Generalized Brillouin Zone Fragmentation

TL;DR

This work shows that in generic non-Hermitian lattices, the generalized Brillouin zone (GBZ) is not unique when multiple skin-accumulation channels compete, leading to GBZ fragmentation where OBC eigenstates are superpositions of several bulk solutions. It develops a boundary-constraint formalism based on a matrix $M$ satisfying $ ext{det}M=0$ to compute OBC spectra and introduces the composition-IPR (cIPR) to quantify fragmentation, along with a relative-entropy framework to compare spectra and assess non-Hermitian skin effects. The authors demonstrate that GBZ fragmentation induces edge localization in biorthogonal observables, enables continuous topological transitions via a weighted winding $W$, and arises universally in multi-mode/non-Hermitian media, including photonic crystals exhibiting antagonistic NHSE channels. Overall, GBZ fragmentation challenges the notion of a single bulk GBZ and a discontinuous topological transition, proposing a new paradigm for understanding band structure, topology, and dynamics in complex non-Hermitian systems.

Abstract

The Generalized Brillouin Zone (GBZ) encodes how lattice momentum is complex-deformed due to non-Hermitian skin accumulation, and has proved essential in restoring bulk-boundary correspondences. However, we find that generically, the GBZ is neither unique nor well-defined if more than one skin localization direction or strength exists, even in systems with no asymmetric hoppings. Instead, open boundary condition (OBC) eigenstates become complicated superpositions of multiple competing skin modes from "fragments" of all possible GBZs solutions. We develop a formalism that computes the fragmented GBZ in a scalable manner, with fragmentation extent quantified through our newly-defined composition IPR and spectral relative entropy. GBZ fragmentation is revealed to fundamentally challenge the notion of discontinuous phase transitions, since topological winding contributions from different GBZ fragments can "melt away" at different rates. Phenomenologically, GBZ fragmentation also leads to edge localization in all observables in energetically weighted ensembles such as thermal ensembles. This contrasts with conventional GBZs where the skin localization completely cancels in biorthogonal expectations. Occurring universally in multi-mode non-Hermitian media, as we concretely demonstrate with photonic crystal simulations, GBZ fragmentation points towards a new paradigm that is essential for understanding the band structure and the topological and dynamical properties of diverse generic non-Hermitian systems.

Figures (21)

• Figure 1: Synopsis of GBZ fragmentation in archetypal models, all at $N=30$. Top Row: OBC spectra colored by cIPR [Eq. \ref{['cIPRmain']}] (departure from red indicates GBZ fragmentation), compared against PBC spectra (black). 2nd Row: Fragmentation of spectra into different GBZ contributions with distinct $\log|z_\mu|$, colored by composition weight $|c_\mu|^2$ [Eq. \ref{['psix']}] evaluated at $x_0=N/2$. 3rd Row: Corresponding GBZs, similarly colored. Bottom Row: Illustrative highest Im$(E)$ eigenstates $\psi_\text{OBC}(x)$ (black) and and their constituent $\phi_\mu(x)$ (colored). Models are (a) $H_\text{HN}(z)=z+h/z$ with $h=2$; $H_\text{coupled-HN}$ [Eq. \ref{['cNHSEg']}] with $g=2$, $\Delta=0.1$, $h=2$ respectively; (c) $5\times 5$ random hopping model $H_\text{rand}$ with coefficients in suppmat. GBZ fragmentation is evident in (b) from the distinct broken GBZ rings (blue) , and becomes inevitable with sufficiently complicated random hoppings (d).
• Figure 2: Edge skin current unique to GBZ fragmentation: Illustrative models are (a) $H_\text{HN}(z)=z+2/z$ with unfragmented GBZ, and (b) $H_\text{coupled-HN}(z)$ [Eq. \ref{['cNHSEg']}] with $h=2,\Delta=0.1,g=1$, that exhibits GBZ fragmentation. Top Row: The biorthogonal spatial density $\rho^{LR}(x)$ of all eigenstates, colored by cIPR [Eqs. \ref{['M']}, \ref{['cIPRmain']}] at boundary reference position $x_0=0$. Bottom: Physical tunneling current $\langle \hat{I}(x)\rangle$ [Eq. \ref{['hatO']}] at inverse temperature $\beta=5$, for various Fermi energies $\varepsilon_F$. In (a), the NHSE completely cancels in a conventional GBZ, yielding uniform eigenstate density profiles (red) and relatively constant tunneling current. In (b), GBZ fragmentation causes most eigenstates (orange) to acquire large edge-localized densities, such that the physical current $\langle \hat{I}(x)\rangle$ in a thermal ensemble also become significantly enhanced at the edges (yellow).
• Figure 3: Intrinsically continuous topological transition of $H_\text{coupled-topo}$ [Eq. \ref{['Cext']}] from GBZ fragmentation. The $U^2(z_\mu)$ trajectories (Top Row) are colored by their Re$(E)$ (Bottom Row), with intensity proportional to their GBZ weight $|c_\mu|^2$. Increasing $\Delta$ causes the $U^2(z_\mu)$ windings to slowly fade away, while the spectrum gradually transitions from a real topological gap to local imaginary gap (red cross near zero mode, Center). Parameters are $\gamma=2$,$t=0.6,t'=0.1$.
• Figure 4: (a) GBZ fragmentation occurs in simple metamaterial structures with multiple eigenmodes per unit cell, as in our illustrative photonic crystals (see suppmat for details). (b) Highly complicated diffuse unit cell eigenmodes lead to multi-component heavily-coupled effective lattices reminiscent of $H_\text{rand}$. (c) The relative entropy $S$, which measures the contrast between PBC/OBC spectra, is much lower for anti-aligned cases (dashed) at small $g$, and also universally decreases with system size $N$. It is far lower than that of $H_\text{HN}$ (black), which experiences clean GBZ deformations.
• Figure S1: Density plot of $\frac{1}{N}\log|\text{Det}M|$ for the illustrative model $H(z)=z^2+z+\frac{1}{z}$, computed from Eq. \ref{['Mmunu']} by sweeping across all complex $E$ in the region. The rescaling factor of $\frac{1}{N}$ exponentially normalizes Det$M$, whose dominant non-vanishing components scale like $z_\mu^N$. In all cases, the $N$ brightest spots with exponentially low Det$M$ correspond to the OBC eigenvalues.