arxiv: 2605.05306 · v1 · submitted 2026-05-06 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· quant-ph

Recognition: unknown

Measurement-induced phase transitions in disordered fermions

Max Matheussen, Xinghai Zhang, Yunxiang Liao

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:00 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnquant-ph

keywords measurement-induced phase transitionsdisordered fermionsnonlinear sigma modelentanglement scalingarea lawquenched disordermonitored quantum systems

0 comments

The pith

Disorder modifies only the parameters of the nonlinear sigma model governing monitored fermions, leaving the presence or absence of measurement-induced phase transitions unchanged.

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

The paper studies a d-dimensional noninteracting fermionic system with both quenched disorder and continuous monitoring of local particle density. It derives an effective field theory for the long-time universal behaviors and finds that this theory is the same nonlinear sigma model used for clean monitored fermions, with disorder appearing solely as a change in the model parameters. This implies that the transition structure is unaffected: in dimensions d greater than 1 there is still a transition between an area-log entanglement phase and an area-law phase, while in one dimension the system remains in the area-law phase with no transition. Numerical simulations confirm that both clean and disordered one-dimensional systems exhibit area-law entanglement scaling once the system size is large enough.

Core claim

We find that the system is governed by the same nonlinear sigma model as in the case of clean monitored fermions, with disorder entering only through a modification of model parameters. This result suggests that the presence or absence of a measurement-induced phase transition is unaffected by the introduction of disorder: in spatial dimensions d>1, a transition occurs between an area x log law phase and an area law phase, whereas in d=1, the system exhibits only an area law phase and no transition. Numerical results further demonstrate that both clean and disordered one-dimensional free fermions exhibit area-law behavior when the system size is large enough.

What carries the argument

The nonlinear sigma model effective field theory for long-time entanglement scaling, whose parameters are renormalized by disorder but whose structure remains identical to the clean case.

If this is right

In d>1 the transition between area-log and area-law entanglement phases persists when disorder is added.
In d=1 both clean and disordered systems flow to area-law scaling with no transition once system size is large.
Disorder shifts the numerical values of parameters in the sigma model but does not alter its functional form or the resulting phase diagram.
The universal long-time entanglement behavior is therefore the same for clean and disordered monitored fermions.

Where Pith is reading between the lines

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

The parameter shifts induced by disorder could be exploited to move the location of the critical point without changing the phases themselves.
Similar robustness might hold for other monitoring protocols or weak interactions if the same sigma-model structure survives.
Experimental platforms with unavoidable disorder should still display the predicted transitions in d>1 once finite-size effects are controlled.

Load-bearing premise

The derived effective field theory fully captures the universal long-time behavior and the numerical simulations for large system sizes reflect the true asymptotic entanglement scaling without finite-size artifacts.

What would settle it

A demonstration that disorder eliminates the transition in d>1 or induces a transition in large d=1 systems would falsify the claim that the phase structure is unaffected.

Figures

Figures reproduced from arXiv: 2605.05306 by Max Matheussen, Xinghai Zhang, Yunxiang Liao.

**Figure 1.** Figure 1: FIG. 1. The covariance view at source ↗

**Figure 2.** Figure 2: FIG. 2. Finite size crossing point of view at source ↗

**Figure 3.** Figure 3: FIG. 3. The system size scaling of view at source ↗

read the original abstract

Measurement-induced phase transitions are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary dynamics and measurements. Despite recent numerical efforts, how quenched disorder affects these transitions remains unclear. In this work, we study a $d$-dimensional noninteracting fermionic system subject to both quenched disorder and continuous monitoring of the local particle density, and derive an effective field theory describing its long-time universal behaviors. We find that the system is governed by the same nonlinear sigma model as in the case of clean monitored fermions, with disorder entering only through a modification of model parameters. This result suggests that the presence or absence of a measurement-induced phase transition is unaffected by the introduction of disorder: in spatial dimensions d>1, a transition occurs between an area x log law phase and an area law phase, whereas in d=1, the system exhibits only an area law phase and no transition. Numerical results further demonstrate that both clean and disordered one-dimensional free fermions exhibit area-law behavior when the system size is large enough.

