Statistical Mechanics of Non-Abelian Learnability Transitions

Abstract: Monitored many-body quantum systems can undergo sharp learnability transitions characterized by how much information can be learned by the observer. When the dynamics conserves a non-Abelian charge, such as an $SU(2)$ spin, understanding how the observer learns the total charge remains an outstanding problem. Unlike the Abelian case, where charge measurements on distinct sites commute, the $SU(2)$-symmetric readouts are noncommuting fusion measurements, making learning a genuinely quantum inference problem. In this work, we propose a theory of $1+1d$ monitored quantum dynamics with $SU(2)$ symmetry, and show that it can be described by an effective replicated loop model comprised of a replica-pairing field and a diffusive ($z=2$) background sector that carries the $SU(2)$ charge and remains gapless throughout the phase diagram. Our theory predicts that the "spin-sharpening'' and entanglement transitions coincide as a single transition. Ordering of the pairing field produces volume-law entanglement and hides the background sector from measurements, leading to a learning time of $t\sim L<sup>{3}$ for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time $t\sim L<sup>{2}$. Our analysis is controlled by a large-loop-fugacity expansion.

• The paper develops a large-loop-fugacity statistical-mechanics theory showing that entanglement and spin-sharpening transitions coincide through an order–disorder transition of a replica-pairing variable.
• The induced pairing interaction is ferromagnetic and effectively short-ranged despite its diffusive algebraic form, placing the transition in the generic measurement-induced transition universality class with a critical spin-wave sector.
• The paper predicts learning times of approximately L³/p in the spin-fuzzy phase and L² in the spin-sharp phase, while logarithmic entanglement persists in the sharp phase because diffusion remains gapless.

The learnability of conserved quantities in monitored quantum dynamics has been well understood for Abelian symmetries but not for non-Abelian ones. The paper under review develops a controlled statistical-mechanics theory for $SU(2)$-symmetric monitored circuits in $1+1$ dimensions and argues that entanglement and spin-sharpening transitions coincide as a single order–disorder transition of a replica-pairing variable, with a diffusive background sector that remains critical throughout the phase diagram (2608.19325).

Model and diagnostics

The system is a periodic chain of $L$ spin-$1/2$ degrees of freedom evolving under a brickwork circuit: on each brick, an $SU(2)$-symmetric two-qubit unitary $U(\theta)=P_s+e^{i\theta}P_t$, with $\theta$ drawn uniformly from $[0,2\pi)$, is applied with probability $1-p$, or the two-spin fusion channel is projectively measured with probability $p$. All gates commute with the total spin $1+1$0. Two diagnostics are studied: (i) trajectory-averaged Rényi entropies computed via the replica trick, and (ii) the classical fidelity $1+1$1 between measurement records conditioned on two total-spin sectors, together with its Rényi generalizations.

Loop model and replica pairings

Contracting the replicated tensor network on the circuit spacetime yields an interacting Temperley–Lieb (TL) loop model with loop fugacity $1+1$2. Averaging over the unitary phase enforces equal singlet counts in forward and backward replica branches; matching these counts introduces a pairing permutation $1+1$3 at each spacetime vertex. The partition function enjoys an $1+1$4 symmetry which breaks spontaneously to the diagonal in the volume-law phase, making $1+1$5—valued in the coset $1+1$6—the order parameter of the measurement-induced transition.

To obtain control, the authors analytically continue to large loop fugacity $1+1$7 at fixed integer $1+1$8, realized by $1+1$9 fundamental/anti-fundamental representations on alternating sublattices, with the physical circuit recovered at $L$0. Expanding the local vertex tensor in $L$1, the leading $L$2-dependence appears at $L$3 through $L$4 coupled to singlet projectors $L$5 on the two branches. A power-counting argument shows all higher $L$6-dependent vertices contribute only at $L$7, so this leading coupling suffices.

Diffusive background and induced interaction

At the measurement-only point $L$8, up to terms of order $L$9, the background sector factorizes into $1/2$0 branches each governed by the ferromagnetic TL chain Hamiltonian $1/2$1. For $1/2$2 this Hamiltonian admits exact local zero modes, which the authors argue are artifacts of the generalization and lift with a small regulator; physical quantities are computed at finite regulator, continued to $1/2$3, then $1/2$4. Coherent-state analysis shows all modes disperse quadratically, so the background is a $1/2$5 diffusive critical sector at all $1/2$6: couplings away from $1/2$7 are irrelevant by power counting. This gaplessness explains why no area-law phase appears anywhere in the phase diagram, consistent with numerics (2608.19325).

Integrating out the background generates, via the second cumulant, an effective ferromagnetic interaction between pairing variables,

$1/2$8

with $1/2$9 and $SU(2)$0. Explicitly, $SU(2)$1, a summable algebraic interaction of diffusive form. By Peierls and Dobrushin arguments for a discrete ferromagnet in two-dimensional spacetime, the pairing sector orders at small $SU(2)$2 and disorders at large $SU(2)$3. The spin-sharpening transition is identified with this order–disorder transition.

