[Paper Review] Comment on "Anomalous Edge State in a Non-Hermitian Lattice"

Ye Xiong, Tianxiang Wang+2 more|arXiv (Cornell University)|Oct 20, 2016

Quantum Mechanics and Non-Hermitian Physics5 citations

TL;DR

This paper challenges the validity of fractional winding numbers and bulk-boundary correspondence breakdown in a non-Hermitian lattice model, arguing that the claimed fractional winding number is gauge-dependent and physically meaningless. It demonstrates that the observed edge states and symmetry properties are finite-size artifacts, and the total winding number across both bands is unity, not half, invalidating the original paper's conclusions on topological protection and edge states.

ABSTRACT

In this comment, we criticize three main conclusions of the letter\cite{Lee2016}. We show that the concept of fractional winding number(FWN) is factitious, Lee's conclusions on Fig. 3 are finite-size effect and the breakdown of bulk-boundary correspondence (BBBC) cannot be explained by "defective".

Motivation & Objective

• To challenge the validity of the fractional winding number (FWN) concept proposed in Lee et al. (2016) for a non-Hermitian lattice model.
• To demonstrate that the observed edge states and preserved PT symmetry are artifacts of finite system size, not intrinsic topological features.
• To argue that bulk-boundary correspondence breakdown is not due to 'defective' equations but due to boundary-induced localization of bulk states.
• To show that the total winding number across both bands is 1, not 1/2, rendering the individual FWNs meaningless as gauge-variant quantities.

Proposed method

• Analyzes the evolution of eigenstates in momentum space, showing that the state trajectory is not closed over a 2π Brillouin zone but requires 4π to return to the initial state.
• Calculates the winding number using a specific gauge for left and right eigenvectors, showing that w₊ + w₋ = 1, not w₊ = 0.5.
• Uses the Berry connection formula for non-Hermitian systems: A± = -i⟨⟨u±|∂k|u±⟩ / ⟨⟨u±|u±⟩, and integrates over k ∈ [0, 4π].
• Compares energy spectra for open boundary conditions (OBC) across different chain lengths (N=30 to N=800), revealing dramatic changes in spectral structure.
• Reinterprets Lee’s equation by showing that integrating over 4π includes contributions from both bands, invalidating the assignment of 0.5 to a single band.
• Applies a thought experiment with a simplified model to demonstrate the logical inconsistency of defining FWN within a 2π BZ when the physical period is 4π.

Experimental results

Research questions

• RQ1 Is the fractional winding number (FWN) of 1/2 for individual bands physically meaningful in this non-Hermitian model?
• RQ2 To what extent are the reported edge states and PT symmetry preservation artifacts of finite system size?
• RQ3 Why does the bulk-boundary correspondence appear to break down, and is it due to 'defective' equations or boundary-induced localization?
• RQ4 What is the correct total winding number when the physical period is 4π and both bands contribute?
• RQ5 Can the winding number be meaningfully assigned to a single band when the state trajectory spans both bands over 4π?

Key findings

• The fractional winding number (FWN) of 1/2 for individual bands is gauge-variant and physically meaningless, as the state trajectory is not closed over a 2π interval.
• The total winding number w₊ + w₋ = 1 over the 4π period, confirming a topological invariant of 1, not 1/2.
• The energy spectrum under OBC changes fundamentally when system size increases from N=30 to N=800, with the band gap closing and spectrum becoming complex, indicating finite-size effects.
• Bulk states transition from extended to localized under OBC, which is the true origin of bulk-boundary correspondence breakdown, not 'defective' equations.
• Lee’s equation in the third reply contradicts his own conclusions: assigning a winding number of 1/2 to a single band ignores that the 4π integral includes contributions from both bands.
• The correct interpretation requires the total winding number w = 1, not w = 1/2, and the individual FWNs are artifacts of an incorrect gauge choice and misinterpretation of the physical period.