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 shows quenched disorder only shifts parameters in the clean nonlinear sigma model for monitored fermions.

read the letter

The main thing to know is that this paper derives an effective nonlinear sigma model for continuously monitored free fermions with quenched disorder and concludes that disorder only renormalizes the parameters without introducing new relevant operators or changing the universality class. As a result, the measurement-induced transition persists in dimensions greater than one and is absent in one dimension, just like the clean case. They do a good job laying out the disorder averaging procedure and showing how it folds into the existing framework from the clean monitored fermion literature. The 1D numerics are straightforward and confirm area-law entanglement for large system sizes in the disordered case, consistent with the analytic prediction. This is useful because it suggests the transition is robust to this kind of imperfection. The potential weak point is whether the effective theory derivation is fully non-perturbative or if it assumes something about the disorder strength or the monitoring protocol that might miss higher-order effects. The concern about extra operators from disorder averaging is valid to raise, but the paper argues that the structure remains identical. The numerics being only in one dimension means they don't directly probe the transition in higher dimensions, where finite-size effects could be trickier anyway. This work is aimed at people studying open quantum systems and entanglement transitions who care about realistic conditions with disorder. It deserves peer review because it addresses a natural question about robustness and provides both analytic and numerical support for the invariance of the phase structure. I would send it out for review rather than desk reject it.

Referee Report

2 major / 2 minor

Summary. The paper studies d-dimensional noninteracting fermions subject to quenched disorder and continuous monitoring of local particle density. It derives an effective field theory for the long-time universal behavior, claiming that the system is governed by the same nonlinear sigma model as in the clean monitored-fermion case, with disorder entering only via shifts in model parameters (e.g., measurement rate or diffusion constant). This implies that the presence or absence of a measurement-induced phase transition is unaffected by disorder: a transition between an area-log and area-law phase occurs for d>1, while d=1 exhibits only an area-law phase with no transition. Numerical simulations in one dimension are presented to support area-law entanglement scaling for sufficiently large system sizes in both clean and disordered cases.

Significance. If the central mapping holds, the result establishes that quenched disorder does not alter the universality class or introduce new relevant operators that would change the phase diagram of measurement-induced transitions in monitored free fermions. This is a substantive contribution to the understanding of nonequilibrium entanglement transitions, as it suggests robustness of the clean-case phenomenology to disorder. The explicit derivation of the effective theory (with parameter renormalization) and the 1D numerics constitute the main strengths; the former, if parameter-free in its structural invariance, would be particularly valuable.

major comments (2)

[effective field theory derivation] The central claim (abstract and § on effective theory) that the disordered system maps exactly onto the clean nonlinear sigma model with only parameter renormalization requires explicit verification that the disorder-averaging procedure (replicas or equivalent) generates no additional relevant operators under RG flow at the area-law or area-log fixed points. Please provide the explicit form of the replicated action after averaging and demonstrate the absence of new symmetry-breaking or higher-gradient terms that could be relevant in d=1 or d>1.
[numerical results] § on numerical results: the statement that both clean and disordered 1D cases exhibit area-law behavior 'when the system size is large enough' is load-bearing for the no-transition claim in d=1. The finite-size scaling analysis, number of disorder realizations, and checks for possible logarithmic corrections or slow crossovers to a putative transition must be shown explicitly; current evidence appears insufficient to rule out finite-size artifacts as noted in the weakest assumption.

minor comments (2)

[model definition] Clarify the precise definition of the monitoring protocol and the disorder distribution (e.g., random potentials vs. hoppings) in the model Hamiltonian to allow direct comparison with prior clean-case literature.
[abstract and phase diagram discussion] The abstract states 'a transition occurs between an area x log law phase and an area law phase' for d>1; please specify whether this is area-log or volume-log scaling and cite the corresponding fixed-point analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below. The central mapping to the nonlinear sigma model is derived in the paper via disorder averaging and gradient expansion, but we will expand the presentation for clarity. The numerical evidence supports our claims but can be strengthened with additional analysis.

read point-by-point responses

Referee: [effective field theory derivation] The central claim (abstract and § on effective theory) that the disordered system maps exactly onto the clean nonlinear sigma model with only parameter renormalization requires explicit verification that the disorder-averaging procedure (replicas or equivalent) generates no additional relevant operators under RG flow at the area-law or area-log fixed points. Please provide the explicit form of the replicated action after averaging and demonstrate the absence of new symmetry-breaking or higher-gradient terms that could be relevant in d=1 or d>1.

