Beyond the non-Hermitian skin effect: scaling-controlled topology from Exceptional-Bound Bands

TL;DR

The paper introduces exceptional-bound (EB) band engineering as a novel, size-controlled topological mechanism in non-Hermitian systems, distinct from the non-Hermitian skin effect. By leveraging defective exceptional points (EPs) and their nonlocal EB states, EB bands are constructed and projected to yield effective 1D descriptions with renormalized, system-size dependent hoppings governed by $oldsymbol{\Omega}_{ riangle Y}(L_y)$. This framework enables topology to switch as a function of the transverse system size $L_y$, with explicit scaling laws for even and odd intra-cell hopping ranges and a general method to design scaling-dependent phase boundaries. The results are applicable across lattice geometries and dimensionalities, with experimental realizations proposed in photonic crystals, metamaterials, and circuit-based quantum simulators, broadening non-Hermitian topology beyond skin-driven scaling and impacting the study of finite-size criticality and entanglement.

Abstract

We establish a novel mechanism for topological transitions in non-Hermitian systems that are controlled by the system size. Based on a new paradigm known as exceptional-bound (EB) band engineering, its mechanism hinges on the unique critical scaling behavior near an exceptional point, totally unrelated to the well-known non-Hermitian skin effect. Through a series of ansatz models, we analytically derive and numerically demonstrate how topological transitions depend on the system size with increasingly sophisticated topological phase boundaries. Our approach can be generically applied to design scaling-dependent bands in multi-dimensional lattices, gapped or gapless, challenging established critical and entanglement behavior. It can be experimentally demonstrated in any non-Hermitian platform with versatile couplings or multi-orbital unit cells, such as photonic crystals, as well as classical and quantum circuits. The identification of this new EB band mechanism provides new design principles for engineering band structures through scaling-dependent phenomena unique to non-Hermitian systems.

Figures (9)

