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arxiv: 2602.12340 · v2 · submitted 2026-02-12 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· quant-ph

Controlled Zeno-Induced Localization of Free Fermions in a Quasiperiodic Chain

Pinaki Singha , Nilanjan Roy , Marcin Szyniszewski , Auditya Sharma This is my paper

Pith reviewed 2026-05-16 05:19 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnquant-ph
keywords measurement-induced localizationquantum Zeno regimeAubry-André-Harper modelnon-Hermitian HamiltonianLyapunov exponentquantum trajectoriesquasiperiodic potentialfree fermions
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The pith

Continuous monitoring of free fermions in a quasiperiodic chain induces localization captured by an effective non-Hermitian Hamiltonian in the Zeno regime.

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

In the quantum Zeno regime of a continuously monitored Aubry-André-Harper chain, an emergent energy scale from strong measurements reduces the dynamics to an effective non-Hermitian problem. This effective description is built from a self-consistent potential extracted directly from individual quantum trajectories without postselection. The reduction allows the localization length to be computed exactly via a transfer-matrix approach that yields the Lyapunov exponent. Numerical simulations of long-time quantum state diffusion trajectories confirm the analytical predictions up to controlled corrections set by the ratio of hopping to measurement strength. The result provides a controlled route to understanding how monitoring interacts with quasiperiodic disorder to produce localization.

Core claim

In the quantum Zeno regime an emergent dominant energy scale reduces the monitored Aubry-André-Harper problem to a transfer-matrix formulation of an effective non-Hermitian Hamiltonian, permitting direct computation of the Lyapunov exponent; this analytic prediction agrees quantitatively with the localization length extracted from long-time steady-state quantum state diffusion trajectories, with corrections of order J² over λ² plus (γ/2)².

What carries the argument

Measurement-induced effective potential constructed self-consistently from individual quantum trajectories, which in the Zeno regime supports a transfer-matrix analysis of the resulting non-Hermitian Hamiltonian.

If this is right

  • The Lyapunov exponent of the effective non-Hermitian transfer matrix directly supplies the localization length.
  • Numerical localization lengths reconstructed from steady-state single-particle wave functions match the analytic prediction.
  • The effective theory captures the combined effects of Zeno localization and quasiperiodic-disorder localization.
  • Corrections remain small and controlled by J² divided by λ² plus (γ/2)² when hopping is weak compared with monitoring.

Where Pith is reading between the lines

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

  • The trajectory-based construction of the effective potential may generalize to other monitoring protocols or lattice geometries.
  • Measurement strength emerges as a tunable knob for engineering localized states in disordered quantum systems.
  • The link between stochastic monitored dynamics and the non-Hermitian transfer matrix could guide studies of measurement-induced phases in interacting systems.

Load-bearing premise

The measurement-induced effective potential can be constructed self-consistently from individual quantum trajectories without postselection and remains valid when the monitoring strength greatly exceeds the hopping amplitude.

What would settle it

A mismatch between the Lyapunov exponent obtained from the transfer matrix of the effective non-Hermitian Hamiltonian and the spatial decay rate measured in long-time quantum state diffusion trajectories would falsify the reduction.

Figures

Figures reproduced from arXiv: 2602.12340 by Auditya Sharma, Marcin Szyniszewski, Nilanjan Roy, Pinaki Singha.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of the Aubry–Andr´e–Harper lat [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Effective-theory Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Localization length [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
read the original abstract

We investigate measurement-induced localization in a continuously monitored one-dimensional Aubry--Andr\'e--Harper model, focusing on the quantum Zeno regime in which the measurements dominate coherent dynamics. The presence of a quasiperiodic potential renders the problem analytically tractable and enables a controlled study of the interplay between monitoring and disorder. We develop an analytical description based on an instantaneous Schr\"odinger equation with a measurement-induced effective potential constructed self-consistently from individual quantum trajectories, without relying on postselection. In the quantum Zeno regime, an emergent dominant energy scale reduces the problem to a transfer-matrix formulation of an effective non-Hermitian Hamiltonian, which allows direct computation of the Lyapunov exponent. Complementarily, we extract the localization length numerically from long-time steady-state quantum state diffusion trajectories by reconstructing the intrinsic localized single-particle wave functions and analyzing their spatial decay. These numerical results show quantitative agreement with the effective theory predictions, with controlled corrections of order $J^2/[\lambda^2+(\gamma/2)^2]$ (where $J$ is the hopping amplitude, $\gamma$ the measurement strength, and $\lambda$ the quasiperiodic potential). Our results underscore the connection between the effective non-Hermitian description and the stochastic monitored dynamics, showing the interplay between Zeno-like localization, coherent hopping, and quasiperiodic-disorder-induced localization, while also laying the groundwork for understanding and exploiting measurement-induced localization as a tool for quantum control and state preparation.

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 measurement-induced localization of free fermions in a continuously monitored one-dimensional Aubry-André-Harper chain, with emphasis on the quantum Zeno regime (γ ≫ J). It constructs an instantaneous Schrödinger equation whose effective potential is built self-consistently from individual quantum trajectories without postselection, reduces the problem in the Zeno limit to a transfer-matrix calculation of the Lyapunov exponent for an effective non-Hermitian Hamiltonian, and reports quantitative agreement between this analytic prediction and localization lengths extracted from long-time quantum-state-diffusion trajectories, with corrections controlled by J²/[λ²+(γ/2)²].

