Universal duality transformations in interacting one-dimensional quantum systems

TL;DR

This paper develops a comprehensive theory of duality transformations in one-dimensional quantum systems, enabling systematic mapping between bosonic and fermionic models with arbitrary interactions and internal structures.

Contribution

It introduces a general framework for local unitary transformations that establish dualities across a wide class of 1D quantum systems, including spinful and multicomponent cases.

Findings

Generated new duality relations linking strong and weak coupling regimes.

Unified treatment of boson-fermion dualities with arbitrary spin and interactions.

Applicable to various dispersion relations, including relativistic cases.

Abstract

One-dimensional quantum systems admit duality relations that put hard core spinless bosons and fermions in one-to-one correspondence via Girardeau's mapping theorem. The simplest models of soft bosons interacting via zero-range potentials can also be mapped onto dual interacting fermions. However, a systematic approach to one-dimensional statistical transmutation for arbitrary low-energy interactions in the spinless and spinful or multicomponent cases has remained elusive. I develop a general theory of local unitary transformations between one-dimensional quantum systems of bosons and fermions with arbitrary spin or internal structure, single-particle dispersion -- including non-relativistic, relativistic or otherwise -- and low-energy interactions in the universal regime. These transformations generate families of new duality relations and models that relate the strong and weak coupling limits of the respective dual theories.