Physical constraints on effective non-Hermitian systems

Aaron Kleger; Rufus Boyack

arxiv: 2508.21067 · v2 · pith:25QCN2Q2new · submitted 2025-08-28 · 🪐 quant-ph · cond-mat.str-el

Physical constraints on effective non-Hermitian systems

Aaron Kleger , Rufus Boyack This is my paper

Pith reviewed 2026-05-22 12:09 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el

keywords non-Hermitian HamiltonianMatsubara Green's functionpseudo-Hermitian quantum mechanicsinteracting quantum systemsopen quantum systemsDirac modelelectromagnetic response

0 comments

The pith

Directly adding anti-Hermitian parts to Matsubara Green's functions violates the standard framework for interacting quantum systems, which instead requires pseudo-Hermitian quantum mechanics.

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

The paper examines physical constraints on effective non-Hermitian Hamiltonians when interactions are present. It shows that the common technique of embedding the anti-Hermitian component directly into the Matsubara Green's function conflicts with the requirements of the standard interacting framework. In its place the authors supply a consistent description that distinguishes these systems from ordinary interacting physics and identifies them as instances of pseudo-Hermitian quantum mechanics. The work further characterizes zero-temperature distribution functions across several non-Hermitian frameworks and applies the results to the electromagnetic response of a non-Hermitian Dirac quasiparticle model in one spatial dimension.

Core claim

Interacting and open quantum systems can be formulated with an effective non-Hermitian Hamiltonian, yet the effective action and associated Green's functions must obey important constraints. Incorporating the anti-Hermitian part of the Hamiltonian directly in the Matsubara Green's function is incompatible with the standard framework for systems with interactions. A consistent physical description is furnished by recognizing that such systems are described by pseudo-Hermitian quantum mechanics, distinct from conventional interacting physics. Zero-temperature distribution functions are characterized within several frameworks, and the electromagnetic response is analyzed for the (1+1)-dimension

What carries the argument

Consistency requirements on the effective action and Green's functions imposed by the standard interacting quantum framework, which rule out direct anti-Hermitian insertion and instead select pseudo-Hermitian quantum mechanics as the appropriate description.

If this is right

The Matsubara formalism cannot be extended to non-Hermitian cases with interactions without violating physical consistency.
Non-Hermitian many-body systems must be treated with pseudo-Hermitian quantum mechanics to remain compatible with interactions.
Zero-temperature distribution functions acquire specific forms that depend on the chosen non-Hermitian framework.
Electromagnetic response functions for non-Hermitian quasiparticles become well-defined only under the consistent pseudo-Hermitian description.

Where Pith is reading between the lines

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

Simulations of open quantum many-body systems may need to adopt pseudo-Hermitian structures to avoid inconsistencies when interactions are included.
The same consistency requirement could apply to other formalisms beyond Matsubara, pointing to broader adjustments in modeling non-Hermitian physics.
This distinction supplies a route to construct new methods for dissipative systems that preserve required physical properties such as causality.

Load-bearing premise

The standard framework for interacting quantum systems imposes specific requirements on the effective action and Green's functions that are violated by directly including the anti-Hermitian part of the Hamiltonian.

What would settle it

An explicit calculation of Green's functions or response functions in an interacting non-Hermitian model that produces unphysical features such as negative spectral weights when the anti-Hermitian term is added directly to the Matsubara function, while remaining physical under the pseudo-Hermitian treatment.

Figures

Figures reproduced from arXiv: 2508.21067 by Aaron Kleger, Rufus Boyack.

**Figure 1.** Figure 1: FIG. 1: Fermionic spectral function in units of [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

read the original abstract

Interacting and open quantum systems can be formulated in terms of an effective non-Hermitian Hamiltonian (NHH), however, there are important constraints that must be satisfied by the effective action and the associated Green's functions. One common approach to many-body non-Hermitian (NH) systems is to incorporate the anti-Hermitian part of the Hamiltonian directly in the Matsubara Green's function. Here, we show that such an approach is incompatible with the standard framework for systems with interactions. Furthermore, we furnish a consistent physical description for such systems by determining their distinction from conventional interacting physics, and find that they are described by pseudo-Hermitian quantum mechanics. Furthermore, we characterize the zero-temperature distribution functions within several frameworks for NH systems. As an application of our results, we consider the electromagnetic response of a NH quasiparticle Hamiltonian based on the (1+1)-dimensional NH Dirac model subject to various physical descriptions.

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.

