Second-order Skin Effect in a Brick-Wall Lattice

Abstract

Non-Hermitian skin effect, which is a unique feature of non-Hermitian systems, exhibits the formation of an extensive number of boundary modes under open boundary conditions. However, its manifestation in higher dimensions remains elusive. In our work, we demonstrate a hybrid skin-topological effect arising from the interplay between first-order band topology and non-reciprocal hopping in an engineered two-dimensional brick-wall geometry. The non-Hermitian brick-wall lattice under open boundary conditions in both directions exhibits several unconventional spectral features. Notably, the eigenvalues associated with the corner skin modes do not exhibit non-trivial windings in the complex energy plane; instead, they exhibit dynamically stable exceptional point-like features that do not originate from eigenvector coalescence. In contrast, the remaining modes accumulate at the opposite pair. Of all the corner skin modes, only the two that originate from the topological corner states of the Hermitian brick-wall lattice remain localized at individual corners, while the rest accumulate at the pair of opposite corners. This spatial distribution contrasts sharply with the second-order skin effect, where corner skin modes are more uniformly distributed. Finally, for the non-Hermitian Brick-wall lattice, we design and implement the corresponding topolectrical circuit (circuit for a square lattice is included for comparison) to directly visualize the hybrid skin-topological modes.

Figures (7)

• Figure 1: (a) A schematic representation of the modified BW lattice is shown. The vertical hopping, which is only allowed between the intracell $A$ and $C$ sublattices and intercell $B$ and $D$ sublattices, is denoted by $t$. The horizontal direction resembles a stacked SSH model with a bipartite hopping structure characterized by hopping strengths $t_1$ and $t_2$. This model is Hermitian, with hopping allowed in both the forward and backward directions. (b) The NH version of the BW lattice, where the hoppings between $A\;(B)$ and $C\;(D)$ sublattices are unidirectional.
• Figure 2: The bulk bandstructure showing (a) the presence of four-fold Dirac cones for $t<t_1+t_2$ and (b) the gapped spectra for $t>t_1+t_2$.
• Figure 3: The ribbon spectra for (a) $t<t_1+t_2$ and (b $t>t_1+t_2$. For $t>t_1+t_2$, the spectrum is completely gapped. However, the existence of an in-gap flat band is observed.
• Figure 4: (a) The real part of the energy spectra of a finite NH square lattice (OBC in both directions) for $t<t_1+t_2$. (b) The spectral distribution in the complex plane. The corner skin modes are arranged in the form of an ellipse. (c) IPR of the corner skin modes is non-zero. (d) The probability distribution of a random second-order skin mode, shown via yellow dots. For clarity, these dots are marked inside red circles.
• Figure 5: (a) TEC design of NH square lattice. (b) Grounded inductors are associated with each node. (c) IP of the TEC, imitating the corner skin modes (yellow dots shown inside the red circles) in Fig. \ref{['Fig4']}(d).