Geometric test for topological states of matter

TL;DR

This paper extends flux insertion techniques to higher-genus surfaces to characterize the topological robustness of fractional quantum Hall states, providing a geometric test that distinguishes localized quasiholes from delocalized ones.

Contribution

It generalizes the flux insertion argument to higher-genus surfaces, computes the Laughlin bundle's topological invariants, and establishes a geometric test for topological state robustness.

Findings

Computed the rank and Chern classes of Laughlin bundles for any genus and quasihole number.

Proved that Laughlin bundles with non-localized quasiholes are not projectively flat.

Demonstrated that quantized Hall conductance occurs only with localized quasiholes.

Abstract

We generalize the flux insertion argument due to Laughlin, Niu-Thouless-Tao-Wu, and Avron-Seiler-Zograf to the case of fractional quantum Hall states on a higher-genus surface. We propose this setting as a test to characterise the robustness, or topologicity, of the quantum state of matter and apply our test to the Laughlin states. Laughlin states form a vector bundle, the Laughlin bundle, over the Jacobian - the space of Aharonov-Bohm fluxes through the holes of the surface. The rank of the Laughlin bundle is the degeneracy of Laughlin states or, in presence of quasiholes, the dimension of the corresponding full many-body Hilbert space; its slope, which is the first Chern class divided by the rank, is the Hall conductance. We compute the rank and all the Chern classes of Laughlin bundles for any genus and any number of quasiholes, settling, in particular, the Wen-Niu conjecture. Then we show that Laughlin bundles with non-localized quasiholes are not projectively flat and that the Hall current is precisely quantized only for the states with localized quasiholes. Hence our test distinguishes these states from the full many-body Hilbert space.