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

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.

💡 Research Summary

The manuscript presents a comprehensive analytical treatment of a two‑dimensional non‑Hermitian Su‑Schrieffer‑Heeger (SSH)–type lattice with four sublattices. By parametrizing the four hopping rates in terms of two real numbers, c (a measure of Hermiticity) and r (a measure of chirality), the authors rewrite the Bloch Hamiltonian in a compact form involving four complex functions A, B, C, D. They first demonstrate that the non‑Hermitian skin effect is absent, allowing the crystal momentum (kₓ, k_y) to be taken as real and the Brillouin zone to remain the ordinary torus T².

Symmetry analysis places the model in the AI class of the non‑Hermitian ten‑fold way (time‑reversal symmetry squaring to +1 together with sublattice symmetry). Additional order‑two and order‑four symmetries (denoted P and U_{C4}) further constrain the spectrum. The energy eigenvalues are expressed as E = ±q a ±p a, where a and b are explicit functions of (c, r, kₓ, k_y). The sign of the derived quantity Φ ≡ a² − b determines whether the band structure contains purely real/imaginary pairs or four complex‑conjugate pairs. By analytically locating the global minimum of Φ, the authors obtain exact conditions for the appearance of second‑order exceptional points (EP2) and fourth‑order exceptional points (EP4). These conditions define a phase diagram: for r > ½ and c > ½ the system is topologically non‑trivial, while r < ½ yields a trivial phase; a blue region marks coexistence of EP2 and EP4.

The eigenvectors are derived in closed form using auxiliary ratios τ and μ, leading to a representation where each state is characterized by a non‑vanishing amplitude r(kₓ, k_y) and a phase θ(kₓ, k_y). The amplitude only vanishes at EP4, ensuring that the phase is globally well‑defined except at isolated singularities. This structure enables the definition of a phase‑based topological invariant: the two‑dimensional Zak phase \