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arxiv: 2606.07275 · v1 · pith:7Z24726Onew · submitted 2026-06-05 · 🪐 quant-ph · cond-mat.stat-mech

Quantum critical properties of non-Hermitian XY models with magnetic field

Jia-Jia Luo , Volker Meden This is my paper

Pith reviewed 2026-06-27 21:58 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords non-Hermitian XY modelquantum criticalityphase diagramcorrelation functionsstandard quantum mechanicsbiorthogonal quantum mechanicsspin chainsmagnetic field
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The pith

Quantum critical properties of non-Hermitian XY spin chains depend on both the formalism and the chosen ground-state analog.

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

This paper examines two non-Hermitian versions of the XY spin chain subject to a magnetic field. Using exact solutions, it computes the energy, magnetization, and long-distance decay of static correlation functions, once in the standard formalism of quantum mechanics and once in the biorthogonal formalism, each applied to two candidate states that might serve as analogs of a Hermitian ground state. The locations of critical points and the structure of the phases change with these choices. A reader would care because non-Hermitian Hamiltonians appear in models of open quantum systems, so the ambiguity affects how one predicts measurable critical behavior. The authors conclude that standard quantum mechanics is the appropriate formalism and that the state used for calculations must match the way the system would be prepared experimentally.

Core claim

Using exact solutions for the non-Hermitian XY models, the energy, magnetization, and asymptotic behavior of static correlation functions are computed in both the standard and biorthogonal formalisms for two different states. The critical properties, including the phase diagram, are found to depend on the formalism and the state. Arguments are provided in favor of standard quantum mechanics, with the state choice depending on hypothetical experimental preparation.

What carries the argument

Exact solutions of the non-Hermitian XY spin chains with magnetic field, used to evaluate expectation values in standard versus biorthogonal quantum mechanics for two candidate ground states.

If this is right

  • The phase diagram shifts when the formalism is switched from standard to biorthogonal quantum mechanics.
  • Long-distance correlations exhibit different decay rates depending on which candidate state is used.
  • Standard quantum mechanics produces critical behavior that reduces to the Hermitian limit in a consistent way.
  • The appropriate state for calculations is determined by the experimental preparation of the system.

Where Pith is reading between the lines

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

  • Theoretical work on non-Hermitian criticality should state the assumed experimental context to make definite predictions.
  • The same dependence on formalism and state may appear in other non-Hermitian lattice models.
  • Direct comparison with laboratory measurements on driven or lossy spin chains could decide which formalism applies.

Load-bearing premise

The two states considered are appropriate analogs of a Hermitian ground state and the exact solutions fully capture the long-distance asymptotic behavior of the correlation functions.

What would settle it

Prepare one of the non-Hermitian XY chains experimentally, measure its magnetization or long-distance correlations, and check whether the data match the standard or biorthogonal predictions.

Figures

Figures reproduced from arXiv: 2606.07275 by Jia-Jia Luo, Volker Meden.

Figure 1
Figure 1. Figure 1: FIG. 1. The ground state phase diagram of the Hermitian [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The vacuum state energy density as a function of Re( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The relevant energy gaps ∆ for the minimal energy [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The magnetization of the minimal energy state as a function of the real and imaginary part of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The phase diagram of the imaginary- [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The magnetization of the vacuum state as a function of [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. The vacuum state RR [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The vacuum state LR [PITH_FULL_IMAGE:figures/full_fig_p017_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. The [PITH_FULL_IMAGE:figures/full_fig_p017_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. The oscillation periods extracted from the numer [PITH_FULL_IMAGE:figures/full_fig_p019_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. The linear-log derivative [PITH_FULL_IMAGE:figures/full_fig_p021_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. The [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. The oscillation periods of RR and LR correlation [PITH_FULL_IMAGE:figures/full_fig_p022_27.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. The real parts of LR [PITH_FULL_IMAGE:figures/full_fig_p022_26.png] view at source ↗
read the original abstract

The characterization of the quantum critical properties of genuine non-Hermitian many-body systems remains ambiguous as neither the state considered nor the definition of expectation values is unique. In this work, we investigate the quantum critical properties of two models of non-Hermitian XY spin chains with magnetic field. Using exact solutions, we systematically investigate the parameter dependence of the energy, the magnetization as well as the long-distance asymptotic behavior of static correlation functions. We compute expectation values within the standard formalism of quantum mechanics as well as within biorthogonal quantum mechanics and take two different states which one might reasonably consider to be the analog of the ground state of a Hermitian model. The critical properties, including such fundamental characteristics as the phase diagram, depend on both the formalism used as well as the state considered. We provide arguments in favor of the use of standard quantum mechanics. Which state to be taken in computations, depends on the (hypothetical) experimental preparation of the system.

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

0 major / 2 minor

Summary. The manuscript investigates the quantum critical properties of two non-Hermitian XY spin chain models with magnetic field. Using exact solutions, the authors compute the energy, magnetization, and long-distance asymptotics of static correlation functions. They compare results obtained within the standard quantum mechanics formalism and biorthogonal quantum mechanics, considering two different states as potential analogs of the Hermitian ground state. The central finding is that the critical properties, including the phase diagram, depend on both the chosen formalism and the state. The authors provide arguments favoring the standard formalism and note that the appropriate state depends on the experimental preparation of the system.

Significance. If the results hold, this paper highlights a fundamental ambiguity in defining quantum criticality for non-Hermitian systems, showing through concrete examples that different formalisms and state choices can lead to qualitatively different phase diagrams and critical behaviors. The use of exact solutions on solvable models is a notable strength, enabling precise comparisons without numerical approximations or fitting parameters. This contributes to clarifying how to approach non-Hermitian many-body physics, particularly in light of potential experimental realizations, and supports the preference for standard quantum mechanics based on the presented arguments.

minor comments (2)
  1. The abstract states that correlation functions and magnetization are computed from exact solutions, but the manuscript should include a brief explicit example (e.g., in the section deriving the long-distance asymptotics) showing how the decay exponents or amplitudes are extracted for at least one parameter point in each formalism to increase transparency.
  2. Notation for the two candidate ground states (standard vs. biorthogonal) should be introduced with a short table or explicit definitions early in the methods section to avoid any ambiguity when results from the two formalisms are contrasted.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. No specific major comments were listed in the report, so there are no individual points requiring point-by-point rebuttal or changes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claims rest on explicit exact solutions of the XY models, with direct computation of energies, magnetizations, and correlation functions under two formalisms and two candidate ground states. These quantities are obtained from the model's spectrum and eigenvectors without parameter fitting or renormalization that would force the outputs to match inputs. No self-citation chains, uniqueness theorems, or ansatzes are invoked to justify the phase diagrams or critical properties; the dependence on formalism and state is demonstrated by explicit comparison rather than by construction. The modeling choices are flagged as assumptions conditioned on experimental preparation, not derived results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no free parameters, invented entities, or non-standard axioms are apparent. Relies on standard assumptions of exact solvability for XY models and the validity of biorthogonal QM as an alternative formalism.

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Reference graph

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    RR versus LR To provide an overview of our results, we summarize the asymptotic behavior of the correlation functions in Table II. In this we also compare the behavior obtained from the RR and LR expectation value. As already emphasized when investigating open quan- tum systems employing the formalism of standard quan- tum mechanics and thus use RR expect...

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    This is misleading as the latter would also transform the spin- 1/2 operatorsσ α l → −σ α l

    Note that in several papers [31–33, 44] the antilinear op- eratorKis referred to as the time-reversal operator. This is misleading as the latter would also transform the spin- 1/2 operatorsσ α l → −σ α l