An introduction to anyons

Sumathi Rao

E. Westerberg

Simon Trebst, Matthias Troyer, Zhenghan Wang +1 more

We discuss how to construct models of interacting anyons by generalizing quantum spin Hamiltonians to anyonic degrees of freedom. The simplest interactions energetically favor pairs of anyons to fuse into the trivial (“identity") channel, similar to the quantum Heisenberg model favoring pairs of spins to form spin singlets. We present an introduction to the theory of anyons and discuss in detail how basis sets and matrix representations of the interaction terms can be obtained, using non-Abelian Fibonacci anyons as example. Besides discussing the “golden chain”, a one-dimensional system of anyons with nearest neighbor interactions, we also present the derivation of more complicated interaction terms, such as three-anyon interactions in the spirit of the Majumdar-Ghosh spin chain, longer range interactions and two-leg ladders. We also discuss generalizations to anyons with general non-Abelian SU(2)k statistics. The k→∞ limit of the latter yields ordinary SU(2) spin chains.

Ijaz Durrani

I. J. R. Aitchison, Nick E. Mavromatos

An elementary introduction is given to the concept of anyons—quantum particles whose ‘statistics’ interpolate smoothly between those of bosons and fermions. Such particles—or quasi-particle excitations—can occur in two-dimensional systems in which the particles experience strong short-range mutual repulsions. These circumstances effectively mean that one has to apply quantum mechanics to a configuration space in which some points are excluded (i.e. it has ‘holes’)—namely, those points where two particles would coincide. The Aharonov—Bohm effect provides the simplest physical example of quantum mechanics in such a non-simply connected space, and is also fundamental to a simple model for anyons. A qualitative introduction is given to two currently proposed physical applications of anyons: high temperature superconductivity, and the (fractional) quantum Hall effect.

Sumathi Rao

In this set of lectures, we give a pedagogical introduction to the subject of anyons. We discuss 1) basic concepts in anyon physics, 2) quantum mechanics of two anyon systems, 3) statistical mechanics of many anyon systems, 4) mean field approach to many anyon systems and anyon superconductivity, 5) anyons in field theory and 6) anyons in the Fractional Quantum Hall Effect (FQHE). (Based on lectures delivered at the VII SERC school in High Energy Physics at the Physical Research Laboratory, Ahmedabad, January 1992 and at the I SERC school in Statistical Mechanics at Puri, February 1994.)