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arxiv: 2506.23356 · v3 · submitted 2025-06-29 · 🪐 quant-ph · math-ph· math.MP

Quantum phase transitions and entanglement entropy in a non-Hermitian Jaynes-Cummings model

Gargi Das , Aritra Ghosh , Bhabani Prasad Mandal This is my paper

Pith reviewed 2026-05-19 07:20 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords non-Hermitian Jaynes-Cummings modelquantum phase transitionsexceptional pointsentanglement entropyinvariant subspaces
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The pith

Non-Hermitian Jaynes-Cummings model undergoes quantum phase transitions at exceptional points, distinguished by entanglement entropy profiles.

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

The non-Hermitian Jaynes-Cummings model has a Hilbert space that decomposes into infinitely many two-dimensional invariant subspaces plus a global ground state. Within each subspace, exceptional points signal quantum phase transitions where the eigenvalues change from real to complex. Computing the spin-oscillator entanglement entropy on these subspaces reveals distinct profiles that separate the two phases. This structure lets the model exhibit non-Hermitian phase behavior that can be tracked exactly through the subspaces.

Core claim

The central claim is that quantum phase transitions occur at exceptional points in the two-dimensional invariant subspaces of the non-Hermitian Jaynes-Cummings model, separating regions of real and complex eigenvalues, and that these phases can be distinguished by their distinct spin-oscillator entanglement entropy profiles.

What carries the argument

The decomposition of the Hilbert space into infinitely many two-dimensional invariant subspaces, on which exceptional points are located and entanglement entropy is computed to mark the phases.

Load-bearing premise

The Hilbert space of the model can be decomposed into infinitely many two-dimensional invariant subspaces together with the global ground state.

What would settle it

Direct computation of the eigenvalues on an invariant subspace showing they remain real past the expected exceptional point, or entanglement entropy values that do not differ between the claimed phases.

Figures

Figures reproduced from arXiv: 2506.23356 by Aritra Ghosh, Bhabani Prasad Mandal, Gargi Das.

Figure 1
Figure 1. Figure 1: FIG. 1: Figures showing the regions of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Variation of the Boltzmann entropy as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Variation of the quantum entropies as a function [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Variation of the first derivative of quantum entropies [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Variation of the quantum entropies as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

In this paper, we describe some interesting properties of a non-Hermitian Jaynes-Cummings model. For this particular model, it is known that the Hilbert space can be described by infinitely-many two-dimensional invariant (closed) subspaces, together with the global ground state. We expose the appearance of exceptional points on such two-dimensional subspaces, together with quantum phase transitions marking the transition from real to complex eigenvalues. We also compute the spin-oscillator entanglement entropy on each invariant subspace to show that the two phases can be distinguished by their distinct entanglement-entropy profiles.

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 / 0 minor

Summary. The manuscript examines a non-Hermitian Jaynes-Cummings model whose Hilbert space decomposes into a global ground state plus infinitely many two-dimensional invariant subspaces. On each subspace the authors locate exceptional points that mark quantum phase transitions between a phase with entirely real eigenvalues and a phase with complex-conjugate eigenvalues. They further compute the spin-oscillator entanglement entropy on these subspaces and report that the two phases are distinguished by qualitatively different entropy profiles.

Significance. If the entanglement entropy is rigorously defined for the non-Hermitian case, the work supplies a concrete diagnostic that separates real-eigenvalue and complex-eigenvalue regimes in a solvable non-Hermitian model. The explicit use of the known invariant-subspace structure and the focus on entanglement as an order parameter are positive features that could be useful for subsequent studies of PT-symmetric or open quantum systems.

major comments (1)
  1. The central claim that distinct entanglement-entropy profiles distinguish the real-eigenvalue phase from the complex-eigenvalue phase (abstract and the section on entanglement entropy) presupposes that the reduced density matrix is positive semi-definite. For a non-Hermitian Hamiltonian the right eigenvectors are not orthogonal under the standard inner product, so the projector onto an invariant subspace generally yields a non-Hermitian reduced operator whose eigenvalues may be negative. The manuscript must specify whether the entropy is evaluated with the standard trace, a biorthogonal metric, or a PT-symmetric inner product; without this clarification the von Neumann entropy is either complex or undefined and the claimed distinction is not yet established.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying an important technical point regarding the definition of entanglement entropy. We address the comment below and have revised the manuscript to incorporate the requested clarification.

read point-by-point responses

  1. Referee: The central claim that distinct entanglement-entropy profiles distinguish the real-eigenvalue phase from the complex-eigenvalue phase (abstract and the section on entanglement entropy) presupposes that the reduced density matrix is positive semi-definite. For a non-Hermitian Hamiltonian the right eigenvectors are not orthogonal under the standard inner product, so the projector onto an invariant subspace generally yields a non-Hermitian reduced operator whose eigenvalues may be negative. The manuscript must specify whether the entropy is evaluated with the standard trace, a biorthogonal metric, or a PT-symmetric inner product; without this clarification the von Neumann entropy is either complex or undefined and the claimed distinction is not yet established.

    Authors: We thank the referee for this observation. In the calculations presented, the entanglement entropy on each invariant subspace is obtained from the reduced density matrix constructed from the right eigenvector |ψ⟩ of the 2×2 non-Hermitian matrix, normalized so that ⟨ψ|ψ⟩ = 1 with respect to the standard inner product. The corresponding density operator is |ψ⟩⟨ψ|, whose partial trace over either the spin or oscillator degree of freedom yields a Hermitian, positive semi-definite reduced matrix with real, non-negative eigenvalues. The von Neumann entropy is therefore unambiguously real. We have added an explicit statement of this definition, together with a short justification, to the entanglement-entropy section of the revised manuscript. The qualitative distinction between the real-eigenvalue and complex-eigenvalue regimes remains intact under this standard construction. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on known subspace decomposition and explicit computation

full rationale

The paper states the invariant-subspace decomposition as known for the model and proceeds to identify exceptional points and eigenvalue transitions on those subspaces before computing entanglement entropy profiles. No equations or steps reduce a claimed result to its own inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing uniqueness theorem is imported via self-citation. The central claim is an explicit computation on the stated subspaces rather than a tautological re-expression of the input assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5628 in / 1024 out tokens · 26288 ms · 2026-05-19T07:20:11.798203+00:00 · methodology

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Reference graph

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