Topology of the generalized Brillouin zone of one-dimensional models

Abstract

The generalized Brillouin zones (GBZs) are integral in the analysis of non-Hermitian band structures. Conventional wisdom suggests that the GBZ should be connected, where each point can be indexed by the real part of the wavevector, similar to the Brillouin zone. Here we demonstrate rich topological features of the GBZs in generic non-Hermitian one-dimensional models. We prove and discuss a set of sufficient conditions for the model to ensure the connectivity of its GBZ. In addition, we show that the GBZ can become disconnected and have more connected components than the number of bands, which results from the point-gap features of the band structure. This novel GBZ topology is applied to further demonstrate a counterintuitive effect, where the line gap of an open-boundary spectrum with sublattice symmetry may be closed without changing its point-gap topology. Our results challenge the current understanding of bands and gaps in non-Hermitian systems and highlight the need to further investigate the topological effects associated with the GBZ including topological invariants and open-boundary braiding.

Figures (5)

• Figure 1: Connectivity of the GBZ. (a) Euler diagram showing that, for single-band models, a simply-connected OBC spectrum implies a connected GBZ, per Theorem I. (b) The model $E = 5z^{-2} -z^{-1} +5z -z^2$ has a simply connected OBC spectrum (top) and a connected GBZ (bottom). (c) The model $E = -2z^{-3} +5z^{-2} -z^{-1} +5z -z^2 +2z^3$ has a multiply connected OBC spectrum (top) and a disconnected GBZ (bottom). (d) The model $E = -2z^{-3} +3z^{-2} -z^2 +2z^3$ has a multiply connected OBC spectrum (top) but a connected GBZ (bottom).
• Figure 2: Example of a disconnected GBZ. The model is Eq. (\ref{['eq:disconn_model']}) with $\epsilon = 0.5$. (a) The PBC (dashed purple) and the OBC (solid black) spectra of the model. (b) A gauge-transformed winding at $|z| = 0.85$, with the regions shaded according to their winding numbers. The $w=0$ region near $E = 1$ is disconnected from the outside. (c) Enlarged view of the multiply-connected part of the OBC spectrum. (d) GBZ of the model (top) and an enlarged view of its disconnected part (bottom). Inside and outside of the GBZ has been colored light red and light blue, respectively. Arrows with Greek letters indicate the direction for traversing the GBZ, and the corresponding path on the OBC spectrum is marked in (c).
• Figure 3: Transitions between connected and disconnected GBZs. The models are Eq. (\ref{['eq:disconn_model']}) with (a) $\epsilon = 1.4$; (b) $\epsilon = 0.5$; (c) $\epsilon = -0.5$ and (d) $\epsilon = -1.5$. Left panels show the OBC spectra near $E = 0$ and right panels show the GBZs near $z = -1$. Inside and outside of the GBZ has been colored light red and light blue, respectively.
• Figure 4: Line-gap transitions in sublattice-symmetric models. (a) The OBC spectra of Eq. (\ref{['eq:disconn_twoband']}) with $\epsilon = -1.5$ [the square root of Fig. \ref{['fig:3']}(d)] possesses a line gap ($\text{Re}\ E =0$). (b) The OBC spectra of Eq. (\ref{['eq:disconn_twoband']}) with $\epsilon = 0.5$ [the square root of Fig. \ref{['fig:3']}(b)] does not possess a line gap. (c) Berry phase $\arg \sqrt{H_+/H_-}$ along the integration contour for the two models. Markers indicate the corresponding states in (a) and (b). Greek letters mark the contour segments for (b) and are the same as in Fig. \ref{['fig:2']}(d).
• Figure 5: (a) The OBC spectrum of the model $E = -2z^{-3} + (3+\sqrt{5})z^{-2} + (-3+\sqrt{5})z^2 +2z^3$. (b) Different sets of $z$ defined for the model $E = -2z^{-3} + (3+\sqrt{5})z^{-2} + (-3+\sqrt{5})z^2 +2z^3$. Black curves indicate that the boundary is included in the sets, and hollow circles indicate the points are missing from the sets.