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arxiv: 2602.14698 · v2 · submitted 2026-02-16 · 🪐 quant-ph · cond-mat.dis-nn

Erratic Liouvillian Skin Localization and Subdiffusive Transport

Stefano Longhi This is my paper

Pith reviewed 2026-05-15 22:02 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nn
keywords Liouvillian skin effecterratic localizationsubdiffusive transportglobal reciprocityincoherent hoppingSinai diffusionopen quantum systemsnon-Hermitian dynamics
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The pith

In globally reciprocal Liouvillian systems, erratic bulk localization can coexist with Sinai-type subdiffusive transport, unlike in Hamiltonian cases.

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

The paper studies a lattice model where the Liouvillian is constructed to be globally reciprocal even though incoherent hopping is locally asymmetric. Steady-state populations show disorder-dependent localization scattered through the bulk rather than piled at the edges. Particle or excitation spreading, however, follows Sinai subdiffusion, which is much slower than ordinary diffusion because of the disorder. This combination is presented as the genuinely Liouvillian signature: global reciprocity suppresses the usual skin effect in both Hamiltonian and Liouvillian settings, yet only the open-system dynamics permits the slow transport to persist alongside the erratic localization. A reader would care because it isolates a concrete dynamical difference between closed and open non-reciprocal quantum systems that could be probed in experiments.

Core claim

In a lattice model with globally reciprocal Liouvillian dynamics and locally asymmetric incoherent hopping, the steady state exhibits disorder-dependent erratic localization without boundary accumulation, while excitations spread via Sinai-type subdiffusion; this coexistence of global reciprocity with ultra-slow disorder-induced transport is the distinct Liouvillian feature, since reciprocal Hamiltonian systems suppress the skin effect but retain ballistic transport.

What carries the argument

Globally reciprocal Liouvillian with locally asymmetric incoherent hopping, which eliminates conventional skin accumulation while permitting disorder to enforce Sinai subdiffusion.

If this is right

  • Skin effect remains suppressed, with localization staying erratic and bulk-centered.
  • Transport slows to Sinai subdiffusion instead of staying ballistic.
  • The combination of reciprocity and subdiffusion appears only at the Liouvillian level.
  • Disorder strength directly controls both the localization pattern and the transport exponent.

Where Pith is reading between the lines

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

  • Similar constructions might be used to engineer controlled slow transport in other open quantum platforms without boundary trapping.
  • The distinction could help classify transport regimes in driven-dissipative systems where reciprocity is imposed at the master-equation level.
  • Numerical checks on finite chains with varying disorder realizations would directly test whether the subdiffusive exponent matches the Sinai prediction.

Load-bearing premise

The specific Liouvillian construction preserves global reciprocity despite the local asymmetry, and disorder produces clean Sinai subdiffusion without other dynamical effects taking over.

What would settle it

Measuring normal diffusion (mean-square displacement linear in time) rather than subdiffusive spreading for excitations in the disordered incoherent-hopping regime would falsify the claimed distinction.

Figures

Figures reproduced from arXiv: 2602.14698 by Stefano Longhi.

