Quantum geometry: New paradigm governing quantum transport observables

Quantum geometry is reshaping our understanding of topological quantum matter and advancing quantum information technologies. A striking example is the recent discovery of Fractional Chern Insulators -- two-dimensional materials that potentially host non-Abelian anyons in flat electronic bands engineered via twisting van der Waals heterostructures. These flat bands possess rich quantum geometry, which impacts all quantum transport observables - from electric conductivities to thermoelectric and thermal response, and quantum noise. In this talk, I will introduce the concept of the quantum metric, highlighting its impact on quantum transport in 2D topological systems with flat topological bands. We will discuss quantum transport formalism in the topological phase, measurable signatures of the quantum metric, and their sum rules. Time permitting, I will also present how the quantum metric imposes a fundamental lower bound on the entanglement entropy in these systems.

[1] Alexander Kruchkov, Phys. Rev. B 107, L241102 (2023)

[2] Alexander Kruchkov and Shinsei Ryu, Phys. Rev. B 110, L041118 (2024)

[3] Alexander Kruchkov and Shinsei Ryu, Phys. Rev. B (accepted, 2025) arXiv:2408.10314

[4] Alexander Kruchkov and Shinsei Ryu, arXiv:2312.17318

[5] Alexander Kruchkov, Phys. Rev. B 105, L241102 (2022)