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arxiv: 2606.05095 · v1 · pith:ZL365RRFnew · submitted 2026-06-03 · ❄️ cond-mat.str-el · cond-mat.supr-con· quant-ph

Soliton-antisoliton pairs in the supersymmetric gapped phase of an interacting Majorana chain

Alberto Nocera , Mobin Shakeri , Armin Rahmani , Ian Affleck This is my paper

Pith reviewed 2026-06-28 03:59 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-conquant-ph
keywords Majorana chainsupersymmetrygapped phasesoliton-antisoliton pairsnonlocal Dirac fermionfermion paritytricritical Ising
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0 comments X

The pith

In the gapped supersymmetric phase of an interacting Majorana chain, the lowest excitations are soliton-antisoliton pairs that bind Majorana modes into nonlocal Dirac fermions distinguishing even and odd parity states.

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

The paper examines how supersymmetry behaves in the gapped phase of a strongly interacting Majorana fermion chain adjacent to the tricritical Ising point. It shows that a standard supersymmetry diagnostic diverges right after the tricritical point but decays continuously to zero deeper in the gapped phase. The lowest excited states consist of soliton-antisoliton pairs separating regions of different order, with each soliton binding a localized Majorana mode so that the pair forms one nonlocal Dirac fermion. The occupation of this Dirac fermion sets whether the overall state has even or odd fermion parity.

Core claim

In the thermodynamic limit within the supersymmetric gapped regime, the excitations consist of soliton-antisoliton pairs separating distinct ordered regions. Each soliton binds an emergent localized Majorana mode, and together the pair forms a nonlocal Dirac fermion. The occupation of this Dirac mode distinguishes eigenstates with even and odd fermion parity. The conventional SUSY diagnostic remains finite at the tricritical point, diverges immediately on the Ising side, and decays continuously to zero deeper in the gapped phase.

What carries the argument

soliton-antisoliton pairs that bind emergent localized Majorana modes into a single nonlocal Dirac fermion

Load-bearing premise

The conventional SUSY diagnostic remains a valid probe of supersymmetry persistence even after entering the gapped phase.

What would settle it

Numerical computation of the low-energy spectrum on long chains that either shows or fails to show soliton-antisoliton pairs whose Majorana modes produce a parity-dependent nonlocal Dirac fermion.

Figures

Figures reproduced from arXiv: 2606.05095 by Alberto Nocera, Armin Rahmani, Ian Affleck, Mobin Shakeri.

Figure 1
Figure 1. Figure 1: The phase diagram of the OF model of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) The numerically computed value of R defined in Eq. (6) as a function of g/t for various system sizes. A kink appears on the Ising side, which extrapolates toward the TCI point. (b) Finite-size scaling of the position of the kink. For a lattice model with periodic boundary conditions, the states |σ〉 and |µ〉 are respec￾tively the ground states in the sector with even and odd fermion parity, while |σ ′ 〉 … view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the ordering of the two degenerate states as [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The energy densities of the ground and first-excited states in the two [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Density of Dirac fermions for positive ∆ (panel a) and negative ∆ (panel b) in each parity sector. The value of |∆| has been taken smaller than the level crossing shown in [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Panel (a): Ground and first excited state energies as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a–b) Ground-state and first excited state energies in the two fermion parity [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A cartoon illustration of a direct-product state containing an SA pair (see [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Difference of Majorana Green’s functions between even and odd fermion [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Energy levels of a ferromagnetic Ising chain with [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
read the original abstract

A strongly interacting chain of Majorana fermions realizes the supersymmetric tricritical Ising phase, with supersymmetry (SUSY) extending into a symmetry-broken ordered phase adjacent to the tricritical point. Although the signatures of SUSY at the tricritical point are well understood, their behavior in the gapped phase remains less clear. Here, we address two key questions: how SUSY manifests in the gapped phase and what is the nature of the excitations in this phase. We show that, in the thermodynamic limit, a conventional SUSY diagnostic that remains finite at the tricritical point diverges immediately on the Ising side, yet decays continuously to zero deeper in the gapped phase, signaling the persistence of SUSY. Focusing on the lowest excited states in the supersymmetric gapped regime, we find that the excitations consist of soliton-antisoliton pairs separating distinct ordered regions. Each soliton binds an emergent localized Majorana mode, and together the pair forms a nonlocal Dirac fermion. The occupation of this Dirac mode distinguishes eigenstates with even and odd fermion parity.

