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Asymptotically exact solution of the non-Hermitian disordered interacting Hatano-Nelson chain

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Pith's one-line read The paper establishes an asymptotically exact low-energy solution of the disordered interacting Hatano-Nelson spin chain, showing that non-Hermitian couplings grow under renormalization and turn the ground state into classicalized…

desk verdict The paper's central claim of a non-Hermitian relevant perturbation collapses under the model's exact similarity to the Hermitian XXZ chain; the headline signatures are basis-dependent. read the letter →

arxiv 2509.16309 v2 pith:RW27UTCJ submitted 2025-09-19 cond-mat.str-el cond-mat.dis-nn

Valéria M. Mattiello , Victor L. Quito , Eduardo Miranda This is my paper · ORCID
classification cond-mat.str-elcond-mat.dis-nn
keywords non-Hermitianmany-bodysystemsHatano-NelsonmodeldisorderedXXZspinchainstrong-disorderrenormalizationgrouprandomsingletphaseentanglemententropymagneticsusceptibilityquantum-to-classicalcrossover
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper gives an asymptotically exact low-energy solution of the disordered interacting Hatano-Nelson model, which is the same as the non-Hermitian spin-1/2 XXZ chain. Its central claim is that non-Hermitian couplings are relevant in the renormalization-group sense: under the strong-disorder flow they grow in both strength and statistical width, driving a quantum-to-classical crossover. The ground state is a random collection of strongly coupled spin pairs, but each pair is a mixture of the singlet and the $M=0$ triplet rather than a pure singlet. Two observable signatures follow: the $xy$ magnetic susceptibility becomes negative and diverges at a finite small temperature, and the entanglement entropy of a size-$L$ partition saturates instead of growing as $\ln L$. If correct, this turns a regime previously treated mainly numerically—disorder, interactions, and non-Hermiticity together—into an analytically controlled problem.

What carries the argument

The engine is the strong-disorder renormalization group, a real-space scheme that repeatedly finds the two neighboring spins with the largest local excitation gap, freezes them in their biorthogonal ground state, and replaces them by one effective bond between the outer neighbors. The load-bearing identity is the decimation rule $\tilde{\gamma}_{i-1} = \gamma_{i-1}+\gamma_i+\gamma_{i+1}$, identical in form to the rule for bond length $\tilde{\ell}_{i-1} = \ell_{i-1}+\ell_i+\ell_{i+1}$ and independent of the flows of $J$ and $\Delta$. This decoupling lets the authors transfer known fixed-point distributions for bond lengths to $\gamma$, and gives the joint fixed-point distribution $Q_n(\eta=0,x,y)$ of scaled couplings, lengths, and $\gamma$ (subscript $n=1$ for symmetric, $n=2$ for asymmetric initial distributions) from which the susceptibility and the entropy follow.

What would settle it

Run the SDRG on chains of millions of sites over many disorder realizations—the same numerical experiment the paper uses for its histograms—and compare the measured joint distribution of scaled couplings, lengths, and gamma at the cutoff with Eqs. (38)-(39); failure to collapse onto those forms would remove the basis for the predicted entropy saturation. A complementary check is to compute the right-right pair entropy of a two-site problem at large gamma and compare with Eq. (11).

Watch

Extended reading notes

Core claim

The paper claims that at low energies the disordered non-Hermitian XXZ chain flows to an infinite-disorder fixed point whose ground state is a random strongly coupled pair phase. In each decimated pair, the right ground state is $|GS^{(R)}\rangle = \cosh(\gamma/2)|0,0\rangle + \sinh(\gamma/2)|1,0\rangle$ (and the left one has $\gamma\to -\gamma$), so pairs are coherent mixtures of singlet and $M=0$ triplet; in the Hermitian limit $\gamma=0$ they reduce to singlets. Because the decimation rule for $\gamma_i$ is additive and decoupled from $J_i$ and $\Delta_i$, the $\gamma$ distribution broadens without bound, in an even universal form for symmetric initial distributions and a one-sided universal form for asymmetric ones. The broadening makes low-energy pairs nearly classical, which produces two concrete predictions: $T\chi_{x,y}$ from already decimated pairs is negative, proportional to $-\sinh^2(\gamma/2)$, and diverges at a finite temperature $T=\Omega_0 e^{-\pi C_\gamma}$ in the symmetric case; and, using the right-right reduced density matrix, the entanglement entropy of a partition saturates for large $L$, in sharp contrast to the $\ln L$ growth of the Hermitian random-singlet chain.

