Spectral Riemann Surface Topology of Gapped Non-Hermitian Systems

TL;DR

This work develops a topological framework for gapped non-Hermitian systems by mapping the complex energy spectrum onto spectral Riemann surfaces with non-contractible branch cuts on a toroidal Brillouin zone. Time-reversal symmetry reduces the classification to four configurations described by a $Z_2 \times Z_2$ invariant, with exceptional points (EPs) acting as excitations that must be threaded through the Brillouin zone to transition between configurations. The authors draw a deep analogy to Kitaev's toric code, linking closed Fermi cuts to non-contractible loops and associating EPs with toric-code–like excitations that carry a half-integer topological charge. They further show how multi-band generalizations yield richer excitation spectra and higher-dimensional invariants, and discuss concrete experimental routes in metasurfaces and interferometric platforms to observe these phenomena.

Abstract

We show topological configurations of the complex-valued spectra in gapped non-Hermitian systems. These arise when the distinctive EPs in the energy Riemann surfaces of such models are annihilated after threading them across the boundary of the Brillouin zone. This results in a non-trivially closed branch cut that is protected by an energy gap in the spectrum. Their presence or absence establishes topologically distinct configurations for fully non-degenerate systems and tuning between them requires a closing of the gap, forming exceptional point degeneracies. We provide an outlook toward experimental realizations in metasurfaces and single-photon interferometry.

Figures (7)

• Figure 1: Illustration showing the real part and absolute value of the Riemann surface structure of non-Hermitian two-band spectra: (a), (b) show an EP2 pair connected by a branch cut identified with the Fermi arc in red. This line is referred to as a Fermi cut; (c), (d) show an open Fermi arc due to band touching of $\Re[\epsilon(\bm{k})]$ in blue. The corresponding imaginary part, and $|\epsilon(\bm{k})|$ accordingly, is gapped along the Fermi arc, shown in (d). In two dimensions the Fermi cut is protected by the presence of the stable EP2 pair, while the Fermi arc may disappear under perturbations.
• Figure 2: Creation of a closed Fermi cut. Shown are the real part and the absolute value of the difference between the eigenenergies $\Delta \epsilon(\bm{k})$ for four different values of the tuning parameter $\beta$ over the full Brillouin zone. The two intermediate values of $\beta$ correspond to gapless spectra, where EPs are highlighted by the red circles. In the first row, the creation of the closed Fermi cut is clearly visible, with the last plot showing the persistence of the Fermi cuts after the merger of the EPs.
• Figure 3: We show the three distinct non-contractible Fermi cuts possible in the two-dimensional Brillouin zone, where Fermi cuts are represented by red lines. Two distinct non-contractible loops, $\mathcal{C}_x$ and $\mathcal{C}_y$, around the different holes of the torus are indicated in blue. Along these loops the eigenenergy braids, shown next to the Brillouin zone, can be measured to determine the topological invariants.
• Figure 4: The real part of the spectral Riemann surface structures over the Brillouin zone is shown for the topologically distinct configurations. (a) Coordinate system of the toroidal Brillouin zone, where the two quasi-momenta $k_x,k_y$ correspond to the two angles in toroidal coordinates and the radius encodes the real part $E$ of the complex-valued spectrum; (b)-(e): Topologically distinct fully non-degenerate configurations of the non-Hermitian model $H_\text{FC}(\bm{k})$ analogous to the ground states of the toric code. The states are distinguished by the topological invariants $(m_x,m_y)$: (b) $(0,0)$, (c) $(0,1)$, (d) $(1,0)$, (e) $(1,1)$.
• Figure 5: Illustration of the toric code on a square lattice given periodic boundary conditions. Physical qubit degrees of freedom are represented by white circles. (a) A star and a plaquette operator, $A_s$ and $B_p$, are shown in blue and red, respectively. Non-contractible loops of flipped spins, which define the ground states, are indicated by red lines. (b) Representatives of the four topologically distinct ground states are shown. They are distinguished by the different possible non-contractible closed loops on the toroidal lattice.