Symmetries, Conservation Laws and Entanglement in Non-Hermitian Fermionic Lattices

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1. Identification of novel phenomena in non-Hermitian systems
2. Synergy of analytical and computational approaches

1. The entanglement phases identified in this article are probed numerically for the subsystem size l=L/4. While the analytical formulas are written in terms of the scaling with l, the numerical results do not distinguish between scaling with l and L. For the logarithmic phase, this is of no importance; however, for the volume-law phase, scaling with L for a fixed ratio l/L may produce a spurious contribution. Indeed, imagine that the entanglement entropy contains a subleading term of the form l^2/L that will vanish in the thermodynamic limit L to infinity. However, when scaling with L is considered for l=L/4, this term will produce a volume-law result. Can the scaling with l for the largest L be presented in the manuscript?

2. Entanglement entropy, while commonly used to describe the entanglement content in the system, contains contribution that are not associated with genuine quantum entanglement. In particular, for systems with the conserved particle number, entanglement entropy also accounts for the ("classical") number entropy. Such non-entanglement contributions to the entanglement entropy can be removed by considering the mutual information; furthermore, genuine entanglement is captured by other entanglement witnesses/measures like concurrence or entanglement negativity (employed by some of the authors in other papers). Importantly, the mutual information can be evaluated for the models considered here just "for free", using the same approach (I guess, other entanglement measures can be obtained in a similar way). It would be interesting to learn whether the volume-law phase as captured by the entanglement entropy indeed corresponds to high entanglement content. I don't insist on evaluating all these entanglement measures here (but strongly encourage the authors to address the mutual information); however, a discussion of these issues should appear in the manuscript to make it even stronger.

3. Figure 10. What is the reason for the apparent particle-hole asymmetry in panel b (for nu=1/4 and 3/4)? By the way, I would suggest the authors using the same colors for the same values of nu in all panels.

4. I don't understand the sentence "This result can be understood within our general framework: volume-law scaling is observed whenever the state filling exceeds the number of quasimomenta k with purely imaginary energies (see panel b) of Fig. 10) and the modes are quasimomenta k are singly occupied" on page 24 below Fig. 10 (possibly, a typo).

5. It seems to me that this reference on a Fisher–Hartwig asymptotic expansion for Toeplitz determinants is sufficiently relevant to the present article for being quoted: https://iopscience.iop.org/article/10.1088/1751-8113/46/8/085003

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This paper investigates steady-state entanglement transitions in non-Hermitian quantum many-body systems, with a focus on the role of conservation laws. The authors develop a theoretical framework for translation-invariant, non-interacting fermionic models with U(1) symmetry. They analyze how the spectrum of the non-Hermitian Hamiltonian and conserved quantities shape the steady-state structure and entanglement properties. The framework is applied to two specific models: the Hatano-Nelson and non-Hermitian Su-Schrieffer-Heeger (SSH) models. The results reveal a rich interplay between the spectrum, conservation laws, and entanglement transitions, supported by both analytical and numerical calculations.

The paper addresses an important and timely topic in the study of non-Hermitian quantum systems, advancing our understanding of steady-state entanglement transitions. The results are novel and intriguing, particularly the identification of filling-driven transitions between critical sub-volume scaling and area-law scaling, as well as the volume-law scaling observed for fully real spectra. The paper is well-written, technically sound, and generally of high quality. I believe it makes a significant contribution to the field and recommend it for publication after very minor revisions.

I have one conceptual question that I believe the authors should address: The paper deals with the long-time behavior of non-Hermitian systems, where the non-Hermitian Hamiltonian is justified as arising in the no-click limit of a monitored system. There seems to be a tension here as the no-click limit will become generally very unlikely in the limit of long times. Could the authors comment? On the other hand, how robust do they believe are the results to deviations from the no-click limit, and how might the framework generalize to systems that are described at the level of the quantum master equation so including fluctuations from the environment? (The authors mention Ref 13 by McDonald, Hanai, and Clerk.)

Minor revisions

1. In several places in Section 5, the authors refer to "amplification" and "damping" of modes. Is this terminology appropriate for fermionic systems? If not, I suggest clarifying or revising the language.

2. In Appendix B, I noticed a few typos and one missing equation number. These should be corrected for clarity and completeness.

I recommend the paper for publication in SciPost Physics after minor revision. It is well-written, technically sound, and addresses a timely topic, advancing our understanding of non-Hermitian quantum systems significantly.

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