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arxiv: 2601.00951 · v3 · pith:ZFMMYC74new · submitted 2026-01-02 · ❄️ cond-mat.mes-hall · cond-mat.stat-mech· quant-ph

Symmetry and Topology in a Non-Hermitian Kitaev chain

Ayush Raj , Soham Ray , Sai Satyam Samal This is my paper

Pith reviewed 2026-05-21 15:33 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.stat-mechquant-ph
keywords non-Hermitian Kitaev chainparticle-hole symmetryMajorana zero modesnon-Hermitian skin effecttopological phase transitionopen versus periodic boundary conditionsZ2 topological invariantphase diagram
0
0 comments X

The pith

Particle-hole symmetry in the non-Hermitian Kitaev chain forces the topological transition of open chains to match the periodic case and makes Majorana zero modes appear as reciprocal pairs that cancel the skin effect.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a Kitaev chain with non-reciprocal hopping and asymmetric pairing, both made complex. It shows that particle-hole symmetry survives for all parameter values. This symmetry makes the point where topology changes identical for a finite open chain and an infinite periodic chain. Explicit construction of the zero-energy states reveals that Majorana modes must form pairs localized on opposite ends; their combined density exactly cancels any non-Hermitian skin accumulation. Excited states still show skin localization, with particle and hole parts piling up at opposite boundaries. A Z2 invariant is defined in limited regimes and used to map the full phase diagram in complex parameter space.

Core claim

Because particle-hole symmetry is preserved for arbitrary complex non-reciprocal hopping and asymmetric pairing, the topological phase boundary of a finite open chain is identical to that of the periodic system. The zero-energy Majorana wave functions occur exclusively as reciprocal localization pairs on opposite boundaries; the sum of their probability densities is free of skin-effect bias. In contrast, generic excited states exhibit skin localization with particle and hole components accumulating at opposite ends. A Z2 bulk invariant correctly labels the topological versus trivial phases inside restricted parameter windows, allowing a complete phase diagram to be drawn across the complex-h

What carries the argument

Particle-hole symmetry of the Hamiltonian, which remains intact for all complex values of the non-reciprocal hopping and asymmetric pairing parameters, together with the explicit analytic construction of the zero-energy wave functions as reciprocal localization pairs.

If this is right

  • The location of the topological phase transition is the same whether the chain is open with finite length or periodic and infinite.
  • Majorana zero modes always appear as a pair localized at opposite ends whose total density shows no net skin accumulation.
  • Non-zero-energy states display skin-effect localization with particle-like and hole-like components at opposite boundaries.
  • A Z2 topological invariant can be constructed in restricted parameter regions and matches the presence or absence of the zero-mode pairs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The exact cancellation for zero modes but not for excited states suggests that only the topological sector is protected against non-Hermitian boundary accumulation in this symmetry class.
  • Realizing the required non-reciprocal terms in a superconducting nanowire or circuit-QED array would allow direct imaging of the reciprocal Majorana pairs via local density measurements.
  • The persistence of the same transition point for open and periodic chains may simplify the design of finite-length devices whose topological properties can be predicted from bulk periodic calculations.

Load-bearing premise

The Hamiltonian is assumed to keep particle-hole symmetry for every value of the complex non-reciprocal hopping and asymmetric pairing parameters.

What would settle it

Measure the combined probability density of the two zero-energy modes on an open chain; if the skin-effect bias does not cancel exactly while the modes remain at zero energy, the central claim is falsified.

Figures

Figures reproduced from arXiv: 2601.00951 by Ayush Raj, Sai Satyam Samal, Soham Ray.

Figure 1
Figure 1. Figure 1: A caricature of the non-Hermitian Kitaev chain [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Energy bands of the finite (L = 50) non-Hermitian Kitaev chain with open boundary conditions (OBC) as func￾tion of the chemical potential, µ. Non-Hermiticity is intro￾duced by choosing non-reciprocal hopping parameters i.e., left hopping parameter t1 = 1 and right hopping param￾eter, t2 = 1.5. The superconducting pairing terms are ∆1 = 2 = ∆2 and φ1 = 0 = φ2 cf., Eq. (3). (a) Abso￾lute value of the energy … view at source ↗
Figure 4
Figure 4. Figure 4: Probability density of zero mode wavefunctions and [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Topological phase diagram for non-Hermitian Ki [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

We investigate the non-Hermitian Kitaev chain with non-reciprocal hopping amplitudes and asymmetric superconducting pairing. We work out the symmetry structure of the model and show that particle-hole symmetry (PHS) is preserved throughout the entire parameter regime. As a consequence of PHS, the topological phase transition point of a finite open chain coincides with that of the periodic (infinite) system. By explicitly constructing the zero-energy wave functions (Majorana modes), we show that Majorana modes necessarily occur as reciprocal localization pairs accumulating on opposite boundaries, whose combined probability density exhibits an exact cancellation of the non-Hermitian skin effect for the zero energy modes. Excited states, by contrast, generically display skin-effect localization, with particle and hole components accumulating at opposite ends of the system. At the level of bulk topology, we further construct a $\mathbb{Z}_2$ topological invariant in restricted parameter regimes that correctly distinguishes the topological and trivial phases. Finally, we present the topological phase diagram of the non-Hermitian Kitaev chain across a broad range of complex parameters and delineate the associated phase boundaries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes symmetry and topology in a non-Hermitian Kitaev chain with non-reciprocal hopping and asymmetric pairing. It establishes that particle-hole symmetry is preserved across the parameter space, implying that the topological phase transition occurs at the same point for open and periodic boundary conditions. Explicit construction of zero-energy wave functions reveals that Majorana modes form reciprocal localization pairs on opposite boundaries, with their combined probability density canceling the non-Hermitian skin effect. Excited states, however, exhibit skin-effect localization with particle and hole components at opposite ends. A Z_2 topological invariant is defined in restricted parameter regimes to distinguish phases, and the full phase diagram is presented for a broad range of complex parameters.

