Anomalous non-Hermitian skin effect: the topological inequivalence of skin modes versus point gap

However, we uncover that the open boundary eigenstates can exhibit the anomalous non-Hermitian skin effect [Fig. 2], where skin modes can be ever present even though the point gap is trivial, while skin modes are absent for the nontrivial point gap. This phenomenon occurs as long as the unidirectional hopping amplitudes leading to decoupling-like behaviors of the system are considered. Following the appearance of the inequivalence between skin modes and point gap, the open boundary eigenvalues will produce a sudden change. Specifically, the multifold exceptional points, whose degree of degeneracy is proportional to the system size, will be displayed in the open boundary eigenvalues which have significant differences compared with the continuum bands. The physical properties of the eigenvalues may have applications in the sensors field. Finally, an experimental setup with electric circuits is implemented to realize our system.

To clarify our results unequivocally, we start with the normal case for comparison. The non-Hermitian skin effect can be generally confirmed by the nontrivial point gap, or equivalently, by non-zero spectral winding $W=\frac{1}{2\pi i}\int_{0}^{2\pi}dk\partial_{k}\ln\det[H(k)-E_{ref}]$ Zhang et al. (2020) ^, Okuma et al. (2020); Gong et al. (2018); Denner et al. (2021); Yi and Yang (2020), in which $E_{ref}$ is an arbitrary reference point. Such as, we can choose $E_{ref}=\frac{1}{2}i$ and $W=1$ is received (see Methods and Supplementary Informationenn for more details). Thus, the system holds the non-Hermtian skin modes, as shown in Fig. 2b.

However, we find that this conclusion is not suitable for our non-Hermitian three-band system when the unidirectional hopping amplitudes are considered. To characterize the localization properties of all wave functions quantitatively, the inverse participation ratio ($IPR$) can be introduced as $IPR^{s}=\big{(}\sum\limits_{i=1}|\Psi_{i}^{s}|^{4}\big{)}/\big{(}\sum\limits_{i=1}|\Psi_{i}^{s}|^{2}\big{)}^{2}$ for the $sth$ eigenstate $|\Psi^{s}\rangle$ Li et al. (2017)^, Li and Das Sarma (2020). In large system size, $IPR^{s}$ is finite for localized eigenstate, while it approaches zero for extended eigenstate. As shown in Fig. 2c2, $\min(IPR^{s})=0.9406$ as the energy value changes under numerical simulation, which means that all eigenstates are localized, rather than extended, even if the spectral winding number is constantly equal to zero. Here, the information about the exceptional points [discussed next] and the corresponding eigenstates have been excluded. We can arbitrarily present the distribution of an open boundary eigenstate in Fig. 2c3. It is obviously localized at the left boundaries of the system. Furthermore, Fig. 2c4 exhibits the distribution of all open boundary eigenstates associated with eigenvalues, where eigenstates have been normalized. Globally, all open boundary eigenstates are pinned at the left boundaries of the system, i.e., the non-Hermitian skin effect is present unexpectedly.

The non-Hermitian skin effect can be unexpectedly absent with the unidirectional hopping amplitudes of $\gamma_{1}=\gamma_{4}=0$, even though the point gap is topologically nontrivial. In this case, three eigenvalue branches in momentum space are $E_{k1}=V_{a}+t_{a}e^{-ik}$, $E_{k2}=V_{c}+t_{c}e^{ik}$ and $E_{k3}=V_{b}+2t_{b}\cos(k)$. The complex energy spectra $E_{k1}$ ($E_{k2}$) will rotate clockwise (counterclockwise) on the complex plane, and $E_{k3}$ just forms a line as $k$ varies from $0$ to $2\pi$. Therefore, as long as $V_{a}\neq V_{c}$ or $t_{a}\neq t_{c}$, there must exist a reference point $E_{ref}$ (Here we can choose $E_{ref2}=\frac{1}{2}i$) being surrounded only once and the area of the curve must be nonzero Yi and Yang (2020), as shown in Fig. 2d1. Hence, it seems that our system holds the non-Hermitian skin effect.

However, we also can examine the localization properties of the open boundary eigenstates from the aspects of $IPR^{s}$. As shown in Fig. 2d2, $IPR^{s}\equiv 0.0075$, which indicates that all eigenstates should be extended, not localized on the open chain. One eigenvalue in Fig. 2d2 can be selected to exhibit the distribution of the eigenstates. As shown in Figs. 2d3, the open boundary eigenstates are indeed extended only on the $B$ sub-chain, which conforms to the vanishment of the non-Hermitian skin effect numerically. Physically, $A$ ($C$) sub-chain has only unidirectional hopping to sub-chain $B$, but the reverse does not happen when $\gamma_{1}=0$ ($\gamma_{4}=0$), and $B$ sub-chain is Hermitian, which can be solved analytically enn . Hence, there is no probability distribution on the sub-chain $A$ ($C$), and Bloch waves will be survived in the $B$ sub-chain. We further plot the distribution of all open boundary eigenstates corresponding to eigenvalues in Fig. 2d4. There is no large number of eigenstates localized at the boundary of the system, i.e., the non-Hermitian skin effect is absent whereas the point gap is nontrivial.

Obviously, the characteristic polynomial $f(\beta,E)$ is a quartic equation. The solutions can be numbered as $\left|\beta_{1}\right|\leq\left|\beta_{2}\right|\leq\left|\beta_{3}\right|\leq\left|\beta_{4}\right|$ for the given E and $\left|\beta_{2}\right|=\left|\beta_{3}\right|$ is necessary to determine the generalized Brillouin zone and the derivative continuum bands. As shown in Fig. 3a, the open boundary energy spectrum can be well recovered by the continuum bands with $\gamma_{2}=\frac{1}{500}$ and $\gamma_{3}=\frac{1}{400}$.

