Geometric Test for Topological States of Matter

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Topology in Physics

Background:

• The flux insertion argument is a key tool for understanding quantum Hall states.
• Generalizing this argument to higher-genus surfaces is crucial for exploring topological properties of matter.
• Laughlin states are a fundamental example of fractional quantum Hall states.

Purpose of the Study:

• To generalize the flux insertion argument to higher-genus surfaces for fractional quantum Hall states.
• To use this generalized framework as a test for the robustness and topologicity of quantum states.
• To characterize Laughlin states and their associated vector bundles, known as Laughlin bundles.

Main Methods:

• Generalization of the flux insertion argument (Laughlin, Niu-Thouless-Tao-Wu, Avron-Seiler-Zograf) to higher-genus surfaces.
• Mathematical formulation of Laughlin states as a vector bundle over the Jacobian of the surface.
• Computation of the rank and Chern classes of these Laughlin bundles for arbitrary genus and number of quasiholes.

Main Results:

• The rank of the Laughlin bundle corresponds to the degeneracy of Laughlin states, including those with quasiholes.
• The first Chern class divided by the rank yields the Hall conductance.
• The Wen-Niu conjecture regarding Chern classes of Laughlin bundles is proven for any genus and number of quasiholes.
• Laughlin bundles with non-localized quasiholes are not projectively flat, unlike those with localized quasiholes.
• Precisely quantized Hall current is demonstrated to occur only for states with localized quasiholes.

Conclusions:

• The generalized flux insertion argument provides a robust test to distinguish between topologically distinct quantum states.
• The study confirms and extends the understanding of Laughlin bundles and their topological invariants.
• Localized quasiholes are essential for the precise quantization of Hall current in these topological states.