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arXiv:2112.14684 · Journal of Physics A Mathematical and Theoretical

Regularization of a strong–weak duality between pointlike interactions in one dimension

Etienne Granet

Published May 26, 2022• 2 citations
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Abstract

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.

DOI
10.1088/1751-8121/ac73c4

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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.
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.
In one dimensional quantum gases there is a well known between hard core bosons and non-interacting 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 point-like interaction between fermions in 1D that induces a discontinuity in the wave function proportional to its derivative. In contrast to all known regularisations 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.
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.
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 nonrelativistic, relativistic, or otherwise---and low-energy interactions in the universal regime. These transformations generate families of duality relations and models that relate the strong- and weak-coupling limits of the respective dual theories.
We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model.
We consider the one-dimensional Lieb-Liniger model (bosons interacting via 2-body delta potentials) in the infinite coupling constant limit (the so-called Tonks-Girardeau model). This model might be relevant as a description of atomic Bose gases confined in a one-dimensional geometry. It is known to have a fermionic spectrum since the N-body wavefunctions have to vanish at coinciding points, and therefore be symmetrizations of fermionic Slater wavefunctions. We argue that in the infinite coupling constant limit the model is indistinguishable from free fermions, i.e., all physically accessible observables are the same as those of free fermions. Therefore, Bose-Einstein condensate experiments at finite energy that preserve the one-dimensional geometry cannot test any bosonic characteristic of such a model.