Wilson loops as probes of phase transitions and conductivity phenomena

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.