Geometric Test for Topological States of Matter

We generalize the flux insertion argument due to Laughlin, Niu-Thouless-Tao-Wu, and Avron-SeilerZograf to the case of fractional quantum Hall states on a higher-genus surface. We propose this setting as a test to characterize 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 the 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 nonlocalized 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.

Automated summary

This study generalizes the flux insertion argument to the case of fractional quantum Hall states on a higher-genus surface, and applies this test to investigate the robustness and topologicity of Laughlin states. The results show that Laughlin states with localized quasiholes have precisely quantized Hall current, while Laughlin states with nonlocalized quasiholes are not projectively flat.