Significance. Should the derivations hold, this work provides analytical insight into how particle-hole symmetry can protect zero modes from the non-Hermitian skin effect in topological systems. The explicit wave-function constructions and phase diagram offer concrete results that could guide further theoretical and experimental studies in non-Hermitian condensed-matter physics.

major comments (2)
  1. [Symmetry analysis section] Symmetry analysis section: The claim that PHS is preserved for arbitrary complex values of the non-reciprocal hopping (t + γ) and asymmetric pairing (Δ) parameters is load-bearing for equating the open-chain and periodic-chain transition points. The manuscript must show explicitly that the standard PHS operator C = τ_x K maps the Hamiltonian to -H without implicit restrictions on Im(γ) or Im(Δ), as conjugation would otherwise replace parameters by their complex conjugates.
  2. [Zero-energy wave functions construction] Zero-energy wave functions construction: The explicit construction of Majorana zero modes as reciprocal localization pairs whose combined probability density cancels the skin effect is central to the main result. The algebraic steps deriving the wave functions and the exact cancellation should be presented in full detail (including any intermediate equations) to permit independent verification.
minor comments (2)
  1. [Phase diagram figure] The phase diagram figure would benefit from explicit labels on the axes indicating the real and imaginary parts of the parameters used.
  2. [Z2 invariant section] Notation for the Z_2 invariant should be cross-referenced to the restricted parameter regimes where it is defined to avoid ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive major comments, which will help strengthen the clarity and rigor of the manuscript. We address each point below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Symmetry analysis section] Symmetry analysis section: The claim that PHS is preserved for arbitrary complex values of the non-reciprocal hopping (t + γ) and asymmetric pairing (Δ) parameters is load-bearing for equating the open-chain and periodic-chain transition points. The manuscript must show explicitly that the standard PHS operator C = τ_x K maps the Hamiltonian to -H without implicit restrictions on Im(γ) or Im(Δ), as conjugation would otherwise replace parameters by their complex conjugates.

    Authors: We agree that an explicit verification is essential given the load-bearing role of this symmetry. In the revised manuscript we will add a dedicated paragraph in the symmetry analysis section that applies the operator C = τ_x K term by term to the full non-Hermitian Hamiltonian (including all complex contributions from γ and Δ) and demonstrates that C H C^{-1} = -H holds without any restriction on the imaginary parts. The calculation will be written out with the action on each bilinear term so that the absence of unwanted complex conjugation on the parameters is manifest. revision: yes

  2. Referee: [Zero-energy wave functions construction] Zero-energy wave functions construction: The explicit construction of Majorana zero modes as reciprocal localization pairs whose combined probability density cancels the skin effect is central to the main result. The algebraic steps deriving the wave functions and the exact cancellation should be presented in full detail (including any intermediate equations) to permit independent verification.

    Authors: We concur that the algebraic transparency of this central result can be improved. In the revised version we will expand the zero-energy wave-function section to include every intermediate step: the zero-energy recurrence relations for the open chain, the characteristic equation yielding the two distinct localization factors, the explicit left- and right-localized solutions, the normalization, and the direct computation showing that the sum of the probability densities is spatially uniform (thereby canceling the skin effect). All equations will be numbered and displayed. revision: yes

Circularity Check

0 steps flagged

No circularity: symmetry analysis and wave-function construction are independent of fitted inputs or self-citations.

full rationale

The paper explicitly works out the symmetry structure to establish PHS preservation for the given non-Hermitian model, then derives the coincidence of open/periodic transition points and the reciprocal Majorana localization as direct consequences. The Z2 invariant is constructed only in restricted regimes, and zero-mode wave functions are built explicitly. None of these steps reduce by construction to a fit, a renaming, or a load-bearing self-citation; the central claims rest on the model's Hamiltonian definition and standard symmetry consequences rather than tautological re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the persistence of particle-hole symmetry across all complex parameters and on the algebraic construction of zero-energy eigenvectors; no new particles or forces are postulated and no parameters are fitted to data.

axioms (2)
  • domain assumption Particle-hole symmetry is preserved for arbitrary complex non-reciprocal hopping and asymmetric pairing amplitudes
    Invoked throughout the abstract to equate transition points and guarantee reciprocal localization
  • standard math The zero-energy eigenvectors can be constructed explicitly from the model Hamiltonian
    Used to demonstrate the exact cancellation of skin effect for Majorana pairs

pith-pipeline@v0.9.0 · 5733 in / 1463 out tokens · 41503 ms · 2026-05-21T15:33:42.712565+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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