Charge and Spin Sharpening Transitions on Dynamical Quantum Trees

TL;DR

This paper investigates measurement-induced phase transitions in monitored quantum systems with non-Abelian symmetries, using analytical and numerical methods on tree-like quantum circuits, revealing distinct behaviors for U(1) and SU(2) symmetries.

Contribution

It provides the first analytical and numerical analysis of sharpening transitions in monitored quantum trees with non-Abelian symmetries, extending understanding beyond Abelian cases.

Findings

Entanglement and sharpening transitions occur at different measurement rates.

Fuzzy phase is prevalent in SU(2) symmetric systems, with sharp phase only at maximal measurement rate.

Analytical solutions match numerical simulations for phase boundary boundaries.

Abstract

The dynamics of monitored systems can exhibit a measurement-induced phase transition (MIPT) between entangling and disentangling phases, tuned by the measurement rate. When the dynamics obeys a continuous symmetry, the entangling phase further splits into a fuzzy phase and a sharp phase based on the scaling of fluctuations of the symmetry charge. While the sharpening transition for Abelian symmetries is well understood analytically, no such understanding exists for the non- Abelian case. In this work, building on a recent analytical solution of the MIPT on tree-like circuit architectures (where qubits are repatedly added or removed from the system in a recursive pattern), we study entanglement and sharpening transitions in monitored dynamical quantum trees obeying U (1) and SU (2) symmetries. The recursive structure of tree tensor networks enables powerful analytical and numerical methods to determine the phase diagrams in both cases. In the U (1) case, we analytically derive a Fisher-KPP-like differential equation that allows us to locate the critical point and identify its properties. We find that the entanglement/purification and sharpening transitions generically occur at distinct measurement rates. In the SU (2) case, we find that the fuzzy phase is generic, and a sharp phase is possible only in the limit of maximal measurement rate. In this limit, we analytically solve the boundaries separating the fuzzy and sharp phases, and find them to be in agreement with exact numerical simulations.