Wilson loops as probes of phase transitions and conductivity phenomena

TL;DR

The paper addresses how to unify Wilson loops across gauge theory, Berry-phase topology, and intrinsic topological order in quantum Hall systems. It develops a cohesive framework where Wilson loops encode nonperturbative gauge structure, Chern numbers, and braiding statistics, showing that the same topological invariant governs both quantized transport and quasiparticle exchange phases. Key results include the Berry-loop formulation of Chern numbers determining $\sigma_{xy}$, the Chern--Simons description linking Wilson lines to anyonic statistics with linking numbers, and the torus degeneracy arising from Wilson loop algebras, collectively illustrating a deep link between topology and observable transport/ statistics. The work highlights the significance of Wilson loops as a unifying diagnostic for phases of matter and points to experimental platforms and theoretical extensions (higher-form symmetries, fractons, and topological quantum computation) where loop holonomies play a central role.

Abstract

Wilson loops are among the most fundamental gauge-invariant observables in quantum field theory, encoding the global structure of gauge fields through their holonomy along closed contours. Originally introduced as order parameters for confinement in non-Abelian gauge theories, they have recently acquired a central role in condensed matter physics, where they characterize topological phases and quantized transport phenomena. In this work we present a unified theoretical picture in which Wilson loops connect nonperturbative gauge dynamics, Berry-phase topology in band theory, and the quantum Hall response of interacting electron systems. We demonstrate explicitly how Wilson loops encode Chern numbers, fractional charge, and anyonic braiding statistics within Chern--Simons effective field theory. Both quantized Hall conductivity and quasiparticle statistics are shown to originate from the same topological invariant -- the linking number of Wilson loops -- establishing a direct correspondence between microscopic topological structure and macroscopic transport.