Measurement-induced phase transitions in disordered fermions

arXiv:2605.05306 · cond-mat.stat-mech, cond-mat.dis-nn, quant-ph · Submitted 2026-05-06 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Measurement-induced phase transitions in disordered fermions".

Mira: Measurement-induced phase transitions (MIPTs) are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary dynamics and local measurements.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we’re looking at this paper, "Measurement-induced phase transitions in disordered fermions," and what it really claims is that these nonequilibrium transitions between different entanglement scaling behaviors are driven by the tug-of-war between unitary evolution and local measurements.

Mira: Exactly. The core thesis of the paper is that they studied a d-dimensional noninteracting fermionic system with both static disorder and continuous monitoring of local particle density, and they derived an effective field theory to describe how these things interact in the long term.

Lev: From my side, I’m interested in how this maps onto what we could actually test on hardware; if there's a transition predicted, what kind of observable signal would we expect to see?

Kai: Well, the paper lays out that they used a replica Keldysh framework to set up the dynamics under unitary evolution and weak measurement. They are essentially looking at how quenched disorder modifies the standard model for monitored fermions by just tweaking some parameters.

Mira: That's what they found; they claim that the system is governed by the same nonlinear sigma model as in the case of clean monitored fermions, but where disorder only shows up as a modification to those model parameters. This suggests that whether a measurement-induced phase transition happens or not doesn't change based on the presence of disorder in spatial dimensions greater than one.

Lev: If it’s just parameter modifications, that simplifies things for error correction research because we don't have to worry about new fundamental physics introduced by the disorder itself. Does this mean the error-correction threshold calculations remain robust under these conditions?

Kai: The paper points out that in one dimension, their result implies that disorder doesn't stabilize a critical phase or induce a measurement-induced phase transition, which contrasts with some earlier numerical studies. They support this by running simulations on a 1D lattice model using the projection protocol to show the absence of a MIPT for various disorder strengths and measurement rates <ref:2605.05306#pg0>.

Mira: I agree that the one-dimensional result is significant because it shows that disorder doesn't create a new transition point in that dimension, which is important for understanding critical phenomena in low dimensions. The authors also discuss how they derived their effective action, focusing on the terms for static disorder potential and measurement noise averaging.

Paper summary: Lev: Regarding those averaged potentials, I wonder how the scaling of those variances, like tau el and g squared proportional to measurement strength kappa, feeds into the final field theory structure <ref:2605.05306#pg0>? Would that mean we have a clean way to tune the disorder's influence?

Kai: They do introduce these variances explicitly in their action terms, where tau el is tied to elastic scattering time from static impurities, and g squared is proportional to the measurement strength kappa <ref:2605.05306#pg0>. This structure helps them build the theory that describes long-time universal behaviors for these systems.

Mira: The saddle point analysis leads them to an effective action involving matrix fields and, which then yields self-consistent equations for their saddle points, like sp and sp. This process ultimately identifies the measurement-induced fermionic self energy as a term they call m, governed by a specific self-consistent Born equation.

Lev: If we look at that Born equation, how sensitive is the resulting structure to changes in the disorder variance versus changes in the measurement strength? That’s where I see potential for running this on actual quantum hardware.

Kai: The resulting effective action incorporates terms for both disorder averaging and measurement noise averaging, which are what they found through Hubbard-Stratonovich transformations applied separately to those effects. They then analyze Gaussian fluctuations around those saddle points to see the dominant contributions.

Mira: The analysis of fluctuations shows that the dominant contributions come from modes delta and delta, which are governed by a total decay rate = gamma m + gamma D. This total decay rate combines the measurement-induced decay rate with the disorder-induced one, which is a key concept here.

Lev: A combined decay rate seems more realistic for modeling real experimental setups where both noise sources are active simultaneously; how does that total rate translate into the final nonlinear sigma model parameters?

Kai: The Goldstone manifold of the theory involves simultaneous unitary rotations of those matrix fields around their saddle points, and the effective action for these modes ends up being identical to the measurement-only case but with gamma m replaced by this total decay rate. This leads to an NL sigma M where the stiffness coupling constant g is determined by the inverse of that total decay rate instead of just the measurement-induced one.

Mira: The final result suggests that in spatial dimensions d > one a transition occurs at G c = one/four pi(d - one), and they conclude that disorder enlarges the area-law regime, pushing the entanglement transition to occur at weaker measurement rates when disorder is present.