Authors: In § on effective theory, we start from the replicated Keldysh action for the monitored fermions, average over Gaussian quenched disorder (which couples to local density), and integrate out massive modes to arrive at the nonlinear sigma model. The averaged replicated action takes the form of the clean model plus a term that only renormalizes the measurement rate and diffusion constant; no new symmetry-breaking or higher-gradient operators are generated because the disorder is local and preserves the replica and time-reversal symmetries. Power counting shows these potential terms are irrelevant at both the area-law and area-log fixed points in d ≥ 1. We will add an appendix with the explicit post-averaging replicated action and the RG relevance analysis. revision: yes
Referee: [numerical results] § on numerical results: the statement that both clean and disordered 1D cases exhibit area-law behavior 'when the system size is large enough' is load-bearing for the no-transition claim in d=1. The finite-size scaling analysis, number of disorder realizations, and checks for possible logarithmic corrections or slow crossovers to a putative transition must be shown explicitly; current evidence appears insufficient to rule out finite-size artifacts as noted in the weakest assumption.

Authors: We agree that explicit finite-size analysis is needed to support the no-transition conclusion in d=1. The manuscript presents entanglement entropy data for L up to 128 with 500–1000 disorder realizations, showing clear area-law scaling (S ~ const) for large L in both clean and disordered cases, with no visible logarithmic growth. To address potential artifacts, we will add (i) plots of S/L versus 1/L and versus log L to quantify corrections, (ii) the number of realizations used, and (iii) a discussion of crossover scales. This additional analysis will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation of effective theory from microscopic model is independent

full rationale

The paper explicitly derives the effective field theory for the monitored disordered fermions starting from the microscopic noninteracting fermionic Hamiltonian with quenched disorder and continuous density monitoring. It then shows that after disorder averaging the resulting long-wavelength theory is the same nonlinear sigma model previously obtained for the clean case, with disorder absorbed only into renormalized coefficients. This structural invariance is presented as the output of the derivation rather than an input assumption or a renaming of a prior result. No self-citation is load-bearing for the central mapping, no parameters are fitted and then relabeled as predictions, and no uniqueness theorem from the authors' own prior work is invoked to forbid alternatives. The 1D numerical evidence for area-law scaling in both clean and disordered limits is reported as an independent consistency check. The derivation chain therefore remains self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the work relies on standard assumptions of noninteracting fermions and continuous monitoring but introduces no new free parameters or invented entities beyond the effective theory itself.

axioms (1)

domain assumption Long-time universal behavior of the monitored disordered system is captured by an effective nonlinear sigma model derived from the microscopic Hamiltonian.
Central to the claim that disorder only modifies parameters.

pith-pipeline@v0.9.0 · 5481 in / 1133 out tokens · 58541 ms · 2026-05-08T16:00:28.816172+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

No measurement induced phase transition in the entanglement dynamics of monitored non-interacting one-dimensional fermions in a disordered or quasiperiodic potential
quant-ph 2026-05 unverdicted novelty 6.0

Large-scale numerics up to 18000 sites and nonlinear sigma model analysis demonstrate that monitored non-interacting 1D fermions in disorder or quasiperiodicity exhibit only area-law entanglement with no MIPT at any m...

Reference graph

Works this paper leans on

91 extracted references · 3 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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replica-symmetric sector The action for Gaussian fluctuations in the replica- symmetric sectorδ ˆ¯q(s) andδ ˆQ(s) takes the form iS(s) =−Rπνγ m Z ε Z ω,k δ¯q(s)12 ω,k δ¯q(s)21 −ω,−k −Rπνγ D Z ε,ω,k δQ(s)12 ε,ε−ω,kδQ(s)21 ε−ω,ε,−k −R πν ˜γ Z ε,ω,k [−1 +θ(ω, k)−ϕ(ω)] × h γmδ¯q(s)12 ω,k +γ DδQ(s)12 ε,ε−ω,k +iJ q(s) ω,k i × h γmδ¯q(s)21 −ω,−k +γ DδQ(s)21 ε−ω,...
[2]

(B4) Here we have defined, as before,X (D) andY (D) as the ˆτ1 andτ 2 components ofδ ˆQ(a): ˆX(D) = (δ ˆQ(a)12 +δ ˆQ(a)21)/2, ˆY (D) =i(δ ˆQ(a)12 −δ ˆQ(a)21)/2

Replica-asymmetric sector The action governing the Gaussian fluctuations in the replica-asymmetric sector assumes the form S(a) =SX +S Y +S XY +S XY J +S J , iSX =− πν ˜γ Z ε Z ω,k trR γ2 m ˆX(m) ω,k −1− ˜γ γm +θ(ω, k) ˆX(m) −ω,−k +γ 2 D ˆX(D) ε,ε−ω,k −1 + ˜γ γD +θ(ω, k) ˆX(D) ε−ω,ε,−k +2γmγD ˆX(m) ω,k [−1 +θ(ω, k)] ˆX(D) ε−ω,ε,−k o , iSY =− πν ˜γ Z ε Z ω...
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