Entanglement scaling

Entanglement maps to the free-energy cost of twisted boundary conditions, receiving contributions from both sectors. In the ordered (spin-fuzzy) phase, the twist forces a domain wall in $SU(2)$4 with tension $SU(2)$5, producing volume-law entanglement. In the disordered (spin-sharp) phase, the twist costs nothing in the pairing sector, but the critical background contributes logarithmically. The coefficient depends on the initial state's spin sector:

• Maximally mixed initial state: late-time dynamics is governed by the ground-state manifold, the partial flag manifold $SU(2)$6, reducing to $SU(2)$7 at $SU(2)$8. Each real coordinate contributes half a logarithm (the standard Goldstone-mode counting), giving $SU(2)$9 with $U(\theta)=P_s+e^{i\theta}P_t$0, i.e., $U(\theta)=P_s+e^{i\theta}P_t$1 at the physical value.
• Fixed total spin $U(\theta)=P_s+e^{i\theta}P_t$2: the ferromagnetic ground state within the spin-$U(\theta)=P_s+e^{i\theta}P_t$3 sector is a Bloch-wall spiral winding at $U(\theta)=P_s+e^{i\theta}P_t$4, breaking all three $U(\theta)=P_s+e^{i\theta}P_t$5 generators. A region much smaller than $U(\theta)=P_s+e^{i\theta}P_t$6 sees an approximately uniform texture and detects only two broken generators, giving $U(\theta)=P_s+e^{i\theta}P_t$7; a system-sized region detects all three, giving $U(\theta)=P_s+e^{i\theta}P_t$8. At general $U(\theta)=P_s+e^{i\theta}P_t$9, fixing total spin adds exactly half a logarithm for regions of order $\theta$0.

These logarithmic coefficients are independent of the Rényi index, as follows from the linearity of the gluing exponent in $\theta$1.

Regarding universality: although the induced interaction is algebraic, it has finite second moments, with leading nonanalyticities $\theta$2 and $\theta$3 lying on the short-range side of the Sak criterion for any positive scaling dimension of the pairing field. The transition therefore belongs to the universality class of the generic MIPT without symmetry, tensored with a spectator critical spin-wave sector. Moreover, the coupling modulation by diffusive charge fluctuations that destabilizes the $\theta$4 MIPT fixed point is irrelevant here: $\theta$5 symmetry forbids linear coupling of the vector spin density to the pairing energy operator, and the leading composite coupling fails the Harris criterion ($\theta$6 required, whereas $\theta$7).

Learning times

Decomposing $\theta$8 into a bulk rate $\theta$9 plus a transient step $[0,2\pi)$0, the learning time is set by their competition; the transient saturates after the inverse diffusive gap $[0,2\pi)$1, the Thouless time.

In the fuzzy phase, perturbation theory about $[0,2\pi)$2 gives $[0,2\pi)$3, where $[0,2\pi)$4 is the local swap expectation in the maximally mixed state of sector $[0,2\pi)$5. Since $[0,2\pi)$6, one obtains $[0,2\pi)$7. Physically, symmetric measurements distinguish two nearby spin sectors only through the Casimir density, differing by $[0,2\pi)$8, requiring $[0,2\pi)$9 outcomes accumulated at a rate $1-p$0.

Crucially, in the fuzzy phase the $1-p$1-ordered vacuum factorizes over replicas, forcing $1-p$2 up to $1-p$3 corrections: the pairing order hides the diffusive background from the record. In the sharp phase the replica symmetry is restored, inter-replica correlations through the shared record are generic, and $1-p$4. Because the total-spin information resides in the uniform ($1-p$5) mode, resolvable only once diffusive dynamics spans the chain, learning occurs at $1-p$6. The paper thus proposes that the entanglement transition and spin-sharpening transition are one and the same, $1-p$7, consistent with the numerical finding that they could not be resolved as distinct [Majidy et al., (Majidy et al., 2023)]—a sharp contrast to the $1-p$8 case, where charge sharpening occurs strictly inside the volume-law phase via a Kosterlitz–Thouless-type mechanism.

Limitations and open questions

The theory rests on several assumptions stated explicitly by the authors. The controlled expansion is in $1-p$9, physically suppressing singlet fusion outcomes; results at $p$0 assume continuity from large $p$1, expected to hold in the scrambling-dominated fuzzy phase but questionable in the measurement-dominated sharp phase. Indeed, at $p$2 the neglected $p$3 terms are precisely those generating inter-replica correlations, so the framework gives no meaningful description of the measurement-only limit. Numerically, spin sharpening shows no saturation after $p$4 in the sharp phase, suggesting a nonperturbative enhancement of $p$5 beyond the present treatment. The constant $p$6 is likewise not determined analytically. Pinning down subleading corrections—and hence the true universality class—is left open, as is the relation to strong-to-weak spontaneous symmetry breaking in $p$7-symmetric dephasing dynamics, which the authors argue cannot be reduced to the monitored problem because noncommuting fusion projectors admit no joint eigenbasis.

Conclusion

This work supplies the first controlled stat-mech description of non-Abelian charge learning in monitored quantum circuits. Its central result is structural: a single replica-pairing order parameter simultaneously controls volume-law entanglement and blindness of the record to a persistent diffusive background, thereby unifying the entanglement and spin-sharpening transitions predicted to occur at $p$8, with learning times $p$9 and $1+1$00 on the two sides. The framework extends naturally to higher spins, other non-Abelian groups, anyonic fusion categories, and monitored anyon chains, where a $1+1$01 sharp phase would be expected.