Applications of Adiabatic Flux Insertion to Quantum Many-Body Systems: A Pedagogical Introduction

Masaki Oshikawa

ISSP, University of Tokyo



A time-dependent magnetic field induces an electric field. While this has been known as Faraday’s law in classical electromagnetism, new aspects  appear when applied to a quantum mechanical system.  Quantum mechanical systems couple to the magnetic field through the vector potential, and they can  be affected by the magnetic field even when the particles do not directly touch the magnetic field (Aharonov-Bohm effect).
A static magnetic field through a hole does not affect the outside particles only when the total magnetic flux is an integral multiple of the unit flux  quantum, which  can be eliminated by a topologically nontrivial gauge
transformation. Considering an adiabatic insertion process of the
magnetic flux, this “periodicity” leads to various topological quantizations in quantum many-body systems, such as Quantum Hall Effects, Lieb-Schultz-Mattis type theorems, and Luttinger’s theorem.

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