Load-bearing premise

The analysis depends on an unproven joint fixed-point distribution of couplings, bond lengths, and non-Hermitian parameters, which the paper says will be derived elsewhere; the entanglement-entropy saturation is computed from it, so an error there would invalidate that central prediction.

Editorial extensions

If this is right

  • The ground state is a random strongly coupled pair phase, not a random singlet phase: pairs of arbitrary length at random positions are singlet-triplet mixtures, so spin-spin correlations and response functions differ from the Hermitian chain.
  • The $xy$ magnetic susceptibility turns negative at low temperature and diverges at a finite temperature set by the initial distribution ($T=\Omega_0 e^{-\pi C_\gamma}$ in the symmetric case), while the $z$ susceptibility keeps the Hermitian quasi-Curie form.
  • The entanglement entropy of a partition saturates at large $L$ instead of growing as $\ln L$; the saturated value is nonuniversal and controlled by the same constants $C_\gamma$ and $C_\ell$ that set the flow.
  • The one-sided growth of the $\gamma$ distribution at low energies gives a many-body, interacting explanation of the non-Hermitian skin effect: long bonds have strongly one-sided hopping, while symmetric initial $\gamma$ distributions show no skin effect.
  • The same SDRG framework can be applied to other disordered non-Hermitian models, such as a non-Hermitian random transverse-field Ising chain, where a similar quantum-to-classical crossover is expected.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the claimed fixed-point distribution is right, the saturation of entanglement entropy should be visible in numerical simulations of finite chains with modest disorder: the crossover scale in $L$ should be controlled by $C_\gamma$, so systems with broader initial $\gamma$ distributions should saturate earlier.
  • The negative, diverging susceptibility is an equilibrium thermodynamic signature of non-Hermiticity; because the paper shows it is not self-averaging, any experiment or numerical estimate would need many disorder realizations before claiming a finite-temperature divergence.
  • The RR versus RL density-matrix ambiguity suggests a sharper test: measuring entanglement with different state-vector normalizations should give either maximal pair entropy (RL) or $\gamma$-dependent suppression (RR); the paper's saturation picture only holds for RR.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper presents a strong-disorder renormalization-group (SDRG) study of the disordered non-Hermitian spin-1/2 XXZ chain, Eq. (1), which is equivalent to the interacting fermionic Hatano-Nelson model. The authors derive decimation rules for the couplings J, Δ, and the non-Hermitian parameter γ, show that J and Δ flow as in the Hermitian case, and find that the distribution of γ broadens without limit. They interpret this broadening as non-Hermiticity being a relevant perturbation, leading to a 'quantum-to-classical crossover' and a random strongly coupled pair phase. The paper predicts a negative transverse magnetic susceptibility that diverges at a finite temperature and an entanglement entropy that saturates with subsystem size, and it supports these predictions with numerical SDRG simulations of chains up to 5 million sites.

Significance. The model is well chosen and the SDRG machinery is applied with care: the decimation rules are derived in biorthogonal perturbation theory, and the fixed-point distributions of γ are checked numerically on very large chains. If the central physical interpretation were correct, the paper would be a significant analytical advance in disordered non-Hermitian many-body physics. However, the main claim—that non-Hermiticity is an RG-relevant perturbation—is not supported because the γ couplings can be removed by an exact similarity transformation, and the two headline observables (susceptibility and entanglement entropy) depend, as the paper itself shows, on the chosen biorthogonal/RR convention. The significance is therefore conditional on resolving these foundational issues.

