[Paper Review] Illuminating the bulk-boundary correspondence of a non-Hermitian stub lattice with Majorana stars

James H. Bartlett, Haiping Hu+1 more|arXiv (Cornell University)|Aug 27, 2021

Quantum Mechanics and Non-Hermitian Physics71 references21 citations

TL;DR

This paper introduces a Z2 topological invariant based on Majorana's stellar representation to restore the bulk-boundary correspondence in a non-Hermitian stub lattice with a flat band, where conventional winding numbers fail due to broken chiral symmetry. The method successfully predicts edge states and reveals the system is not a square-root topological insulator despite a block-diagonalizable parent Hamiltonian.

ABSTRACT

Topological characterization of non-Hermitian band structures demands more than a straightforward generalization of the Hermitian cases. Even for one-dimensional tight-binding models with nonreciprocal hopping, the appearance of point gaps and the skin effect leads to the breakdown of the usual bulk-boundary correspondence. Luckily, the correspondence can be resurrected by introducing a winding number for the generalized Brillouin zone for systems with even number of bands and chiral symmetry. Here, we analyze the topological phases of a nonreciprocal hopping model on the stub lattice, where one of the three bands remains flat. Due to the lack of chiral symmetry, the biorthogonal Zak phase is no longer quantized, invalidating the winding number as a topological index. Instead, we show that a $Z_2$ invariant can be defined from Majorana's stellar representation of the eigenstates on the Bloch sphere. The parity of the total azimuthal winding of the entire Majorana constellation correctly predicts the appearance of edge states between the bulk gaps. We further show that the system is not a square-root topological insulator, despite the fact that its parent Hamiltonian can be block diagonalized and related to a sawtooth lattice model. The analysis presented here may be generalized to understand other non-Hermitian systems with multiple bands.

Motivation & Objective

• To restore the bulk-boundary correspondence in non-Hermitian systems lacking chiral symmetry.
• To develop a topological invariant for non-Hermitian three-band models with a flat band.
• To investigate the interplay between flat bands, the non-Hermitian skin effect, and topological edge states.
• To determine whether the system is a square-root topological insulator based on its parent Hamiltonian structure.
• To provide a generalizable framework for topological characterization in multi-band non-Hermitian systems.

Proposed method

• Introduces a generalized Brillouin zone (GBZ) defined by complex quasimomentum β = e^{ik} to handle non-Bloch band theory.
• Applies biorthogonal Zak phase over the GBZ, though it is not quantized due to absence of chiral symmetry.
• Employs Majorana's stellar representation to map eigenstates to points on a Bloch sphere and computes the total azimuthal winding of the Majorana constellation.
• Defines a Z2 invariant as the parity of the total azimuthal winding, which predicts edge state emergence.
• Analyzes the squared Hamiltonian H²(β) to relate the model to a sawtooth lattice and rule out square-root topological insulator behavior.
• Uses inverse participation ratio to quantify localization and compare continuum band vs. edge state behavior.

Experimental results

Research questions

• RQ1 Can a topological invariant be defined in non-Hermitian systems without chiral symmetry?
• RQ2 Does the Majorana stellar representation enable a robust topological invariant in multi-band non-Hermitian lattices?
• RQ3 How does the presence of a flat band affect the non-Hermitian skin effect and edge state localization?
• RQ4 Is the non-Hermitian stub lattice a square-root topological insulator despite its block-diagonalizable parent Hamiltonian?
• RQ5 Can the generalized bulk-boundary correspondence be restored using the GBZ and stellar representation?

Key findings

• A Z2 invariant based on the parity of the total azimuthal winding of the Majorana constellation correctly predicts the presence of edge states in the bulk gaps.
• The biorthogonal Zak phase over the GBZ is not quantized due to the absence of chiral symmetry, invalidating the winding number as a topological invariant.
• The system is not a square-root topological insulator, as the parent Hamiltonian’s block-diagonal structure does not lead to the expected topological invariants.
• The flat band remains robust against the non-Hermitian skin effect, as compact-localized states resist edge accumulation.
• The inverse participation ratio confirms that edge states are localized at boundaries while continuum bands exhibit extended character.
• The knot-theoretic analysis of the band structure confirms the topological phase transitions predicted by the Majorana stellar method.