Non relativistic limit of integrable QFT and Lieb–Liniger models

Abstract

In this paper we study a suitable limit of integrable QFT with the aim to identify continuous non-relativistic integrable models with local interactions. This limit amounts to sending to infinity the speed of light c but simultaneously adjusting the coupling constant g of the quantum field theories in such a way to keep finite the energies of the various excitations. The QFT considered here are Toda Field Theories and the O(N ) non-linear sigma model. In both cases the resulting non-relativistic integrable models consist only of Lieb-Liniger models, which are fully decoupled for the Toda theories while symmetrically coupled for the O(N ) model. These examples provide explicit evidence of the universality and ubiquity of the Lieb-Liniger models and, at the same time, suggest that these models may exhaust the list of possible non-relativistic integrable theories of bosonic particles with local interactions.

Non relativistic limit of integrable QFT with fermionic excitations

“…In a previous paper by us [1] we were driven by the curiosity of understanding the striking unbalance which exists between the very large number of Integrable Quantum Field Theories (IQFT) and, in contrast, the very short list of Non-Relativistic Integrable Models 1 (NRIM). To analyze the origin of this richness/paucity difference between the two classes of models in Ref.…”

“…To analyze the origin of this richness/paucity difference between the two classes of models in Ref. [1] we pursued a study of the non-relativistic limit of a large set of IQFT (along the lines of the strategy adopted in earlier papers to study the non-relativistic limit of the Sinh-Gordon model [3] and of the Thirring model [4]): for nonrelativistic limit we mean not only sending the speed of limit c to infinity but simultaneously tuning the relevant coupling constant of the model in order to keep the energy of its excitations finite. In Ref.…”

“…In Ref. [1] we were concerned with purely bosonic theories such as the Sinh-Gordon model [6,7], the Bullough-Dodd model [6,[8][9][10][11] or, more generally, the Toda Field Theories [6,[12][13][14][15][16][17][18][19][20] but also with the vectorial O(N ) non-linear sigma model [21].…”

“…Interestingly, the duality between the Thirring model and the purely bosonic Sine Gordon model [27] is reflected also in their NR limits [4,26,28], being the NR limit of the Sine Gordon the attractive Lieb Liniger model [4,29]. The purpose of the present investigation is to enlarge the landscape of the known NR limits, in particular we consider the supersymmetric Sinh-Gordon model (SShG) [30,31], the O(N ) Gross-Neveu model (GN) [21,32,33] and 1 In Ref. [1] we were concerned and we are also concerned here only with models which are local, galilean invariant and with field derivatives non higher that second order in their Hamiltonian density.…”

“…The purpose of the present investigation is to enlarge the landscape of the known NR limits, in particular we consider the supersymmetric Sinh-Gordon model (SShG) [30,31], the O(N ) Gross-Neveu model (GN) [21,32,33] and 1 In Ref. [1] we were concerned and we are also concerned here only with models which are local, galilean invariant and with field derivatives non higher that second order in their Hamiltonian density. Note that other NRIM such as the Calogero-Sutherland or the Haldane-Shastry models [2], as well as all lattice models, do not belong to this class, since they are either non-local or they are not continuum models, so they break relativistic invariance.…”