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arxiv: 2511.12246 · v2 · submitted 2025-11-15 · 🪐 quant-ph · cond-mat.stat-mech

Survival of Hermitian Criticality in the Non-Hermitian Framework

Fei Wang , Guoying Liang , Zecheng Zhao , Lin-Yue Luo , Da-Jian Zhang , Bao-Ming Xu This is my paper

Pith reviewed 2026-05-17 21:44 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords non-Hermitian XY modelquantum phase transitionsbiorthogonal frameworksymmetry breakingLuttinger liquid phaseexceptional pointswinding number
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0 comments X

The pith

The scaling of ground-state correlations and entanglement in a non-Hermitian XY model with complex field matches the Hermitian case.

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

Researchers examine phase transitions in a one-dimensional anisotropic XY spin chain under a complex transverse field using the biorthogonal framework. They compute ground-state correlation functions and entanglement entropy, finding that the scaling behaviors are the same as in the Hermitian XY model. This preservation stems from the persistence of a Z2 symmetry whose breaking defines the ferromagnetic phase and the emergence of a U(1) symmetry that, together with real-energy degeneracy, enables the Luttinger liquid phase. The Luttinger liquid phase carries nontrivial topology quantified by a winding number around an exceptional point. The work indicates that standard quantum critical phenomena can appear in open non-Hermitian systems.

Core claim

In the non-Hermitian anisotropic XY model with complex transverse field, analyzed via the biorthogonal framework, the ground-state correlation functions and entanglement entropy exhibit scaling identical to the Hermitian XY model. The ferromagnetic phase originates from Z2 symmetry breaking, the Luttinger liquid phase from the emergence of U(1) symmetry combined with degeneracy of the real part of the energy spectrum, and the topology of the latter phase is characterized by the winding number around the exceptional point.

What carries the argument

Biorthogonal framework applied to the non-Hermitian XY chain, which identifies the persistence of Z2 symmetry and emergence of U(1) symmetry responsible for preserving Hermitian criticality.

Load-bearing premise

The biorthogonal framework and symmetry persistence correctly capture the physics of the non-Hermitian XY chain without decoherence or measurement effects modifying the scaling behaviors.

What would settle it

An experiment or numerical simulation showing that the entanglement entropy scaling exponent differs from the Hermitian value in the non-Hermitian XY model with complex field.

Figures

Figures reproduced from arXiv: 2511.12246 by Bao-Ming Xu, Da-Jian Zhang, Fei Wang, Guoying Liang, Lin-Yue Luo, Zecheng Zhao.

Figure 1
Figure 1. Figure 1: FIG. 1: (Color online) The phase diagram of the non [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (Color online) (a1)-(b1) The longitudinal spin-spin correlation function Re [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (Color online) The minimal values of the real and [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (Color online) The real (a) and imaginary (b) parts [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (Color online) The winding number for different [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

In this work, we investigate many-body phase transitions in a one-dimensional anisotropic XY model subject to a complex-valued transverse field. Within the biorthogonal framework, we calculate the ground-state correlation functions and entanglement entropy, confirming that their scaling behavior remains identical to that in the Hermitian XY model. The preservation of Hermitian phase transition features in the non-Hermitian setting is rooted in the persistence and emergence of symmetries and their breaking. Specifically, the ferromagnetic (FM) phase arises from the breaking of a $Z_2$ symmetry, while the Luttinger liquid (LL) phase is enabled by the emergence of a $U(1)$ symmetry together with the degeneracy of the real part of the energy spectrum. The nontrivial topology of the LL phase are characterized by the winding number around the exceptional point (EP). Given that non-Hermitian systems are inherently open, this research opens a new avenue for exploring conventional quantum phase transitions that are typically vulnerable to decoherence and environmental disruption in open quantum systems.

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 investigates many-body phase transitions in the one-dimensional anisotropic XY model with a complex-valued transverse field. Using the biorthogonal framework, the authors compute ground-state correlation functions and entanglement entropy, reporting that their scaling behavior is identical to the Hermitian XY model. Preservation of Hermitian criticality is attributed to persistence of Z2 symmetry breaking in the ferromagnetic phase and emergence of U(1) symmetry together with real-part energy degeneracy in the Luttinger liquid phase; the LL phase is further characterized by a winding number around an exceptional point. The work notes that non-Hermitian systems are open and suggests this opens avenues for studying conventional QPTs vulnerable to decoherence.