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 / 0 minor

Summary. The manuscript studies the supersymmetric gapped phase of an interacting Majorana chain adjacent to the tricritical Ising point. It reports that a conventional SUSY diagnostic, finite at the tricritical point, diverges immediately on the Ising side yet decays continuously to zero deeper inside the gapped phase, taken as evidence that SUSY persists. The lowest excitations in this regime are identified as soliton-antisoliton pairs that separate distinct ordered regions; each soliton binds an emergent localized Majorana mode, the pair forms a nonlocal Dirac fermion, and the occupation of this mode distinguishes even- and odd-fermion-parity eigenstates.

Significance. If the central claims hold, the work supplies a concrete microscopic picture of how supersymmetry can survive in a translationally broken gapped phase and links it to a specific excitation structure (soliton pairs carrying bound Majorana modes). This would be of interest to the condensed-matter community working on interacting Majorana systems and supersymmetric lattice models.

major comments (2)
  1. [Abstract] Abstract: The central claim that the continuous decay of the SUSY diagnostic to zero signals persistence of supersymmetry inside the gapped phase rests on the assumption that the diagnostic retains its meaning once the gap opens and translational symmetry is broken by ordered regions. The manuscript must demonstrate, via an independent cross-check (e.g., correlation with the soliton-pair spectrum or a SUSY-protected quantity), that the diagnostic is not instead tracking domain-wall density or parity mixing; without this, the decay-to-zero no longer implies unbroken SUSY.
  2. [Abstract] Abstract: The identification of the lowest excited states as soliton-antisoliton pairs is presented without reference to the concrete diagnostic (energy spectrum, wave-function overlap, or parity-sector analysis) used to establish this assignment. The link between this construction and the reported behavior of the SUSY diagnostic is therefore not independently verified and remains load-bearing for the persistence claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, agreeing that additional explicit cross-checks and references will strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the continuous decay of the SUSY diagnostic to zero signals persistence of supersymmetry inside the gapped phase rests on the assumption that the diagnostic retains its meaning once the gap opens and translational symmetry is broken by ordered regions. The manuscript must demonstrate, via an independent cross-check (e.g., correlation with the soliton-pair spectrum or a SUSY-protected quantity), that the diagnostic is not instead tracking domain-wall density or parity mixing; without this, the decay-to-zero no longer implies unbroken SUSY.

    Authors: We agree that an independent cross-check is needed to confirm the diagnostic tracks supersymmetry rather than domain-wall density or parity mixing. In the revised manuscript we will add a new analysis correlating the diagnostic's decay with the soliton-antisoliton excitation gap (a SUSY-protected quantity) across the gapped phase, explicitly ruling out alternative interpretations. revision: yes

  2. Referee: [Abstract] Abstract: The identification of the lowest excited states as soliton-antisoliton pairs is presented without reference to the concrete diagnostic (energy spectrum, wave-function overlap, or parity-sector analysis) used to establish this assignment. The link between this construction and the reported behavior of the SUSY diagnostic is therefore not independently verified and remains load-bearing for the persistence claim.

    Authors: The identification is based on the low-energy spectrum in even/odd parity sectors together with wave-function overlaps to trial soliton-pair states. We will revise the manuscript to explicitly reference these diagnostics (including the parity-sector analysis that distinguishes the nonlocal Dirac fermion) and to state how they connect to the SUSY diagnostic's behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and provided context present the SUSY diagnostic behavior and soliton-antisoliton construction as outcomes of thermodynamic-limit analysis and examination of lowest excited states, without any quoted equations, self-citations, or fitting procedures that reduce a claimed prediction or result to an input by construction. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citation chains are exhibited. The derivation therefore remains self-contained against external benchmarks and does not match any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be extracted from the abstract alone.

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