=\alpha_x=\pi/2$. Lattice size in the $y$ direction is $L=500$. (a,b) Energy spectrum versus quasi-momentum $k_xa_X$ under $y$-OBC [panels (a)] and $y$-PBC [panels (b)]. Real and imaginary parts of $E$ are shown in the upper and lower panels, respectively. The curves in the gaps in panel (a) correspond to non-Hermitian extension of usual chiral edge states in the 2D quantum Hall model. (c,d) Energy spectrum in complex energy plane for $k_xa_X=0$ and for $y$-PBC [panel (c)] and $y$-OBC [panel (d)]. (e) Mean eigenvector distribution $I_n$ under $y$-OBC, clearly showing the persistence of the NHSE. The inset in (e) shows the IPR of the $L$ eigenstates under $y-$OBC.
• Figure 3: Same as Fig.2, but for a weak magnetic flux $\Phi=\pi/50$. Note that the skin area in panel (c) is reduced as compared to the case of Fig.2(c), and the corresponding mean eigenvector distribution $I_n$ in (e) tends to spread toward the bulk owing to the competing Landau localization.
• Figure 4: Same as Fig.2, but for an irrational magnetic flux $\Phi=(\sqrt{5}-1)/2$. Other parameter values are $J_X=2$, $J_Y=1$, $h_X=0$ and $h_Y=0.2$. In numerical simulations, the inverse of the golden ratio has been approximated by the rational number $\alpha \simeq 377/610$, i.e. ratio of Fibonacci numbers, and a lattice of size $L=610$ has been assumed. The energy spectrum is the same for $y$-PBC and $y$-OBC [apart for chiral edge states under $y$-OBC, visible in panel (a)], the eigenstates are bulk localized via Anderson localization and the NHSE is fully suppressed.
• Figure 5: Suppression of the skin effect in the isotropic Harper-Hofstadter model ($J_X=J_Y=1$) under an irrational magnetic flux ($\Phi=(\sqrt{5}-1)/2$) by the application of non-reciprocal hopping parameter $h_X$ in the $X$ direction. The panels show the numerically-computed behavior of the mean eigenstate distribution $I_n$ of the Hamiltonian under $y-$OBC for $h_Y=0.2$, quasi-momentum $k_xa_X=0$, and for a few increasing values of $h_X$: (a) $h_X=0$, (b) $h_X=h_Y=0.2$, and (c) $h_X=0.3$.