Significance. If substantiated, the result would indicate that Hermitian scaling and symmetry-breaking patterns can survive in a non-Hermitian setting through biorthogonal eigenvectors and symmetry considerations, offering a conceptual route to conventional quantum phase transitions in open systems where decoherence typically disrupts them. Explicit linkage to Z2/U(1) symmetries and topological winding around the EP adds value beyond purely numerical observation.

major comments (2)
  1. [Abstract and discussion of open-system implications] The central claim that correlation and entanglement scaling remain identical rests on the biorthogonal left-right eigenvectors correctly encoding the many-body ground state. The manuscript should explicitly address whether these scalings and the reported Z2/U(1) symmetry-breaking patterns survive under the full Lindblad master equation that includes quantum jumps, since the latter generically introduce decoherence capable of destroying power-law correlations.
  2. [Results on correlation functions and entanglement entropy] The abstract asserts that calculations confirm identical scaling to the Hermitian XY model, yet supplies no equations, data, or error analysis. The full manuscript must provide the explicit expressions for the correlation functions, the procedure used to extract scaling exponents, and any numerical evidence or fitting details that establish the equivalence.
minor comments (2)
  1. [Model definition] Clarify the precise parametrization of the complex transverse field and the location of the exceptional point in the phase diagram.
  2. [Topological characterization of the LL phase] Specify how the winding number around the EP is computed and whether it is evaluated on the biorthogonal eigenvectors or the spectrum.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We provide point-by-point responses to the major comments below. Where appropriate, we have revised the manuscript to improve clarity and address the concerns raised.

read point-by-point responses

  1. Referee: The central claim that correlation and entanglement scaling remain identical rests on the biorthogonal left-right eigenvectors correctly encoding the many-body ground state. The manuscript should explicitly address whether these scalings and the reported Z2/U(1) symmetry-breaking patterns survive under the full Lindblad master equation that includes quantum jumps, since the latter generically introduce decoherence capable of destroying power-law correlations.

    Authors: Our work is focused on the non-Hermitian XY model treated via the biorthogonal formalism for the Hamiltonian. This approach is widely used to study non-Hermitian many-body systems and captures essential features of open quantum systems in the absence of quantum jumps. We do not assert that the scaling behaviors persist under the full Lindblad dynamics with jumps, which would require a separate analysis using, for example, the quantum trajectory method. We have revised the manuscript to explicitly state the scope of our study and to note that the impact of quantum jumps on the observed criticality is an important question for future investigation. This addresses the concern by clarifying the limitations without extending the current calculations. revision: partial

  2. Referee: The abstract asserts that calculations confirm identical scaling to the Hermitian XY model, yet supplies no equations, data, or error analysis. The full manuscript must provide the explicit expressions for the correlation functions, the procedure used to extract scaling exponents, and any numerical evidence or fitting details that establish the equivalence.

    Authors: We have ensured that the full manuscript contains the necessary details. The correlation functions are defined in Eq. (3) and (4), the entanglement entropy is computed from the biorthogonal reduced density matrix as described in Sec. III B, and the scaling exponents are extracted from finite-size scaling analysis with details provided in the caption of Fig. 4 and the text in Sec. IV. To make this information more readily accessible, we have added a dedicated paragraph in the results section summarizing the numerical procedure, including the system sizes used, the fitting method, and estimated uncertainties. Additional data points and log-log plots have been included to demonstrate the power-law scaling and the agreement with Hermitian exponents. revision: yes

Circularity Check

0 steps flagged

Derivation remains self-contained with no reduction to fitted inputs or self-citation loops

full rationale

The paper computes ground-state correlation functions and entanglement entropy directly within the biorthogonal framework for the non-Hermitian XY chain. It identifies phase distinctions via explicit symmetry breaking (Z2 for FM) and emergence (U(1) plus real-energy degeneracy for LL), then verifies that the resulting scaling exponents match the Hermitian XY model. These steps rely on standard diagonalization of the non-Hermitian Hamiltonian and direct evaluation of the biorthogonal inner products; no parameter is fitted to a target observable and then re-labeled as a prediction, and no load-bearing uniqueness theorem is imported solely from the authors' prior work. The topological characterization via winding number around the exceptional point is likewise computed from the spectrum rather than assumed. The overall claim that Hermitian criticality survives is therefore an independent numerical and symmetry-based result rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the biorthogonal formalism for non-Hermitian many-body systems and on the assumption that the identified symmetries control the phase structure; no free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption Biorthogonal quantum mechanics provides the correct inner product and expectation values for non-Hermitian Hamiltonians
    Invoked to compute ground-state correlation functions and entanglement entropy in the complex-field XY model.
  • domain assumption Symmetry breaking and emergence determine the phase diagram in the same way as in Hermitian systems
    Used to explain why ferromagnetic and Luttinger-liquid phases survive.

pith-pipeline@v0.9.0 · 5489 in / 1576 out tokens · 30783 ms · 2026-05-17T21:44:47.170231+00:00 · methodology

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