major comments (4)
  1. [Main text, around Eq. (4) and SM Section II] The Hamiltonian in Eq. (1) is exactly similar to the Hermitian disordered XXZ chain. With U = ∏_i exp(θ_i S^z_i) and θ_{i+1} – θ_i = γ_i/2, one has U H U^{-1} = ∑_i J_i[(S^+_i S^-_{i+1} + S^-_i S^+_{i+1})/2 + Δ_i S^z_i S^z_{i+1}], so the full spectrum and the SDRG decimation order are those of the Hermitian model. This is consistent with SM Eq. (16), which shows the two-site eigenvalues are γ-independent, and with the sentence after SM Eq. (25) stating that the decimation 'follows the same hierarchy of decimations as the Hermitian one would.' The additive rule for γ in Eq. (4) therefore accumulates the gauge phase in the same way that the bond length ℓ does. The broadening of P(γ) is a kinematic consequence of cluster formation, not the flow of a coupling that controls any spectral scale or similarity-invariant physical property. The central claim in the abstract and Conclusions that non-Hermiticity is 'relevant in the RG sense' is thus not established. The paper should either identify a similarity-invariant observable that distinguishes the non-Hermitian model from the Hermitian one or substantially revise the interpretation.
  2. [Eq. (8) and SM Section IV; Eq. (11) and SM Section V] Both headline predictions are convention-dependent. The negative transverse susceptibility of a single pair, Eq. (8) and SM Eq. (46), is obtained as the biorthogonal matrix element ⟨ψ_L|(S^α)^2|ψ_R⟩ of a positive operator; this quantity can be negative because the right and left eigenstates of a non-Hermitian Hamiltonian are not orthogonal. The paper does not justify why this matrix element is the physical magnetic response of a non-Hermitian thermodynamic system. Similarly, the entanglement-entropy saturation in Fig. 3 is based on the RR reduced density matrix, whereas SM Eq. (69) shows that the RL scheme gives S_pair = 1 for all γ and hence no saturation. Since the claimed quantum-to-classical crossover is identified through these two quantities, the main physical conclusions are not robust to the choice of convention. A physical argument for the RR/biorthogonal choice is required before these results can be interpreted as evidence of a crossover.
  3. [SM Eqs. (38)–(39)] The joint fixed-point distribution Q_n(η, x, y) is stated without derivation; the text says the derivation is lengthy and details will be provided elsewhere. This distribution is essential for the entanglement-entropy calculation, since the EE in SM Eq. (72) and the analytical curves in Fig. 3 are computed from it. The 'asymptotically exact' claim for the EE therefore rests on an unverified input. The derivation should be included in the paper, or the EE result should be labeled as a numerical observation.
  4. [SM Fig. 4 and main-text Figs. 2–3] The quantitative comparison between the analytical expressions and the numerics is not parameter-free. The constants C_γ and C_ℓ are determined by fitting the numerical SDRG flow to the universal fixed-point distributions (SM Fig. 4), and the same constants enter the analytical curves for χ and S that are overlaid on the numerical data in Figs. 2 and 3. Because the fitted parameters come from the same simulations that are being compared, the agreement demonstrates consistency but does not independently validate the theory. Please separate the fitting step from the test, for example by extracting C_γ and C_ℓ from one quantity and using them to predict another.
minor comments (5)
  1. [Fig. 2 caption and text] The text refers to red, yellow, green, and blue curves, but the color scheme in the figure as rendered is not consistent; please verify the labels against the published figure.
  2. [Main text, after Eq. (9)] The quantity Γ_T is used in Eq. (9) but only defined in the following sentence; please define it before first use.
  3. [SM Eqs. (26)–(28)] Please clarify the dimensions and definitions of C_γ and C_ℓ explicitly, and state whether they are dimensionless constants depending on the initial distributions.
  4. [Main text, paragraph after Eq. (5)] The phrase 'infrared stable fixed points of the Δ distribution are Δ_i = 0 ... and Δ_i → ∞' is imprecise; the Δ → ∞ fixed point is the Ising fixed point, and the flow to Δ = 0 occurs for 0 < Δ_i < 1, while Δ_i > 1 flows to the Ising fixed point. Please rephrase for clarity.
  5. [Introduction and references] Several references are recent preprints; please update citation data where journal versions are available, and define 'biorthogonal basis' on first use in the main text.

