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arxiv: 2512.24528 · v3 · pith:DROBFN2Gnew · submitted 2025-12-31 · 🪐 quant-ph · hep-th· nucl-th

Geometric phase from encircling an exceptional point of a quantum resonance in the complex-scaling method

Okuto Morikawa , Shoya Ogawa , Soma Onoda This is my paper

Pith reviewed 2026-05-21 17:15 UTC · model grok-4.3

classification 🪐 quant-ph hep-thnucl-th
keywords exceptional pointsgeometric phasequantum resonancescomplex scalingnon-Hermitian physicsscattering theoryBerry phaseChern number
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The pith

In a one-dimensional scattering model, complex scaling embeds a resonance pole as an exceptional point whose encirclement produces a geometric phase of π.

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

The paper shows that resonance poles of the S-matrix in a scattering problem can be realized as discrete eigenvalues of a non-Hermitian Hamiltonian through complex scaling. When such a pole coalesces with a scattering state to form an exceptional point, the eigenstates become self-orthogonal and a Berry phase appears upon encircling the point in parameter space. This construction directly links the branch-point structure of exceptional points to the poles that govern quantum resonances and decay. A reader would care because it supplies a concrete computational route to topological features of resonances inside standard scattering theory rather than abstract non-Hermitian models alone.

Core claim

In the complex-scaled non-Hermitian Hamiltonian for a one-dimensional scattering system, resonance poles become discrete eigenvalues that coalesce with continuum states to form exceptional points. Encircling these points yields a geometric phase together with a nonzero Chern number, with the phase arising from the self-orthogonality and branch structure of the resonance poles.

What carries the argument

The complex-scaling transformation, which rotates the integration contour to convert resonance poles of the S-matrix into discrete eigenvalues of the dilated Hamiltonian so they can coalesce with scattering states into an exceptional point.

If this is right

  • The Berry phase and Chern number become computable quantities for quantum resonances without additional regularization.
  • Exceptional-point branch structure emerges directly from the analytic continuation of scattering resonances.
  • Non-Hermitian spectral topology is thereby placed inside the conventional theory of quantum resonances and the S-matrix.
  • Self-orthogonality at the exceptional point follows from the coalescence of the resonance pole with a scattering eigenstate.

Where Pith is reading between the lines

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

  • The same complex-scaling route could be tested on two-dimensional or three-dimensional scattering problems to check whether the geometric phase survives increased dimensionality.
  • If the phase remains robust, it offers a new diagnostic for resonance lifetimes in open quantum systems such as atomic or molecular scattering.
  • Experimental platforms that already tune scattering lengths or potentials might look for signatures of this phase by varying parameters in a closed loop.

Load-bearing premise

Complex scaling applied to a one-dimensional scattering Hamiltonian can embed a resonance pole into the continuum spectrum to create a genuine exceptional point whose self-orthogonality and geometric phase can be computed directly.

What would settle it

Numerical diagonalization of the complex-scaled Hamiltonian around the identified parameter values for the exceptional point either fails to produce self-orthogonal states or yields a geometric phase different from π when parameters are looped around the point.

read the original abstract

Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.

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

1 major / 2 minor

Summary. The manuscript formulates the geometric phase associated with encircling an exceptional point (EP) in a one-dimensional scattering model by employing the complex-scaling method. Resonance poles of the S-matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, enabling coalescence between a resonant state and a scattering state to form an EP in the complex energy plane. The work analyzes self-orthogonality near the EP, computes the Berry phase, and examines the Chern characteristic, with the goal of connecting non-Hermitian spectral topology directly to the traditional theory of quantum resonances.

Significance. If the construction holds, the results would provide a concrete bridge between EP physics in finite non-Hermitian systems and scattering-theoretic resonances in unbounded Hamiltonians. This could be significant for understanding geometric holonomy arising from resonance poles, with potential relevance to open quantum systems, optical resonators, and non-Hermitian topological phenomena where continuum effects are central. The use of complex scaling to embed poles as discrete eigenvalues is a standard tool, and explicit demonstration of the resulting Berry/Chern quantities would strengthen the claimed connection.

major comments (1)
  1. The central construction relies on the complex-scaling transformation embedding a resonance pole exactly into the continuum spectrum of the dilated operator to produce a genuine EP with self-orthogonality. The manuscript should explicitly verify (e.g., via the definition of the biorthogonal inner product and the location of the pole relative to the rotated contour) that no additional principal-value regularization or contour deformation is required; otherwise the claimed self-orthogonality and subsequent geometric phase are not those of a standard defective EP but of a regularized resonance-continuum hybrid.
minor comments (2)
  1. Clarify the precise definition of the inner product used for the scattering states in the dilated space, including any cutoff or limiting procedure, to make the self-orthogonality calculation reproducible.
  2. In the discussion of the Chern characteristic, specify whether it is computed over a closed loop in parameter space or a surface, and confirm consistency with the branch structure of the EP.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The major comment raises an important point about the rigor of the EP construction, which we address below. We will revise the manuscript to incorporate the requested explicit verification.

read point-by-point responses

  1. Referee: The central construction relies on the complex-scaling transformation embedding a resonance pole exactly into the continuum spectrum of the dilated operator to produce a genuine EP with self-orthogonality. The manuscript should explicitly verify (e.g., via the definition of the biorthogonal inner product and the location of the pole relative to the rotated contour) that no additional principal-value regularization or contour deformation is required; otherwise the claimed self-orthogonality and subsequent geometric phase are not those of a standard defective EP but of a regularized resonance-continuum hybrid.

    Authors: We thank the referee for this clarification request. In our complex-scaling construction, the dilation angle θ is chosen such that the resonance pole lies strictly below the rotated branch cut of the dilated Hamiltonian and is therefore realized as an isolated discrete eigenvalue, not lying on the integration contour. The biorthogonal inner product is defined directly via integration along this rotated contour using the scaled eigenfunctions; because the pole is off-contour, the inner product is well-defined without principal-value regularization or further contour deformation. Self-orthogonality then follows in the standard manner for a defective eigenvalue of a non-Hermitian operator. We will add an explicit verification paragraph (with a schematic of the rotated contour and pole location) in the revised manuscript to make this transparent. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from complex-scaled Hamiltonian without reduction to inputs

full rationale

The paper constructs the geometric phase by applying complex scaling to embed resonance poles of a 1D scattering Hamiltonian as discrete eigenvalues of the dilated non-Hermitian operator, allowing coalescence with continuum states to form an EP. Self-orthogonality, Berry phase, and Chern numbers are then obtained directly from the resulting defective eigenstates and their biorthogonal inner products. No equation or step equates the target holonomy to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose validity presupposes the claimed result. The chain remains independent of the final geometric quantities and is grounded in the spectral properties of the scaled operator.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of complex scaling for resonances and on the existence of self-orthogonal states at EPs; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Complex scaling maps resonance poles of the S-matrix to discrete eigenvalues of a non-Hermitian dilated Hamiltonian.
    Invoked to realize resonances as eigenvalues that can coalesce with continuum states.
  • domain assumption An exceptional point can be formed by coalescence of a resonance eigenvalue with a scattering state in the complex energy plane.
    Central modeling choice that enables the geometric phase calculation.

pith-pipeline@v0.9.0 · 5731 in / 1433 out tokens · 58168 ms · 2026-05-21T17:15:03.793719+00:00 · methodology

discussion (0)

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