Disorder Operators and their Descendants

TL;DR

This paper reviews the concept of disorder operators, exploring their role in various physical systems and phases, including topological matter, dualities, and fermionic systems, highlighting their broad theoretical significance.

Contribution

It provides a comprehensive overview of disorder operators, their generalizations, and their connections to dualities, Majorana fermions, and topological phases, expanding understanding across multiple physics domains.

Findings

Disorder operators are crucial in understanding phase distinctions.

They connect to dualities and topological phases.

Their role in fermion-boson dualities is significant.

Abstract

I review the concept of a {\em disorder operator}, introduced originally by Kadanoff in the context of the two-dimensional Ising model. Disorder operators acquire an expectation value in the disordered phase of the classical spin system. This concept has had applications and implications to many areas of physics ranging from quantum spin chains to gauge theories to topological phases of matter. In this paper I describe the role that disorder operators play in our understanding of ordered, disordered and topological phases of matter. The role of disorder operators, and their generalizations, and their connection with dualities in different systems, as well as with Majorana fermions and parafermions, is discussed in detail. Their role in recent fermion-boson and boson-boson dualities is briefly discussed.