Geometric Test for Topological States of Matter

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 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.

Figure 1

The adiabatic transport of quantum states in the parameter space $M$ along the curves ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ starting and ending at the same point $x_0$.

Figure 2

$AB$ phases on the genus-$g$ Riemann surface.

Figure 3

Representation of the $N$th symmetric power of the Riemann surface as projective spaces ${\mathbb{P}}^{N-g}$ fibered over the Picard variety.