Circularity Check

2 steps flagged · score 6.0 of 10

The γ-flow is the Hermitian bond-length flow relabeled, and the quantitative susceptibility/EE curves use constants fitted to the same SDRG flow they are compared with.

  1. renaming known result [Main text, 'SDRG flows' section (after Eq. (5)); SM Sec. II A, Eq. (16)]
    "The asymptotic form of the joint distribution of Ji and bond lengths ℓi was obtained analytically for the Hermitian XXZ chain [47]. That result can be used to immediately determine the joint distribution of Ji and γi, due their identical decimation rules. ... Note that the eigenvalues are independent of γi and, therefore, the same as those of the Hermitian XXZ model."

    The paper's own two-site spectrum is γ-independent, so the SDRG decimation hierarchy is exactly the Hermitian one. The γ distribution is then taken, by the paper's admission, directly from Fisher's known Hermitian bond-length distribution because γ and ℓ obey identical additive rules. Moreover, γ enters Eq. (1) only through an exact non-unitary gauge transformation U=∏_i exp(θ_i S^z_i) with θ_{i+1}-θ_i=γ_i, which conjugates the Hamiltonian to the Hermitian XXZ chain. Hence the claimed 'growth' of γ is the accumulation of a removable gauge phase/cluster size, not the flow of a physical coupling that affects any spectral scale.

  2. fitted input called prediction [Supplemental Material, Sec. III and Fig. 4; main text Fig. 3 caption]
    "We could then fit the numerical histograms at late flow stages to the fixed-point distributions of Eqs. (32) and (33), and thus determine the constant Cγ ... By fitting to the distributions from the late stages of the SDRG flow [(c) and (f)], we determined the non-universal constants for (c) Cγ = 20.5 and (f) Cγ = 4.8. ... The lines correspond to fits to the analytical expressions derived in the asymptotic large-L regime."

    The non-universal constants Cγ and Cℓ enter the analytical susceptibility (SM Eq. (55), including the divergence at T=Ω0 e^{-πCγ}) and the analytical entanglement-entropy curves (SM Eqs. (79) and (86)). These constants are fitted to the same numerical SDRG flow whose late-stage distributions the analytical expressions are then compared with; the main-text Fig. 3 caption explicitly labels the curves as fits. Thus the agreement in Figs. 2 and 3 is a fit to the same flow, not an external check. The qualitative broadening of P(γ) follows directly from the additive rule Eq. (4) without fitting, which limits the circularity, but the quantitative predictions are not independent.

full rationale

The SDRG decimation rules themselves are derived self-consistently in the biorthogonal basis, and the J and Δ flows genuinely reproduce the Hermitian XXZ structure, so the paper is not wholly circular. However, two load-bearing reductions weaken the central claims. First, because the two-site spectrum is γ-independent and γ obeys the same additive decimation rule as the bond length, the paper imports the Hermitian bond-length distribution as the γ distribution; together with the exact non-unitary gauge removal of γ, the 'relevance of non-Hermiticity' is a relabeling of the known growth of cluster sizes rather than an independent physical result. Second, the quantitative curves for the susceptibility divergence and the entanglement-entropy saturation use Cγ and Cℓ fitted from the same numerical SDRG flow with which they are compared, so the agreement in Figs. 2 and 3 is partly a fit. The joint fixed-point distribution Q_n in SM Eqs. (38)-(39) is also assumed without derivation ('details will be provided elsewhere'), adding fragility, though this is not itself circularity. On balance the qualitative additive growth is derived rather than fitted, so the score is not maximal; but the central interpretation and the quantitative predictions are substantially built from the Hermitian flow and fitted constants, giving a score of 6.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The solution rests on standard SDRG assumptions plus two non-universal constants fitted from the same numerical flow. The joint distribution Q_n is a load-bearing ingredient that is not derived. The physical predictions also depend on conventions for non-Hermitian expectation values and density matrices.