Paper summary: Lev: That implies that if we are designing an experiment aiming for a specific MIPT signature, we need to account for how static impurities modify the required measurement strength setting. That’s a practical consideration for experimental design.

Kai: And in one dimension, the system only exhibits an area law phase and no transition, which is what supports their conclusion about disorder not stabilizing critical phases there. Their numerical simulations using the quantum projection protocol on a 1D disordered fermion chain confirmed these analytical predictions <ref:2605.05306#pg0>.

Mira: Furthermore, their numerical work showed that for large measurement rates gamma, the particle-number covariance G AB decreases with system size L, confirming the area-law entanglement behavior, while for weak measurement rates, the finite-size crossing point gamma L c drifts towards zero as L increases in disordered fermions.

Lev: The numerical verification is strong then; showing that the absence of a genuine MIPT holds even when you introduce disorder and vary the measurement rate significantly. Where does this leave us regarding real quantum systems?

Kai: The authors state plainly that their work suggests static disorder doesn't affect the presence or absence of a MIPT in arbitrary spatial dimensions, which is what they found in their study on "Measurement-induced phase transitions in disordered fermions."

Mira: So, to summarize the big picture for us, the main implication is that we can describe these complex nonequilibrium transitions using a single nonlinear sigma model whose parameters are just modified by disorder, meaning disorder doesn't fundamentally alter whether a transition occurs in dimensions greater than one.

Lev: From an error correction viewpoint, this means we can treat the noise sources separately in our theoretical models for the transition dynamics rather than having to solve for them simultaneously from scratch every time.

Kai: That’s right; the paper provides a framework that connects static disorder and measurement noise into a single, manageable effective theory, which is really useful for predicting how these systems behave under continuous observation.

Mira: And it highlights that the competition between unitary dynamics and local measurements remains the central driver of these transitions, regardless of whether there's some quenched randomness in the system or not.

Lev: We should watch how this effective field theory is applied to more complex, interacting models where static disorder might couple differently to particle interactions.

Kai: Exactly; while this work focuses on noninteracting fermions, the structure they derived for the NL sigma M with modified parameters gives us a solid starting point for exploring those more complicated scenarios.

Conclusion: Kai: So, we’ve just finished going through the technical details of this paper on measurement-induced phase transitions in disordered fermions, and now it's time to wrap up with a look at what this all means for us.

Mira: I think it's important that we circle back to the title and the authors because they really set the stage for understanding how these non-equilibrium dynamics play out. The core idea revolves around how continuous local monitoring, when combined with static disorder, drives transitions in entanglement scaling.

Lev: From a hardware standpoint, I’m still trying to visualize exactly what physical system this describes—are we talking about superconducting circuits or something more fundamental? I need to know if the math maps onto anything we can cool and measure right now.

Kai: Well, the authors are using a replica Keldysh framework, which is a complex way of modeling time evolution under both unitary dynamics and measurement noise. They are essentially mapping out how adding static disorder changes the rules for these transitions compared to just monitoring without impurities.

Mira: Precisely; the authors argue that this system behaves like a known nonlinear sigma model, but with parameters that reflect both the measurement strength and the disorder variance. The implication is that while disorder modifies the specific conditions for a phase transition, it doesn't fundamentally change whether one exists in dimensions greater than one.

Lev: That distinction about dimensions is crucial for me; if disorder just shifts a critical point rather than eliminating a phase entirely, that gives us much more room to design error-correction protocols. I’m curious how sensitive the resulting NL sigma M stiffness coupling constant is to those disorder parameters they introduced.

Kai: Exactly what Lev is asking; the paper shows that in one dimension, things are much simpler and there's no transition at all, which is a neat contrast to higher dimensions. The authors confirm this with numerical simulations on a 1D disordered chain <ref:2605.05306#pg0>.

Mira: And for the real impact, I think it suggests a unified way to approach these transitions by integrating disorder into the field theory itself, rather than treating it as an external perturbation. This offers a more robust theoretical foundation for condensed matter physics in non-equilibrium settings.

Lev: So, if this theory holds up under scrutiny with simulations and the analytical framework, what’s the next step for running this on actual quantum hardware? Is there a specific noise profile we need to engineer to probe these effects?

Kai: The paper lays out a clear path forward by suggesting that incorporating disorder into the decay rate is key. This gives us a concrete target for experimentalists trying to tune the measurement strength against impurity concentrations.

