Enhancement of Second-Order Non-Hermitian Skin Effect by Magnetic Fields

The non-Hermitian skin effect is a unique phenomenon in which an extensive number of eigenstates are localized at the boundaries of a non-Hermitian system. Recent studies show that the non-Hermitian skin effect is significantly suppressed by magnetic fields. In contrast, we demonstrate that the second-order skin effect (SOSE) is robust and can even be enhanced by magnetic fields. Remarkably, SOSE can also be induced by magnetic fields from a trivial non-Hermitian system that does not experience any skin effect at zero field. These properties are intimately related to to the persistence and emergence of topological line gaps in the complex energy spectrum in the presence of magnetic fields. Moreover, we show that a magnetic field can drive a non-Hermitian system from a hybrid skin effect, where the first-order skin effect and SOSE coexist, to pure SOSE. Our results describe a qualitatively new magnetic field behavior of the non-Hermitian skin effect.

Figure 1

(a) Schematic of the model in Eq. (2) with OBC in $x$ and $y$ directions. The red thick and thin bonds indicate dimerized hoppings in the $y$ direction; the blue arrow lines indicate nonreciprocal hoppings in the $x$ direction, and the purple dots mark the SCMs. (b) Sketch of the energy spectrum (colored boxes) in the absence (top) and presence (bottom) of a magnetic field. Solid (dashed) lines indicate the spectrum of SCMs (edge modes). (c) $\log (w_{\mathrm{sc}})/\log (L)$ at $B{a}^2=0$ (blue) and $0.2\pi$ (orange) as a function of $\gamma$ for $L=160$ and $\xi =30$, respectively. Inset: skin corner weight $w_{\mathrm{sc}}$ as a function of $L$ at $B=0$ for $\xi =1$, $\gamma =0.4$, 0.5, 0.6, 1.0, and 1.6 (from top to bottom). (d) $w_{\mathrm{sc}}/L$ as a function of $\gamma$ and $B$ for $L=60$. $\lambda =1$ for (c),(d).

Figure 2

(a) $w_{\mathrm{sc}}/L$ as a function of $B$ for $\xi =1$ at different $\gamma$. (b) $|{\psi }^R(\mathbf{r}){|}^2$ of a SCM at $B=0$ under OBC. Inset: energy spectra under PBC (cyan) and OBC (orange), respectively. The arrow marks the mode energy $E=-0.46-1.03i$. (c) the same as (b) but at $B{a}^2=\pi /30$. The arrow marks the mode energy $E=-0.53-1.10i$. (d) Energy spectrum at $B{a}^2=0.2\pi$ under PBC. The Chern numbers of the Hofstadter bands are indicated. (e) $|{\psi }^R(\mathbf{r}){|}^2$ of an edge mode at $B{a}^2=0.2\pi$ under PBC (OBC) in $y(x)$ direction. The arrow marks the mode energy $E=-0.68-0.96i$. (f) $|{\psi }^R(\mathbf{r}){|}^2$ of a corner mode at $B{a}^2=0.2\pi$ under OBC. Insets in (e),(f): the corresponding energy spectra. The arrow marks the mode energy $E=-1.16-1.19i$. $\gamma =0.5$ in (b)–(f), and other parameters are $L=60$, $\lambda =1$.

Figure 3

(a) Energy spectra at $B=0$ under OBC (orange) and PBC (cyan), respectively. (b) $|{\psi }^R(\mathbf{r}){|}^2$ of a SCM at $B{a}^2=0.2\pi$ under OBC. Inset: the corresponding energy spectrum. The arrow marks the mode energy $E=-1.00-0.73i$. (c) $w_{\mathrm{sc}}$ as a function of $L$ at $B{a}^2=0.16\pi$ for different $\gamma ⩾1$. (d) $w_{\mathrm{sc}}/L$ as a function of $B{a}^2$ for $\gamma ⩾1$. $L=60$ for (a),(b),(d); $\gamma =1.2$ for (a),(b); and other parameters are $\lambda =1$ and $\xi =1$.

Figure 4

(a) Flow of the OBC spectrum of Eq. (3) as $B{a}^2$ increases from 0 (cyan) to $0.06\pi$ (red). (b) $|{\psi }^R(\mathbf{r}){|}^2$ of a SCM at $B{a}^2=0$ and $0.06\pi$, respectively. (c) $w_{\mathrm{sc}}$ as a function of $B{a}^2$ for $L=30$, 40, and 60, respectively. (d) $\alpha$ as a function of $B{a}^2$, extracted by fitting $w_{\mathrm{sc}}(L)\propto L^{\alpha }$ with $L\in [10,100]$, [20, 110], and [30, 130], respectively. Inset: $w_{\mathrm{sc}}$ as a function of $L$ at $B{a}^2=0$ and $0.2\pi$, respectively. $L=12$ in (a),(b) and other parameters are $t=\chi =1$, $h_x=h_y=0.1$, and $\xi =1$.