Geometric Adiabatic Transport in Quantum Hall States

Area of Science:

• Condensed Matter Physics
• Quantum Hall Effect
• Topological Phases of Matter

Background:

• The quantum Hall effect exhibits quantized conductance, but other transport coefficients remain less understood.
• Existing theories primarily focus on Hall conductance and viscous transport.
• A deeper understanding of transport phenomena in topological states is crucial.

Purpose of the Study:

• To identify and characterize a third, precisely quantized, independent transport coefficient in quantum Hall systems.
• To establish the topological origin and mathematical description of this new coefficient.
• To explore its physical implications, particularly its connection to gravitational anomalies.

Main Methods:

• Theoretical analysis within linear response theory.
• Investigation of topological properties using Chern numbers.
• Mathematical formulation for systems with genus 2 or higher surfaces.
• Development of methods for computing transport coefficients in quantum Hall states.

Main Results:

• Discovery of a third quantized transport coefficient, constant along quantum Hall plateaus.
• Identification of this coefficient as the Chern number of a vector bundle over moduli space for genus ≥ 2 surfaces.
• Demonstration that this coefficient is not present on spheres or tori.
• Connection of the coefficient to intensive forces from geometric deformations and gravitational anomalies.

Conclusions:

• A new fundamental transport coefficient exists in quantum Hall systems, offering insights into topological properties.
• The topological nature of this coefficient explains its quantized and invariant behavior.
• This finding provides a new perspective on the interplay between geometry, topology, and quantum transport.