Mira: Indeed, the main point is that these measurements show how competition between unitary evolution and local observables dictates phase transitions, and this framework provides a way to quantify that competition across different noise sources.

Lev: It’s compelling because it moves us from just observing transitions in clean systems to understanding them in the messy reality of noisy, disordered devices. That connection is what I want to see realized on the bench.

Yunxiang Liao, * Max Matheussen, Xinghai Zhang

Department of Physics, KTH Royal Institute of Technology · Department of Physics, Technical University of Denmark

cond-mat.stat-mech, cond-mat.dis-nn, quant-ph

Submitted: 2026-05-06

Updated: 2026-10-05

Comments: v2: Added two-dimensional numerical results. 22 pages, 5 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 90/100

The gist: Measurement-induced phase transitions (MIPTs) are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary

Key concepts

Measurement-induced phase transitions (MIPT): These are nonequilibrium transitions between phases in a system driven by continuous local measurements competing with unitary evolution. They are characterized by distinct scaling behaviors of entanglement, revealing how monitoring affects the system's long-time behavior.
Replica Keldysh Framework: This mathematical framework is used to describe the dynamics of fermions under both static disorder and continuous monitoring. It allows for the calculation of long-time universal properties by integrating unitary evolution, static disorder potential, and measurement noise into a single path integral.
Nonlinear Sigma Model (NL$oldsymbol{ ext{SM}}$): The effective low-energy description of the system is a nonlinear sigma model. Its stiffness coupling constant is determined by the total decay rate ($ ilde{\gamma}$) rather than just the measurement rate ($\gamma_m$). This model predicts that disorder alters how entanglement transitions occur based on measurement strength.
Entanglement Scaling: This refers to how the entanglement entropy of a quantum system scales with its size. The analysis shows that in dimensions $d>1$, disorder enlarges the area-law regime, meaning entanglement persists over larger distances, which is linked to the MIPT prediction.

Terminology

Entanglement — 59× in this paper · explained in 3 episodes
Phase Transitions — 20× in this paper · explained in 2 episodes
Entanglement Entropy — 16× in this paper · explained in 2 episodes
Quantum Circuits — 15× in this paper · explained in 2 episodes
Area Law — 11× in this paper · explained in 1 episode
Decoupling — 9× in this paper · explained in 2 episodes
Quantum Trajectories — 8× in this paper · explained in 2 episodes
Unitary Dynamics — 6× in this paper · explained in 2 episodes
Core — 3× in this paper · explained in 2 episodes
Trace — 3× in this paper · explained in 2 episodes
Verification — 2× in this paper · explained in 2 episodes
Quantum Zeno Effect — 1× in this paper · explained in 2 episodes
Renormalization Group (RG) Analysis — 1× in this paper · explained in 2 episodes
Singular Value Decomposition (SVD) — 1× in this paper · explained in 2 episodes
Topological Order — 1× in this paper · explained in 2 episodes

Summary

Measurement-induced phase transitions (MIPTs) are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary dynamics and local measurements. This work investigates how quenched disorder affects these transitions in a d-dimensional noninteracting fermionic system subject to continuous monitoring of local particle density, deriving an effective field theory that describes long-time universal behaviors.

The gist: 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.

Theoretical Framework and Model Formulation

The study employs a replica Keldysh framework to describe the dynamics of a d-dimensional noninteracting fermionic system subject to both static disorder and continuous monitoring. The dynamics evolve according to unitary evolution generated by the Hamiltonian (kinetic energy plus static disorder) and weak measurement described by Kraus operators. The long-time behavior is captured by a replica Keldysh path integral, where the action includes terms for unitary evolution, static disorder potential, and measurement noise.

The key components of the continuum limit involve:

The bare Green’s function describing unitary evolution (Eq. 7).
The action term accounting for static disorder potential, with variance proportional to elastic scattering time τel (Eq. 8).
The action term encoding measurement noise, with variance proportional to the measurement strength κ (Eq. 8), where g2 ∝ κ characterizes the measurement strength.

Effective Field Theory Derivation

Through Hubbard-Stratonovich transformations applied separately to disorder and measurement averaging, an effective action is derived (Eq. 14). This action is then integrated over fermions, leading to a theory in matrix fields Qˆ and ˆq. The saddle point analysis yields self-consistent equations for the saddle points Qˆ sp and ˆq sp (Eqs. 19a, 19b).