However, the situation will have remarkable mutations when $\gamma_{2}=\gamma_{3}=0$, corresponding to the presence of the anomalous non-Hermitian skin effect. As shown in Fig. 3b, there is very little change for the continuum bands, while the open boundary eigenvalues instead will produce mutations and lay on the real axis. Hence, the open boundary energies and the continuum bands will not be compatible with each other. Further, it takes for granted that if every open boundary eigenvalue is brought into the characteristic polynomial $f(\beta,E)$, and the second and third largest $|\beta|$ are selected, the data should belong to the generalized Brillouin zone obtained from the non-Bloch band theory Yao and Wang (2018); Yokomizo and Murakami (2019); Zhang et al. (2020); Yang et al. (2020)^, Yokomizo and Murakami (2020). However, unlike the established scenarios, this rule is nullified in the singular phenomenon. Detailedly, we can choose an open eigenvalue in Fig. 3b, such as $E_{OBC}=4$, and bring it into the characteristic polynomial $f(\beta,E)$ of our system. After calculation, one can receive $\big{\{}|\beta_{1}|,|\beta_{2}|,|\beta_{3}|,|\beta_{4}|\big{\}}_{E=4}=\big{\{}0.6667,1.000,1.000,4.000\big{\}}$, i.e., $|\beta_{2}|=|\beta_{3}|$. This equality is exactly the condition of obtaining the generalized Brillouin zone and the corresponding continuum bands. Therefore, $E_{OBC}=4$ can be reproduced by the continuum bands based on the non-Bloch band theory Yokomizo and Murakami (2020). Yet, another open boundary eigenvalue $E_{OBC}=\frac{6}{5}$ can be considered as well. Similarly, $\big{\{}|\beta_{1}|,|\beta_{2}|,|\beta_{3}|,|\beta_{4}|\big{\}}_{E=\frac{6}{5}}=\big{\{}1.000,1.000,1.600,10.00\big{\}},$ i.e., $|\beta_{2}|\neq|\beta_{3}|$ but $|\beta_{1}|=|\beta_{2}|$. Evidently, this equality is in contradiction to the condition of obtaining the generalized Brillouin zone, i.e., this open boundary eigenvalue must not belong to the continuum bands determined by $|\beta_{2}|=|\beta_{3}|$. Fig. 3b visually confirms those results.

The singular energy spectrum can also be examined from a more general perspective. In ref. Yang et al. (2020), authors proposed a new concept of the auxiliary generalized Brillouin zone, which is obtained by solving the characteristic equation $f(\beta,E)=0$ when $E$ takes every value of the complex number field, and selecting $|\beta_{m}|=|\beta_{n}|$ ($\{m,n\}=\{1,2,3,4\}$). However, as shown in Fig. 3(c), even though the auxiliary generalized Brillouin zone possesses all equality information of $f(\beta,E)=0$, it still can not recover the generalized Brillouin zone obtained from the open boundary eigenvalues. This further confirms the emergence of the singular energy spectrum.

Meanwhile, we analytically exhibit that the determinant of the open boundary Hamiltonian has the form $det\big{[}H_{OBC}-E_{OBC};N\times N]=2^{-N}(V_{a}-E_{OBC})^{N}(V_{c}-E_{OBC})^{N}y(V_{b},t_{b},E_{OBC})$ with the unidirectional hopping being satisfied enn , with N being the total number of the system size. Hence, $E_{OBC}=V_{a}$ and $E_{OBC}=V_{c}$ must be the solutions of $det\big{[}H_{OBC}-E_{OBC};N\times N]=0$, and be the multifold exceptional points Delplace et al. (2021)^,Bergholtz et al. (2021). In Fig. 4d, we exhibit the number of the open boundary eigenvalues of $E_{OBC}=V_{a}$ and $E_{OBC}=V_{c}$ with different system size. Explicitly, the number of degeneracy points is proportional to the system size.

Additionally, we know that the sensors have already penetrated every field of our life, in which one of the core parts is the transformation circuit, which enlarges the weak signals Degen et al. (2017); Alù and Engheta (2009); Wang et al. (2021). Interestingly, it has been displayed that the minute perturbations of the parameters will induce the mutations of the eigenvalues in our system. Moreover, Refs. Yin et al. (2022); Tuniz et al. (2022); Qin et al. (2021); Koch and Budich (2022); Nikzamir and Capolino (2022); Zhang and You (2019) also show that the higher-order exceptional points have advantages in sensors, and it is significant to seek higher-order exceptional points in various systems. Coincidentally, we also have shown that the emergence of the singular phenomenon is accompanied by the multifold exceptional point. Hence, our non-Hermitian systems may have huge applications in the sensors field.

Such as, one can see that five poles are spread on the complex plane for $E_{ref}=\frac{1}{6}i$ based on the definition, as shown in Fig. 5a. Further, three of them are surrounded by the unit circle, which induces $W=0$. Namely, the point gap is topologically trivial. [Precisely, the spectral winding $W\equiv 0$ for any reference point $E_{ref}$ as long as $V_{a}=V_{c}$, $t_{a}=t_{c}$ and $\gamma_{1}\gamma_{2}=\gamma_{3}\gamma_{4}$ enn ]. In addition, fives poles of order one are distributed on the complex plane for a given $E_{ref1}=\frac{1}{6}i$ in Fig. 5b, where four of them are contributed to the spectral windings enn and $W=1$ can be obtained, i.e., the point gap is topologically nontrivial.