Analytical topological invariants for 2D non-Hermitian phases using Morse theory

Abstract

As energy dissipation and gain are ubiquitous in the real world, such phenomena demand the generalization of Hermitian methods such as the analysis of topological properties for non-Hermitian systems. However, as non-Hermitian systems typically contain more degrees of freedom, this poses a challenge for analytical approaches to understand their topology and invariants. In this work, we analytically calculate the 2D Zak phase for a 2D non-Hermitian SSH-type Hamiltonian that supports a rich structure and edge currents. Closed-form expressions for eigenstates and divisions of the phase diagram are obtained, including for regions in the phase diagram where different types of exceptional points exist. We use Morse theory to determine the topology of exceptional points in momentum space. Although the band structure breaks down at exceptional points, we show that a specific phase-based topological invariant remains well-defined. Furthermore, our work yields an analytic derivation for counting edge states in the Hermitian limit. These results provide new conceptual and analytical tools for the study of complex topological systems.

Figures (6)

• Figure 1: a) Representation of the physical model, with four sublattices and transition rates labelled. b) The phase diagram shows the topological regime for $r>\frac{1}{2}$ and the trivial regime for $r<\frac{1}{2}$. The red region marks where there exist $2^\text{nd}$ order exceptional points (EP2s) within the Brillouin zone and the blue region marks where there exist both $2^\text{nd}$ and $4^\text{th}$ order exceptional points (EP2s/EP4s) within the Brillouin zone.
• Figure 2: These figures show the function $2\theta$ defining the phase of eigenstate entries as in Eq. \ref{['topgauge']} globally defined over the Brillouin zone without any discontinuities, for $(r,c)$ values in the trivial regime without exceptional points. Zero winding numbers of $2\theta$ lead to zero Zak phase. Zero contours of $\theta$ are highlighted with white lines for clarity.
• Figure 3: These figures show the function $2\theta$ with nontrivial winding numbers in the topological regime without exceptional points. A $2\pi$-discontinuity is highlighted by yellow lines and zero sets of $2\theta$ are highlighted by white lines for clarity. Note that for each $k_x$ slice and $k_y$ slice, the discontinuity in $2\theta$ is $2\pi$. This gives rise to nontrivial winding numbers and therefore a nontrivial Zak phase in each direction.
• Figure 4: These figures show the phases of different entries of the eigenstates written in Eq. \ref{['topgauge']} for $(r,c)$ in the topological regime without exceptional points, showing that the phases only have discontinuities of multiples of $2\pi$ and therefore give smooth eigenstates.
• Figure 5: An illustration of how the sublevel set $\Phi^{-1}(-\infty,c]$ changes as $c$ is increased. For every critical point of index $n$, an $n$-handle is added. i.e. a $2$-handle is added at each maximum (red points), a $1$-handle is added at each saddle (orange crosses) and a $0$-handle is added at each minimum (blue crosses).