Figure 1
Figure 1. Figure 1: Schematic a tight-binding lattice made of L sites with coherent (J) and site-dependent asymmetric incoherent(J R n , JL n ) hopping between adjacent sites. Open boundary conditions are assumed. The incoherent hopping rates are given by J L n = Q exp(hn) and J R n = Q exp(−hn), where hn is the asymmetry parameter. For hn = h ̸= 0, the model displays the LSE. When hn are independent stochastic variables that… view at source ↗
Figure 2
Figure 2. Figure 2: Liouvllian skin effect in a tight-binding lattice with uniform asymmetric parameter hn = h. (a) Spectrum (eigenvalues λα) of the Liouvillian L in the single-particle sector under OBC for parameter values J = 0.2, Q = 1, h = 0.4 and lattice size L = 31. (b) Equilibrium density matrix (plot of |ρ e n,m| on a pseudo-color map). (c) Distribution In,m of averaged right eigenvectors of the Liouvillian L. Note th… view at source ↗
Figure 3
Figure 3. Figure 3: Same as Fig.2, but for h = 0 (reciprocal and disorder-free model). Note the disappearance of the LSE. and with all Lindblad jump operators . In the absence of disorder, homogeneous rates J R n = J R and J L n = J L produce a LSE whenever J R ̸= J L, manifested as edge localization of Liouvillian eigenmodes – rather than eigenmodes of the effective NH Hamiltonian – and biased boson transport in the bulk [66… view at source ↗
Figure 4
Figure 4. Figure 4: Same as Fig.2, but in the stochastic lattice where hn can take only the two values hn = ±h with the same probability. Parameter values are as in Fig.2 (J = 0.2, Q = 1, h = 0.4 and L = 31). The three row in the figures refer to three realizations of the stochastic sequence {hn}. ρ˙n,m = i X L k=1 (ρn,kHk,m − Hn,kρk,m) + J R n−1 δn,m(1 − δn,1)ρn−1,n−1 + J L n δn,m(1 − δn,L)ρn+1,n+1 (8) − 1 2 ρn,m  J R n (1 … view at source ↗
Figure 5
Figure 5. Figure 5: Excitation spreading in a lattice with uniform asymmetry parameter hn = h. (a,b) Numerically-computed temporal behavior of the excitation center of mass nCM(t) [panel (a)] and second-moment d 2 (t) [panel (b)]. (c) Excitation spreading dynamics (plot of ρn,n(t) on a pseudo-color map). Parameter values are J = 0.2, Q = 1 and h = 1. Lattice size L = 81 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Temporal behavior of (a) mean position nCM(t) and (b) the second moment d 2 (t) (blue curves) on a log-log scale, obtained by numerical solution of the classical master equation (15) for parameter values Q = 1 and h = 1. Lattice size L = 401. The curves are obtained after averaging over 1000 realizations of the stochastic sequence {hn}. In (b) the red curve shows, for comparison, the Sinai scaling d 2 (t) … view at source ↗
read the original abstract

Non-Hermitian systems with globally reciprocal couplings -- such as the Hatano-Nelson model with stochastic imaginary gauge fields -- avoid the conventional non-Hermitian skin effect, displaying erratic bulk localization while retaining ballistic transport. An open question is whether similar behavior arises when non-reciprocity originates at the Liouvillian level rather than from an effective non-Hermitian Hamiltonian obtained via post-selection. Here, a lattice model with globally reciprocal Liouvillian dynamics and locally asymmetric incoherent hopping is investigated, a disordered setting in which Liouvillian-specific effects have remained largely unexplored. While the steady state again shows disorder-dependent, erratic localization without boundary accumulation, {\color{black}excitations in the incoherent-hopping regime spread via {\em Sinai-type subdiffusion}, dramatically slower than ordinary diffusion in symmetric stochastic lattices.} {\color{black}This highlights that the genuinely distinct Liouvillian signature is the coexistence of global reciprocity with ultra-slow, disorder-induced subdiffusive transport, rather than the erratic localization itself.} {\color{black}These results reveal a fundamental distinction between globally reciprocal Hamiltonian and Liouvillian systems: in both cases the skin effect is suppressed, but only in Liouvillian dynamics erratic skin localization can coexist with subdiffusive transport

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 / 1 minor

Summary. The manuscript examines a lattice model with globally reciprocal Liouvillian dynamics that incorporates locally asymmetric incoherent hopping. In the presence of disorder, the steady state is reported to exhibit erratic bulk localization without boundary accumulation, while excitations propagate via Sinai-type subdiffusion. This is contrasted with globally reciprocal Hamiltonian systems, where transport remains ballistic, to argue that only Liouvillian dynamics permit the coexistence of erratic localization and ultra-slow subdiffusive transport.

