Universality of Measurement-Induced Criticality under Symmetry-Breaking Measurements

Abstract: We study the critical properties of random quantum circuits with a $U(1)$ symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored $U(1)$-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored $U(1)$-symmetric Haar and stabilizer random circuits.

• The paper demonstrates analytically and numerically that any nonzero symmetry-breaking measurement component is a relevant perturbation, driving U(1)-symmetric circuits to the generic measurement-induced criticality universality class.
• Large-d analysis maps the transition to percolation with p_c = 1/2 and ν = 4/3, while finite-dimensional Haar and Clifford simulations reproduce the non-symmetric critical scaling.
• Symmetry-breaking measurements keep charge correlations short-ranged and eliminate the separate charge-sharpening transition, although weak symmetry breaking can create long crossover lengths in finite systems.

Overview and main result

This paper addresses whether local projective measurements that explicitly break a conserved $U(1)$ symmetry of the unitary dynamics constitute a relevant perturbation at the measurement-induced phase transition (MIPT). The authors consider brickwork random circuits whose two-site gates are drawn from the Haar ensemble within fixed magnetization sectors, so that the total qubit charge $\mathcal{Q} = \sum_j Z_j$ is strictly conserved by the unitaries, while each site is measured with probability $p$ in the $X$ eigenbasis — a basis that does not commute with the conserved charge. Their central finding is that such symmetry-breaking measurements are relevant: at large scales the entanglement transition flows to the same universality class as the MIPT of generic monitored Haar-random circuits with no conservation law (2607.08589).

This contrasts sharply with the case of symmetry-preserving ($Z$-basis) measurements in the same $U(1)$-symmetric circuits, where the MIPT belongs to a distinct universality class and is accompanied by an additional charge-sharpening transition within the volume-law phase. The result is established analytically in the limit of large local Hilbert-space dimension $d \to \infty$, via an exact mapping to a classical statistical mechanics model, and numerically at finite $d$ using both Haar-random and Clifford (stabilizer) circuits.

Statistical-mechanics model in the large-$d$ limit

Following the replica approach with Choi–Jamiołkowski vectorization and Weingarten calculus, the trajectory-averaged Rényi entropy is expressed through partition functions of a two-dimensional tilted square lattice. Vertices carry permutation degrees of freedom $\sigma \in S_Q$ associated with the replicated qudit sector; unmeasured sites correspond to bonds that force adjacent permutations to coincide and constrain the binary charge variables $\mathcal{Q} = \sum_j Z_j$0 on each bond. Measurements correspond to broken bonds. In the replica limit $\mathcal{Q} = \sum_j Z_j$1, the qudit contribution to the entropy reduces to the minimal-cut length $\mathcal{Q} = \sum_j Z_j$2 times $\mathcal{Q} = \sum_j Z_j$3, and its disorder average exhibits a transition in the classical 2D percolation universality class with $\mathcal{Q} = \sum_j Z_j$4 and $\mathcal{Q} = \sum_j Z_j$5.

The key simplification specific to $\mathcal{Q} = \sum_j Z_j$6 measurements is that their Boltzmann weight is independent of the measurement outcome: the measured-bond contraction equals $\mathcal{Q} = \sum_j Z_j$7 regardless of the outcome, leaving incoming and outgoing charges completely unconstrained. Consequently the Born probability over trajectories is uniform, $\mathcal{Q} = \sum_j Z_j$8, and the qubit contribution to the entropy depends only on the joint distribution of charges along the minimal cut.

Disordered exclusion process and absence of charge sharpening

In the one-replica limit, the charge dynamics maps exactly onto a symmetric simple exclusion process (SSEP) with diffusive transport, in which $\mathcal{Q} = \sum_j Z_j$9 measurements act as disordered defects implementing the outcome-independent operator $p$0. Using the self-duality of the SSEP, the connected charge correlator for a given measurement configuration equals one quarter of the survival probability of a single Brownian particle propagating backward in time; each measurement annihilates the spin operator at the measured site. This yields the rigorous bound

$p$1

with $p$2 finite for any $p$3. Via Markov's inequality and the Borel–Cantelli lemma, the authors further prove that almost surely over measurement realizations the correlator decays exponentially in time with inverse correlation length bounded below by $p$4, so the typical correlation length is finite as well.

