Fermionic duality approach to strongly-interacting lattice models

J. P. Kestner, Luming Duan

Müller, Tobias

Etienne Granet

Abstract Pointlike interactions between bosons in 1D are related to pointlike interactions between fermions through the Girardeau mapping. This mapping is a strong–weak duality since the coupling constants in the bosonic and fermionic cases are inversely proportional to each other. We present a regularization of these pointlike interactions that preserves the strong–weak duality, contrary to previously known Hermitian regularizations. This is proven rigorously. This allows one to use this duality perturbatively and we illustrate it in the Lieb–Liniger model at strong coupling.

Ulf-G. Meißner

Etienne Granet, Bruno Bertini, Fabian H. L. Essler

In one-dimensional quantum gases there is a well known "duality" between hard core bosons and noninteracting fermions. However, at the field theory level, no exact duality connecting strongly interacting bosons to weakly interacting fermions is known. Here we propose a solution to this long-standing problem. Our derivation relies on regularizing the only pointlike interaction between fermions in one dimension that induces a discontinuity in the wave function proportional to its derivative. In contrast to all known regularizations our potential is weak for small interaction strengths. Crucially, this allows one to apply standard methods of diagrammatic perturbation theory to strongly interacting bosons. As a first application we compute the finite temperature spectral function of the Cheon-Shigehara model, the fermionic model dual to the celebrated Lieb-Liniger model.

Satoshi Ohya

We study boson–fermion dualities in one-dimensional many-body problems of identical particles interacting only through two-body contacts. By using the path-integral formalism as well as the configuration-space approach to indistinguishable particles, we find a generalization of the boson–fermion duality between the Lieb–Liniger model and the Cheon–Shigehara model. We present an explicit construction of n-boson and n-fermion models which are dual to each other and characterized by n−1 distinct (coordinate-dependent) coupling constants. These models enjoy the spectral equivalence, the boson–fermion mapping, and the strong–weak duality. We also discuss a scale-invariant generalization of the boson–fermion duality.