free parameters (3)
  • C_gamma (symmetric case) = 20.5 (for uniform gamma in [-0.1, 0.1])
    Fitted to SDRG histograms of the gamma distribution at late flow stages (SM Fig. 4c). Used in T* = Omega0 exp(-pi C_gamma) and in the analytic EE expressions.
  • C_gamma (asymmetric case) = 4.8 (for uniform gamma in [0.1, 0.2])
    Fitted to SDRG histograms at late flow stages (SM Fig. 4f). Sets the scale of the asymmetric fixed-point variable y = C_gamma gamma / Gamma^2.
  • C_ell = Not explicitly reported; set to 1 in Fig. 6, fitted for Fig. 3 lines
    Non-universal scale for bond lengths. The EE expressions in SM Eqs. (79) and (86) contain 1/C_ell; comparison to SDRG data in Fig. 3 requires it.
assumptions (6)
  • domain assumption The SDRG decimation rules derived from second-order perturbation theory in the biorthogonal basis are asymptotically exact at low energies.
    This is the standard strong-disorder RG assumption; the flow to infinite disorder (Fisher 1994) justifies it, but it is not proven from the Hamiltonian.
  • domain assumption The fixed-point distributions for J and Delta are the same as in the Hermitian random XXZ chain (Fisher 1994).
    The decimation rules for J and Delta do not involve gamma, so the gamma flow is slaved to the Hermitian J/Delta flow. This relies on the eigenvalues being gamma-independent, which is shown in SM Eq. (16).
  • ad hoc to paper The joint fixed-point distribution Q_n(eta,x,y) in SM Eqs. (38)-(39) is correct; its derivation is deferred.
    It is central to the EE calculation but not derived in the paper ('details will be provided elsewhere').
  • domain assumption For non-Hermitian thermodynamics, the trace is evaluated in the biorthogonal (left-right) basis, which is complete.
    SM Eq. (41). This choice yields negative x,y susceptibility. Other choices are possible and would change the result.
  • domain assumption The RR density matrix is the appropriate one for the entanglement entropy of the ground state.
    SM Section V.A. The RL scheme gives Spair = 1 always and no saturation; the claimed saturation requires the RR scheme.
  • domain assumption Delta_i > -1/2 and J_i > 0 to avoid ferromagnetic phases and keep a real spectrum.
    Stated in the model section; restricts the parameter space of the solution.

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Cite this review

Pith. "Pith review of Asymptotically exact solution of the non-Hermitian disordered interacting Hatano-Nelson chain." pith.science (2026). https://pith.science/paper/RW27UTCJ

@misc{pith2026250916309,
  author       = {Pith},
  title        = {Pith review of: Asymptotically exact solution of the non-Hermitian disordered interacting Hatano-Nelson chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RW27UTCJ}},
  note         = {Machine review of arXiv:2509.16309}
}
abstract

We present an asymptotically exact solution of a paradigmatic non-Hermitian model: the disordered interacting fermionic Hatano-Nelson model, or equivalently, the non-Hermitian spin-1/2 XXZ model. We use a renormalization group method suited for disordered systems and show that non-Hermitian couplings are relevant perturbations to the Hermitian model, which ultimately leads to a quantum-to-classical crossover. The ground state of the model consists of a collection of strongly coupled pairs of spins of arbitrary size at random positions which, unlike the Hermitian case, do not form singlets, but a mixture of the singlet and the $M=0$ triplet state. As a result, the magnetic susceptibility in the $x,y$-directions becomes negative and diverges at a finite small temperature. Additionally, in sharp contrast to the $\ln(L)$ increase observed in disordered Hermitian chains, the entanglement entropy of a partition of size $L$ saturates for large $L$, as the strongly coupled pairs become classical and stop contributing at large length scales.

Figures

Figures reproduced from arXiv: 2509.16309 by the authors.

Figure 1
Figure 1. (a) The SDRG decimation step. The strongest [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Magnetic susceptibility multiplied by the tempera [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Entanglement entropy as a function of the partition [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Three representative stages of the SDRG flow of the marginal distribution of the coupling [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The magnetic susceptibility for nine different real [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Entanglement entropy for symmetric (blue) and [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

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