Bose-Fermi dualities for arbitrary one-dimensional quantum systems in the universal low energy regime

TL;DR

This paper establishes a comprehensive framework for Bose-Fermi dualities in one-dimensional quantum systems, enabling a unified low-energy description of interacting particles with arbitrary properties and dispersion relations.

Contribution

It introduces a unitary algebra of generalized functions for statistical transmutation, providing a universal low-energy Bose-Fermi mapping applicable to diverse 1D quantum models.

Findings

Bose-Fermi duality holds at low energies for arbitrary 1D systems.

Explicit statistical interactions are derived for various models.

The duality mapping can be expressed as a gauge interaction.

Abstract

I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (non-relativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the non-relativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.