The implication is direct: since both average and typical charge correlation lengths remain finite at all measurement rates, no charge-sharpening transition can occur — in contrast to symmetry-preserving measurements, where correlations decay algebraically below a critical rate $p$5. The only diverging length scale is therefore the percolation one, the charge distribution along the minimal cut coarse-grains into effectively independent variables, and both qubit and qudit sectors undergo an entanglement transition governed solely by $p$6, i.e., percolation universality.

The authors extend this conclusion to generic rotated measurement bases $p$7 with $p$8: the spin operator survives measurements with a $p$9 prefactor, giving $X$0, still finite for any $X$1. Hence any infinitesimal symmetry-breaking component in the measurement basis drives the critical behavior to the non-symmetric universality class, although for small $X$2 the parametrically large correlation length can produce crossover effects in finite systems. They also prove that measuring each site in an independently Haar-random basis yields exactly the same Boltzmann weight as $X$3 measurements in the large-$X$4 limit, connecting the result to noise-induced symmetry breaking studied elsewhere.

Numerical evidence at finite Hilbert-space dimension

For $X$5 (a qubit chain), where no analytical treatment exists, exact trajectory simulations up to $X$6 sites were analyzed via finite-size scaling of the tripartite mutual information $X$7 with dynamical exponent $X$8. The estimates are $X$9 and $Z$0. While $Z$1 carries errors too large to discriminate between candidate universality classes, the universal amplitude $Z$2 matches the value reported for generic Haar-random circuits. More stringently, the Rényi-index dependence of the critical logarithmic coefficient, fitted to $Z$3, gives $Z$4 and $Z$5, in striking agreement with the non-symmetric Haar values ($Z$6, $Z$7) and clearly distinct from the symmetric-measurement values ($Z$8, $Z$9). This constitutes the main finite-$U(1)$0 evidence that symmetry-breaking measurements act as a relevant perturbation also away from the large-$U(1)$1 limit.

Stabilizer circuit results

The hypothesis is tested independently in a $U(1)$2-symmetric Clifford circuit, whose gates form the 64-element subgroup parametrized as $U(1)$3, with $U(1)$4-basis measurements preserving the stabilizer structure. Two results stand out. First, whereas $U(1)$5-basis measurements in this model produce an area-law steady state for any $U(1)$6 and hence no MIPT at all, symmetry-breaking measurements stabilize a volume-law phase at small finite $U(1)$7 and generate an MIPT. Second, finite-size scaling with system sizes up to $U(1)$8 (using Stim) yields $U(1)$9 and $d \to \infty$0, with $d \to \infty$1 compatible with the non-symmetric Clifford value $d \to \infty$2, and the scaling functions of the symmetric and non-symmetric circuits collapsing onto each other after a non-universal rescaling of the scaling variable by a factor $d \to \infty$3. The same critical theory is thus reproduced despite the drastically smaller gate set, providing strong support for the relevance claim in an independent model.

Limitations and open questions

Several caveats qualify these conclusions. The analytical percolation result holds strictly only in the $d \to \infty$4 limit; at finite $d \to \infty$5 the claim rests on numerical evidence from limited sizes ($d \to \infty$6 for the exact Haar simulations), and the authors explicitly state they cannot fully exclude systematic finite-size effects in the extracted critical parameters. For tilted measurement bases near the symmetry-preserving limit, the growing correlation length $d \to \infty$7 enhances finite-size effects and delays the asymptotic scaling regime, so the rotated-basis check could not be performed reliably at accessible sizes. In the stabilizer case, the estimate of $d \to \infty$8 shows a mild dependence on the minimum system size included in the fit, which dominates its uncertainty. Open questions raised by the paper include the extension to deterministic or Hamiltonian (notably free-fermionic) dynamics, competing measurements interspersing $d \to \infty$9- and $d$0-basis projections with different rates, symmetries beyond $d$1 including discrete and non-Abelian groups, the role of entanglement asymmetry as a diagnostic of the broken symmetry, the universal properties of the volume-law phase away from criticality, and a renormalization-group description of measurements in symmetric unitary evolutions.

Conclusion

The paper establishes that symmetry-breaking local measurements are a relevant perturbation at the measurement-induced critical point of $d$2-symmetric random circuits, driving the entanglement transition into the universality class of non-symmetric monitored circuits. Analytically, this follows from the finiteness of the charge correlation length at all measurement rates in the large-$d$3 SSEP description, which eliminates the charge-sharpening transition and leaves percolation as the sole critical mechanism; numerically, it is corroborated by the universal amplitudes and Rényi scaling of both Haar and stabilizer circuits at finite dimension.