The saddle point structure reveals that the measurement-induced fermionic self energy can be identified as Σm, which is governed by a self-consistent Born equation (Eq. 22). The resulting effective action in Eq. 18 incorporates terms for disorder averaging and measurement noise averaging.

Gaussian Fluctuations and Entanglement Entropy

The Gaussian approximation around the saddle points reveals that the dominant contributions arise from fluctuations δqˆ = (ˆq - qˆsp)/(iγm) and δQˆ = Qˆ - Qˆsp, governed by an action involving total decay rate ˜γ = γm + γD.

The analysis focuses on two sectors:

Replica-symmetric sector: The mode δQ̂+(s) is massless, with a propagator identical to that of the measurement-only theory except that the measurement-induced decay rate γm is replaced by the total decay rate ˜γ (Eq. 39).
Replica-asymmetric sector: The entanglement entropy SA is determined by the connected two-point density correlation function C(r, r′;t, t′), which is found to be determined by the massless mode Yˆ (+) propagator (Eq. 48).

Nonlinear Sigma Model and Phase Transition Prediction

The Goldstone manifold of the theory is generated by simultaneous unitary rotations of the matrix fields Qˆ and qˆ¯ around their respective saddle points. The effective action for these Goldstone modes (Eq. 61) is identical to that of the measurement-only case, but with γm replaced by ˜γ.

The final result leads to a nonlinear sigma model (NLσM) where the stiffness coupling constant g is determined by the inverse of the total decay rate ˜γ rather than the measurement-induced decay rate γm. The RG equation for this NLσM suggests that in spatial dimensions d > 1, a transition occurs at Gc = 1/4π(d − 1), implying that disorder enlarges the area-law regime, causing entanglement transition to occur at weaker measurement rates in the presence of disorder. In one dimension (d=1), the system exhibits only an area law phase and no transition.

Numerical Verification

Numerical simulations using the quantum projection protocol on a 1D disordered fermion chain confirm these analytical predictions. The steady-state particle-number covariance GAB is used to characterize MIPT, and the results show that for large measurement rates γ, GAB decreases with system size L, confirming area-law entanglement. For weak measurement rates, the finite-size crossing point γ L c drifts towards zero as L increases in disordered fermions, clearly indicating the absence of a genuine MIPT in either clean or dirty 1D fermions.

Conclusion and Outlook

The monitored disordered fermionic system is described by the same NLσM with modified parameters as the measurement-only theory, implying that static disorder does not affect the presence or absence of a MIPT.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements that can be made to AI systems, along with what these improved systems could achieve:

)AI System Improvement Suggestions:

The core theoretical framework involves a Two-Matrix Field Theory within the Replica Keldysh formalism to describe non-equilibrium dynamics driven by competition between unitary evolution and local measurements (Measurement-induced Phase Transitions, MIPT).
The effective field theory derived is a Nonlinear Sigma Model (NLσM) that governs the long-time universal behaviors of entanglement entropy and charge fluctuations.

)What the Improved AI System Can Do:

The improved system can perform high-fidelity simulations and predictions for complex quantum systems exhibiting non-equilibrium dynamics, specifically those involving both static disorder (quenched impurities) and continuous monitoring (measurement).
It can accurately predict the presence or absence of Measurement-induced Phase Transitions (MIPT) in fermionic systems across different spatial dimensions.
It can model the entanglement scaling behavior of these complex systems, distinguishing between Area Law phases and Logarithmic phases, based on system size and measurement strength.

)Specific Capabilities for the Improved AI System:

Predict MIPT occurrence: The system can determine if a phase transition exists between an area-law phase and a logarithmic phase based on the dimension of the system (e.g., predicting it only occurs in spatial dimensions d > 1, not d = 1).
Entanglement Scaling Analysis: It can calculate the entanglement entropy scaling (e.g., Area Law saturation) as a function of disorder strength and measurement rate, providing quantitative predictions for how disorder modifies the critical behavior compared to clean systems.
Parameter Renormalization Tracking: The system can track how physical parameters (like the stiffness coupling constant, which is inversely proportional to the total decay rate) are renormalized by local-in-time massive modes arising from both measurement and disorder, allowing it to predict the resulting universal scaling behavior in disordered environments.
Phase Transition Critical Points: It can calculate critical values for measurement strength and disorder strength at which MIPT occurs (e.g., calculating the critical measurement rate based on dimension and disorder strength).