Non-linear sigma model approach to quantum spin chains

TL;DR

This paper reviews various methods for studying quantum spin chains, focusing on the $J_1$-$J_2$-$$ model, and discusses open problems in the field.

Contribution

It categorizes existing approaches to quantum spin chains and provides detailed discussion on the $J_1$-$J_2$-$$ model, highlighting open research questions.

Findings

Classified methods into exact, field theory, and numerical approaches.

Detailed analysis of the $J_1$-$J_2$-$$ model.

Discussion of open problems in quantum spin chains.

Abstract

We introduce and motivate the study of quantum spin chains on a one-dimensional lattice. We classify the varieties of methods that have been used to study these models into three categories, - a) exact methods to study specific models b) field theories to describe fluctuations about the classical ordered phases and c) numerical methods. We then discuss the $J_1$-$J_2$-$\delta$ model in some detail and end with a few comments on open problems.