Desk Editor's Note private letter to a colleague

The paper argues that sticking the anti-Hermitian part directly into Matsubara Green's functions breaks the standard interacting framework, so pseudo-Hermitian quantum mechanics is the consistent description instead.

read the letter

The main thing here is that a common way of handling non-Hermitian many-body systems by putting the anti-Hermitian part straight into the Matsubara Green's function doesn't line up with the standard approach for interacting systems. The authors show this incompatibility and argue that these systems are better described using pseudo-Hermitian quantum mechanics. They also look at zero-temperature distribution functions across different frameworks and apply the results to the electromagnetic response of a non-Hermitian Dirac model in 1+1 dimensions. They earn credit for spotting a potential modeling issue that could affect how people treat open quantum systems with interactions, and the Dirac model example gives a practical illustration. That part feels like a solid step forward within this subfield. The soft spot is that the incompatibility claim hinges on some requirement in the standard framework that isn't spelled out clearly. The stress-test note is right to flag that without naming the exact violated condition, like a particular identity or the structure of the effective action, it's hard to judge if the approach is truly incompatible or just needs tweaks. If the paper has the detailed derivations to back this up, it would address that. This is for people working on non-Hermitian physics in condensed matter or quantum tech. A reader who models systems with gain and loss would get value from the constraints they propose. It looks like it deserves a serious referee to go over the arguments and checks. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper claims that directly incorporating the anti-Hermitian part of an effective non-Hermitian Hamiltonian into the Matsubara Green's function for many-body systems is incompatible with the standard framework for interacting quantum systems. It provides a consistent description by distinguishing these systems from conventional interacting physics and showing they are described by pseudo-Hermitian quantum mechanics. The work also characterizes zero-temperature distribution functions in several NH frameworks and applies the results to the electromagnetic response of a (1+1)-dimensional NH Dirac quasiparticle model.

Significance. If the incompatibility is rigorously derived from specific constraints in the interacting framework (such as the structure of the effective action or self-energy), this would clarify proper handling of effective non-Hermitian descriptions in open many-body systems and strengthen the link to pseudo-Hermitian QM. The characterization of distribution functions and the concrete EM response application are useful contributions. The manuscript attempts to ground the discussion in physical distinctions rather than ad-hoc modifications.

major comments (1)

[Abstract and section on standard framework for interacting systems] The central incompatibility claim (abstract and the section deriving constraints on the effective action/Green's functions) asserts that direct inclusion of the anti-Hermitian part violates the standard interacting framework, but does not explicitly identify the violated requirement (e.g., a specific Ward identity, the form of the self-energy, or the contour/integration structure in the effective action). Without naming and demonstrating this violation via derivation or counter-example, the claim that the approach 'is incompatible' remains unanchored and load-bearing for the subsequent pseudo-Hermitian description.

minor comments (2)

[Introduction or methods] Notation for the Matsubara Green's function and the anti-Hermitian component should be defined explicitly at first use to avoid ambiguity when comparing to conventional Hermitian cases.
[Application section] The application section on the NH Dirac model would benefit from a brief statement of how the pseudo-Hermitian description alters the electromagnetic response relative to a naive anti-Hermitian inclusion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment below and have revised the manuscript to strengthen the explicit identification of the incompatibility.

read point-by-point responses

Referee: [Abstract and section on standard framework for interacting systems] The central incompatibility claim (abstract and the section deriving constraints on the effective action/Green's functions) asserts that direct inclusion of the anti-Hermitian part violates the standard interacting framework, but does not explicitly identify the violated requirement (e.g., a specific Ward identity, the form of the self-energy, or the contour/integration structure in the effective action). Without naming and demonstrating this violation via derivation or counter-example, the claim that the approach 'is incompatible' remains unanchored and load-bearing for the subsequent pseudo-Hermitian description.

Authors: We appreciate the referee's observation that the specific violated requirement should be named more directly to anchor the central claim. The manuscript derives the incompatibility from the structure of the effective action in the interacting case, where the Matsubara Green's function must obey standard contour integration and analytic properties that are incompatible with direct insertion of anti-Hermitian terms (as this disrupts consistency with the self-energy and the required fluctuation-dissipation relation). To address the concern, we have revised the relevant section to explicitly identify the violated contour/integration structure in the effective action and include a short derivation with a counter-example illustrating the resulting inconsistency. This change clarifies the foundation without altering the overall conclusions or the link to pseudo-Hermitian quantum mechanics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation contrasts external standard framework with pseudo-Hermitian QM

full rationale

The paper asserts incompatibility between direct inclusion of anti-Hermitian terms in Matsubara Green's functions and the standard framework for interacting systems, then distinguishes the systems via pseudo-Hermitian quantum mechanics. This contrast relies on external conventions of effective actions, Green's functions, and Ward identities rather than any self-definition or fitted input within the paper. No equations or claims reduce by construction to the paper's own inputs, and the pseudo-Hermitian description is presented as an established external framework rather than a self-citation chain or ansatz smuggled from prior author work. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the paper relies on background assumptions from quantum many-body theory without introducing new free parameters or invented entities.

axioms (1)

domain assumption The standard framework for interacting quantum systems requires specific forms for the effective action and Green's functions that exclude direct anti-Hermitian inclusion.
Invoked to establish the incompatibility of the common Matsubara approach.

pith-pipeline@v0.9.0 · 5680 in / 1402 out tokens · 78533 ms · 2026-05-22T12:09:09.245744+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

One common approach to many-body non-Hermitian (NH) systems is to incorporate the anti-Hermitian part of the Hamiltonian directly in the Matsubara Green's function. Here, we show that such an approach is incompatible with the standard framework for systems with interactions.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the Matsubara Green's function evaluates to G(iωn) = (iωn − h0 − iΓ sgn(ωn))^{-1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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