The $W_{1 + \infty }$ effective theory of the Calogero- Sutherland model and Luttinger systems.

TL;DR

This paper develops an effective $ ext{W}_{1+ ext{infinity}}$ conformal field theory for the Calogero-Sutherland model, providing a universal description of low-energy density fluctuations in one-dimensional fermionic systems, extending beyond integrability.

Contribution

It introduces a $ ext{W}_{1+ ext{infinity}}$ effective theory that describes density fluctuations in the Calogero-Sutherland model and related Luttinger systems, independent of integrability.

Findings

Exact description of density fluctuations via $ ext{W}_{1+ ext{infinity}}$ algebra.

Extension of results to any order in $1/N$ expansion.

Classification of fermionic systems using $ ext{W}_{1+ ext{infinity}}$ representations.

Abstract

We construct the effective field theory of the Calogero-Sutherland model in the thermodynamic limit of large number of particles $N$. It is given by a $\winf$ conformal field theory (with central charge $c=1$) that describes {\it exactly} the spatial density fluctuations arising from the low-energy excitations about the Fermi surface. Our approach does not rely on the integrable character of the model, and indicates how to extend previous results to any order in powers of $1/N$. Moreover, the same effective theory can also be used to describe an entire universality class of $(1+1)$-dimensional fermionic systems beyond the Calogero-Sutherland model, that we identify with the class of {\it chiral Luttinger systems}. We also explain how a systematic bosonization procedure can be performed using the $\winf$ generators, and propose this algebraic approach to {\it classify} low-dimensional non-relativistic fermionic systems, given that all representations of $\winf$ are known. This approach has the appeal of being mathematically complete and physically intuitive, encoding the picture suggested by Luttinger's theorem.