Significance. If the self-consistent single-trajectory construction holds, the work supplies a controlled analytic route from stochastic monitored dynamics to an effective non-Hermitian transfer-matrix problem in a quasiperiodic setting, together with falsifiable quantitative predictions. This strengthens the link between measurement-induced localization and non-Hermitian spectral theory while offering a concrete handle on Zeno-enhanced localization for quantum-control applications.

major comments (2)
  1. [Section describing the effective-potential construction and the instantaneous Schrödinger equation] The central claim that an effective potential can be constructed self-consistently from each individual quantum trajectory (without postselection or ensemble averaging) is load-bearing for the transfer-matrix reduction. The stochastic Schrödinger equation contains a state-dependent, stochastic back-action term; the manuscript must show explicitly how the self-consistency loop closes for a single realization (e.g., by writing the algebraic or differential condition that determines the potential at each time step from the instantaneous state alone). If this closure implicitly invokes trajectory statistics, the Lyapunov exponent obtained from the transfer matrix would not directly govern the localization seen in the unconditioned dynamics.
  2. [Comparison between analytic Lyapunov exponent and numerical localization lengths] The quantitative agreement is stated to hold with corrections of order J²/[λ²+(γ/2)²]. The manuscript should derive this scaling explicitly from the Zeno-regime expansion rather than fitting it post hoc, and should demonstrate that the same scaling appears in the numerical extraction of the localization length from the reconstructed single-particle wave functions.
minor comments (2)
  1. Notation for the measurement-induced potential and the effective non-Hermitian Hamiltonian should be introduced with a single consistent symbol set and clearly distinguished from the original stochastic Schrödinger equation.
  2. The reconstruction procedure for the intrinsic localized single-particle wave functions from the many-body quantum-state-diffusion trajectories should be spelled out in an appendix or dedicated subsection, including any averaging or filtering steps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and derivations.

read point-by-point responses

  1. Referee: The central claim that an effective potential can be constructed self-consistently from each individual quantum trajectory (without postselection or ensemble averaging) is load-bearing for the transfer-matrix reduction. The stochastic Schrödinger equation contains a state-dependent, stochastic back-action term; the manuscript must show explicitly how the self-consistency loop closes for a single realization (e.g., by writing the algebraic or differential condition that determines the potential at each time step from the instantaneous state alone). If this closure implicitly invokes trajectory statistics, the Lyapunov exponent obtained from the transfer matrix would not directly govern the localization seen in the unconditioned dynamics.

    Authors: We agree that an explicit demonstration of single-trajectory closure is necessary for clarity. In the construction, the effective potential is fixed instantaneously by the local density of the current state through the back-action term of the stochastic Schrödinger equation, yielding an algebraic relation V_eff(t) = (γ/2) n(x,t) (or the appropriate measurement operator applied to |ψ(t)⟩) that depends only on the instantaneous wave function. This closes the loop for each realization independently of ensemble statistics. We will add an explicit equation and a short paragraph detailing this update rule in the revised manuscript. revision: yes

  2. Referee: The quantitative agreement is stated to hold with corrections of order J²/[λ²+(γ/2)²]. The manuscript should derive this scaling explicitly from the Zeno-regime expansion rather than fitting it post hoc, and should demonstrate that the same scaling appears in the numerical extraction of the localization length from the reconstructed single-particle wave functions.

    Authors: The indicated scaling follows from a controlled perturbative expansion around the Zeno limit: the leading non-Hermitian term dominates localization while the hopping J enters at second order, producing corrections ∼ J² / [λ² + (γ/2)²] in the Lyapunov exponent. We will include this explicit derivation (via expansion of the effective non-Hermitian Hamiltonian and the resulting transfer-matrix eigenvalues) in the revised text. We will also add a supplementary analysis of the numerically extracted localization lengths versus J, γ, and λ that confirms the same functional dependence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper starts from the stochastic Schrödinger equation for continuous monitoring, introduces an instantaneous effective potential constructed per trajectory, reduces it in the Zeno limit (γ ≫ J) to a deterministic transfer-matrix problem for a non-Hermitian operator, computes the Lyapunov exponent directly from that operator, and then compares the resulting localization length to independent numerical extraction from long-time quantum-state-diffusion trajectories. No step reduces by construction to a fitted parameter, self-citation, or ensemble-averaged input; the numerical check uses the same stochastic dynamics but extracts the observable without feeding the analytical result back into the construction. The self-consistency loop is therefore an internal derivation step, not a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum mechanics of continuous monitoring and the assumption that the quasiperiodic potential renders the Zeno-limit problem analytically tractable; no free parameters or new entities are introduced.

axioms (2)
  • standard math Continuous monitoring is described by quantum state diffusion without postselection
    Invoked throughout the abstract as the underlying stochastic dynamics.
  • domain assumption Quasiperiodic potential allows self-consistent effective-potential construction in the Zeno regime
    Stated as the feature that renders the problem analytically tractable.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Measurement-induced phase transitions in disordered fermions

    cond-mat.stat-mech 2026-05 unverdicted novelty 5.0

    Disorder does not alter the presence or absence of measurement-induced phase transitions in noninteracting fermions; the long-time behavior is controlled by the same nonlinear sigma model with renormalized parameters.

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