Probing chiral topological states with permutation defects

TL;DR

This paper introduces a novel bulk wavefunction measure using permutation defects to directly probe the chirality of two-dimensional topological states, enabling extraction of key topological invariants.

Contribution

It develops a new entanglement-based method involving permutation defects that captures chiral properties directly from the bulk wavefunction, bridging bulk and edge phenomena.

Findings

The method correctly identifies the chiral conformal field theory partition function.

Numerical verification shows excellent agreement for free-fermion and interacting states.

Allows extraction of chiral central charge and Hall conductance from finite wavefunction replicas.

Abstract

The hallmark of two-dimensional chiral topological phases is the existence of anomalous gapless modes at the spatial boundary. Yet, the manifestation of this edge anomaly within the bulk ground-state wavefunction itself remains only partially understood. In this work, we introduce a family of multipartite entanglement measures that probe chirality directly from the bulk wavefunction. Our construction involves applying different permutations between replicas of the ground state wavefunction in neighboring spatial regions, creating "permutation defects" at the boundaries between these regions. We provide general arguments for the robustness of these measures and develop a field-theoretical framework to compute them systematically. While the standard topological field theory prescription misses the chiral contribution, our method correctly identifies it as the chiral conformal field theory partition function on high-genus Riemann surfaces. This feature is a consequence of the bulk-edge correspondence, which dictates that any regularization of the theory at the permutation defects must introduce gapless boundary modes. We numerically verify our results with both free-fermion and strongly-interacting chiral topological states and find excellent agreement. Our results enable the extraction of the chiral central charge and the Hall conductance using a finite number of wavefunction replicas, making these quantities accessible to Monte-Carlo numerical techniques and noisy intermediate-scale quantum devices.