Gapless Symmetry-Protected Topological States in Measurement-Only Circuits

TL;DR

This paper explores gapless symmetry-protected topological states in measurement-only quantum circuits, revealing new critical phases with topological edge states through large-scale simulations and theoretical mappings.

Contribution

It generalizes gSPT to critical steady states, uncovers a symmetry-enriched percolation point, and demonstrates a steady-state gSPT phase in a $\

Findings

Identified a symmetry-enriched non-unitary critical point with topological edge states.

Realized a steady-state gSPT phase in a $\

Mapped the system to the Majorana loop model for theoretical insights.

Abstract

Measurement-only quantum circuits offer a versatile platform for realizing intriguing quantum phases of matter. However, gapless symmetry-protected topological (gSPT) states remain insufficiently explored in these settings. In this Letter, we generalize the notion of gSPT to the critical steady state by investigating measurement-only circuits. Using large-scale Clifford circuit simulations, we investigate the steady-state phase diagram across several families of measurement-only circuits that exhibit topological nontrivial edge states at criticality. In the Ising cluster circuits, we uncover a symmetry-enriched non-unitary critical point, termed symmetry-enriched percolation, characterized by both topologically nontrivial edge states and string operator. Additionally, we demonstrate the realization of a steady-state gSPT phase in a $\mathbb Z_4$ circuit model. This phase features topological edge modes and persists within steady-state critical phases under symmetry-preserving perturbations. Furthermore, we provide a unified theoretical framework by mapping the system to the Majorana loop model, offering deeper insights into the underlying mechanisms.