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arXiv:2610.18965

Measurement-induced criticality for non-Abelian anyons: The monitored golden chain

Zhaohui Zhi, Bo Han, Romain Vasseur, Andreas W. W. Ludwig, Simon Trebst, Guo-Yi Zhu

cond-mat.str-elcond-mat.dis-nncond-mat.stat-mechquant-phSubmitted Oct 13, 2026• Announced Oct 15, 2026• 0 citations
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Abstract

Non-Abelian anyon chains whose interactions favor a single fusion channel host quantum critical ground states described by minimal-model conformal field theories (CFTs) protected by a topological symmetry. We show that their measurement-only counterpart, where non-commuting fusion charge measurements act locally on the anyons of a chain of $L$ sites, provides a fundamental example of a non-unitary CFT controlling a non-Abelian learning problem. Mapping the monitored dynamics to a random tensor network, we numerically find $1/L$ scaling Lyapunov gaps, logarithmic entanglement scaling with the chord length, and an effective central charge in the Shannon entropy of the measurement record. The conserved topological charge sharpens with dynamical exponent $z= 1$ and is learnable by a decoder on an $O(L)$ time scale. As the topological symmetry is a non-invertible symmetry of the $SU(2)_k$ anyon theories, we establish the monitored golden chain $(k=3)$ as a non-Abelian generalization of the weakly (Kramers-Wannier) self-dual quantum critical point of the toric code $(k=2)$, and the first of a family for $k\geq 3$.

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Non-Abelian anyon chains whose interactions favor a single fusion channel host quantum critical ground states described by minimal-model conformal field theories (CFTs) protected by a topological symmetry. We show that their measurement-only counterpart, where non-commuting fusion charge measurements act locally on the anyons of a chain of LLL sites, provides a fundamental example of a non-unitary CFT controlling a non-Abelian learning problem. Mapping the monitored dynamics to a random tensor network, we numerically find 1/L1/L1/L scaling Lyapunov gaps, logarithmic entanglement scaling with the chord length, and an effective central charge in the Shannon entropy of the measurement record. The conserved topological charge sharpens with dynamical exponent z=1z= 1z=1 and is learnable by a decoder on an O(L)O(L)O(L) time scale. As the topological symmetry is a non-invertible symmetry of the SU(2)kSU(2)_kSU(2)k​ anyon theories, we establish the monitored golden chain (k=3)(k=3)(k=3) as a non-Abelian generalization of the weakly (Kramers-Wannier) self-dual quantum critical point of the toric code (k=2)(k=2)(k=2), and the first of a family for k≥3k\geq 3k≥3.
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)SU(2)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)SU(2)SU(2)-symmetric readouts are noncommuting fusion measurements, making learning a genuinely quantum inference problem. In this work, we propose a theory of 1+1d1+1d1+1d monitored quantum dynamics with SU(2)SU(2)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=2z=2z=2) background sector that carries the SU(2)SU(2)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∼L3t\sim L^{3}t∼L3 for the total spin. When the pairing field disorders, the background sector alone gives logarithmic entanglement and a diffusive learning time t∼L2t\sim L^{2}t∼L2. Our analysis is controlled by a large-loop-fugacity expansion.
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Quantum gates for the manipulation of topological qubits rely on interactions between non-Abelian anyonic quasiparticles. We study the collective behavior of systems of anyons arising from such interactions. In particular, we study the effect of favoring different fusion channels of the screened Majorana spins appearing in the recently proposed topological Kondo effect. Based on the numerical solution of a chain of SO(5)2SO{(5)}_{2}SO(5)2​ anyons we identify two critical phases whose low-energy behavior is characterized by conformal field theories with central charges c=1c=1c=1 and c=8/7c=8/7c=8/7, respectively. Our results are complemented by exact results for special values of the coupling constants which provide additional information about the corresponding phase transitions.
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Topological quantum computation relies on braiding non-Abelian anyons, but requires the underlying topological order to survive imperfect state preparation and environmental noise. We show that the instability of topological order to wavefunction deformations and to decoherence, with the latter probed by syndrome distributions, is generically captured by stat-mech models whose G×Rep(G)G\times \textrm{Rep}(G)G×Rep(G) symmetry naturally exposes the corrupting anyonic excitations. As an example, we combine this framework with Monte-Carlo simulations to resolve the stability of D4D_4D4​ topological order under deformations and quantum channels that proliferate multiple non-Abelian anyon species that are individually unable to condense. We show that beyond a finite threshold, proliferation of two non-Abelian anyon species parasitically condenses a shared Abelian-anyon fusion outcome---destroying the topological order. Our symmetry-based approach suggests the resulting trivial phase is distinct from that obtained by condensing all Abelian anyons; in other words, the trivial phase "remembers" which anyons condensed. This framework provides a first step into identifying the relevant symmetry for optimal decoders, conditioned on syndrome measurements, of non-Abelian topological order.
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