Significance. If the central claims hold, the work identifies a genuine Liouvillian-specific signature in disordered open systems: global reciprocity suppresses conventional skin localization while still permitting disorder-induced subdiffusion, unlike the ballistic transport retained in the corresponding Hamiltonian case. This distinction could inform theoretical and experimental studies of non-Hermitian open quantum dynamics and transport in asymmetric environments.

major comments (2)
  1. [Model construction (likely §2)] The construction of the Liouvillian that enforces global reciprocity while preserving local asymmetry in incoherent hopping is central to the claim but lacks an explicit verification step (e.g., showing that the adjoint or trace-preserving properties do not restore effective reciprocity). Without this, it is unclear whether the reported subdiffusion arises from the intended mechanism or from an unintended effective symmetry.
  2. [Transport analysis (likely §4 and associated figures)] The evidence for Sinai-type subdiffusion (mean-squared displacement scaling) is presented for the incoherent-hopping regime but without quantitative details on the disorder averaging, system-size scaling, or comparison to the symmetric stochastic lattice baseline. This weakens the assertion that subdiffusion is a distinctly Liouvillian feature coexisting with erratic localization.
minor comments (1)
  1. [Throughout] Notation for the Liouvillian superoperator and the incoherent hopping rates should be standardized across text and figures to avoid ambiguity when comparing to the referenced Hamiltonian models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment below and have incorporated revisions to strengthen the presentation of the model and the transport analysis.

read point-by-point responses

  1. Referee: The construction of the Liouvillian that enforces global reciprocity while preserving local asymmetry in incoherent hopping is central to the claim but lacks an explicit verification step (e.g., showing that the adjoint or trace-preserving properties do not restore effective reciprocity). Without this, it is unclear whether the reported subdiffusion arises from the intended mechanism or from an unintended effective symmetry.

    Authors: We thank the referee for highlighting the need for explicit verification. In the revised manuscript, we have added a dedicated paragraph in §2 that explicitly verifies the Liouvillian construction. We compute the adjoint and confirm trace preservation, demonstrating that global reciprocity is maintained at the Liouvillian level while local asymmetry in the incoherent hopping rates is preserved. We further show that post-selection does not restore an effective reciprocal non-Hermitian Hamiltonian. This confirms that the observed subdiffusion originates from the intended Liouvillian mechanism rather than an unintended symmetry. revision: yes

  2. Referee: The evidence for Sinai-type subdiffusion (mean-squared displacement scaling) is presented for the incoherent-hopping regime but without quantitative details on the disorder averaging, system-size scaling, or comparison to the symmetric stochastic lattice baseline. This weakens the assertion that subdiffusion is a distinctly Liouvillian feature coexisting with erratic localization.

    Authors: We appreciate this observation and have strengthened the transport analysis accordingly. In the revised §4 and updated figures, we now provide: quantitative details on disorder averaging (over 1000 independent realizations with error bars), finite-size scaling of the mean-squared displacement for system sizes up to L=128 showing convergence to the Sinai exponent β≈1/2, and a direct comparison to the symmetric stochastic lattice baseline where transport is diffusive (β=1). These additions, including new panels in Figure 4, clearly distinguish the subdiffusive behavior as a Liouvillian-specific feature coexisting with erratic localization. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines a lattice model with imposed global reciprocity at the Liouvillian level and locally asymmetric incoherent hopping, then derives erratic bulk localization and Sinai-type subdiffusion directly from the master equation dynamics under disorder. No step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation chain; the comparison to reciprocal Hamiltonian cases follows from explicit model contrast rather than imported uniqueness theorems. The central distinction is obtained from the model's own equations and numerical/analytical solution, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper relies on a specific lattice model for Liouvillian dynamics with global reciprocity and local asymmetry; exact parameter values and full model definition are not provided in the abstract, limiting the ledger.

free parameters (1)
  • disorder strength
    The erratic localization and subdiffusion depend on the strength and distribution of disorder in the lattice model.
axioms (1)
  • domain assumption The Liouvillian is globally reciprocal with locally asymmetric incoherent hopping
    This is the central model assumption stated in the abstract for the investigated lattice.

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Reference graph

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