• Figure 1: Construction of exceptional-bound (EB) band lattice models from parent exceptional points (EPs). (a) From the band projector $P$ (exterior blue square) of a parent EP, one can construct a $\bar{P}$ operator [Eq. \ref{['eq:barP']}] by truncating the degrees of freedom outside of a chosen interval $y\in [y_L,y_R]$ (orange square). (b1) To craft lattice models featuring intrinsic EB bands, one defines $y_R-y_L+1$-atom unit cells whose connectivity structure (green) is given by the $\bar{P}$ matrix elements. Their long-ranged nature lee2022exceptionalzou2024experimental, inherited from the defectiveness of the parent EP, gives rise to unconventional scaling properties. (b2) Also due to this EP defectiveness, the spectrum of the lattice in (b1) features a pair of robust eigenvalues beyond the usual bounds of 0 and 1, which we call EB bands. (c) Illustrative EB band basis eigenstates corresponding to different parent EP orders $B=1,2,3$, with solid/dashed surfaces representing state amplitudes on odd/even sites, computed with $y_L=1,y_R=L_y-1$. Pronounced scaling behavior arises from the very different basis eigenstates profiles at different system sizes $L_y=10,20,30,40$ (red, blue, green, purple, respectively).
• Figure 2: Topological characterization of the NN EB-SSH warm-up model in Eq. \ref{['eq:ham_SSH_NN']}, as a prelude to the scale-dependent transition scenario in Fig. \ref{['fig:phasediagram']}. (a) Lattice structure, showing alternating hopping amplitudes $t_1$ and $t_2$ between $\bar{P}$ unit cell building blocks. (b) Band structure obtained from $H_{\text{NN}}^{\text{(EBB)}}\Psi_i(x,y)=E_i\Psi_i(x,y)$, with angle $\theta=\pi/2$ between lattice (x) and EB (y) degrees of freedom. Mid-gap topological states (green), which appear only for $t_1<t_2$, can be distinguished from the bulk states (pink) through IPR$_x$ [Eq. \ref{['eq:IPR_x']}]. EB bands (circled) do not play special roles in the topological transition within this warm-up scenario. (c) Spatial state distribution $|\Psi(x,y)|$ for a topological EB eigenstate at $E\approx 1.5$, which simultaneously incorporates topological (blue) and EB (orange) localization with dissimilar exponential $(t_1/t_2)^x$ and linear $y$ profiles, respectively. (d) In this warm-up model with trivially horizontal NN couplings, the topological phase boundary remains at the conventional $t_1=t_2$ line, with no scaling ($L_y$)-dependence. Here, we used $B=2, a_0=10, y_R=L_y-1=19$.
• Figure 3: The system size $L_y$ as a decisive control knob for topological phase transitions within our extended EB band model [Eq. \ref{['eq:ham_SSH_NNN']}]. (a) Schematic of its lattice, whose sensitivity to length $L_y$ arises from the criss-crossing NNN couplings $\delta$ between unit cells $\bar{P}$. (b) Topological phase diagram as a function of hopping strength $t_1$ and $L_y=y_R+1$, for fixed $t_2$. Due to the unconventional scaling of EB bands, a strongly $L_y$-dependent phase boundary curve appears, as determined by the ratio $|t_1'/t_2'|$ [Eq. \ref{['t1pt2p']}]. As such, increasingly $L_y$ can enigmatically drive the topological transition in either direction, depending on $t_1/t_2$. Consequently, at a fixed $L_y$, there exists non-monotonic topologically trivial$\rightarrow$non-trivial$\rightarrow$ trivial transitions (or vice-versa). (c,d) Scaling-induced topological transitions in either direction: (c) trivial$\rightarrow$nontrivial and (d) nontrivial$\rightarrow$trivial, as indicated by the white arrows in (b). These transitions can be accurately explained as crossings of the effective $\delta$-shifted EB couplings $t_1'$ and $t_2'$ [Eq. \ref{['eq:effectivet1t2']}], as shown in panels (c5) and (d5).
• Figure 4: Generalized $L_y$-scaling and diverse topological transition boundaries for extended EB band models [Eq. \ref{['eq:ham_delta12']}]. (a) Scaling of the "renormalizing" factor $\Omega_{\Delta Y}(L_y)$ [Eq. \ref{['eq:OmegaLy']}], with excellent agreement between numerical results (circles) and theoretical expressions (solid curves) from Eq. \ref{['eq:OmegaAnsatz']}. Distinct $\sim L^{3/2}_y$ and $\sim L_y^{-1}+\text{const.}$ scaling behaviors are respectively observed for odd and even intra-cell hopping ranges $\Delta Y$. (b) Qualitatively distinct topological phase diagrams in our extended model [Eq. \ref{['eq:ham_delta12']}] with effective hoppings that scale in prescribed manners [Eq. \ref{['eq:generalized_t1t2']}], for (b1) $\delta_1 = 0$, (b2) $\delta_1 = 0.015$ and (b3) $\delta_1 = 0.15$. We fix $t_1=t_2=0.1$ and $\delta_2=0.1$. The resulting topological boundaries are highly sensitive to $L_y$ and the tuning parameters $\delta_1$ and $\delta_2$, demonstrating the versatility and richness of EB band engineering beyond the simplest SSH-like lattice models.
• Figure 5: Lattice configuration and spectral characteristics of the non-Hermitian model described by Eq. \ref{['eq:HN_kx']}. (a1) Lattice geometry comprising $L_x$ unit cells of $\bar{P}$ along the $y$-direction, interconnected via NN symmetric couplings $t e^{i\theta}$ along the $x$-direction. (a2) Representative energy band structure $E_{\text{NN}}$ calculated from $H_{\text{NN}}$ under open boundary conditions in the $x$-direction ($x$-OBCs). (b1) Computed energy spectrum revealing exceptional-bound bands (EB band, red, $\chi_y=1$) and non-exceptional bulk bands (blue, $\chi_y\neq1$), distinguished by the projection fraction $\chi_y$ (Eq. \ref{['eq:chi']}). Parameters: $t=1$, $a_0=10$, $B=2$, $\theta=\pi/2$, $L_x=20$, $L_y=20$, $y_R=L_y-1$, $N=2$. (b2) Two-dimensional averaged spatial distributions $\Psi^{\text{ave}}(x,y)$ in Eq. \ref{['eq:spatial_averaged']}, shown separately for states outside (left) and within (right) the EB band. (b3) $x$-integrated spatial distributions $x\text{-}\Psi^{\text{ave}}(y)$ in Eq. \ref{['eq:x-spatial_averaged']} demonstrating distinct scaling behaviors: bulk-like distribution for non-EB bands versus linear decay with spatial localization for EB bands. Here, only the amplitudes on odd sites are plotted, as those on the even sites are almost vanishing.