Non-relativistic limit
“Nothing can travel faster than light but there is an entire world down there, especially going closely to zero speed”
-- Achilles and the Tortoise
Nowadays there are known many integrable quantum field theories (QFT) which have relativistic invariance as their distinguished feature: the list includes purely bosonic models, as the Sinh-Gordon, or more generally Toda models, fermionic systems, as the Gross-Neveu model as well as supersymmetric versions of all these examples.
This list increases even further if one also adds to it theories with soliton excitations, as the Sine Gordon model, or various sigma models based on group manifolds.
Compared to this extraordinary richness of examples concerning relativistic integrable models, the paucity of non-relativistic integrable models (NRIM) with interactions strictly local and galilean invariant is rather dazzling, their list being essentially given by the Lieb-Liniger model! The fact that the Lieb-Liniger model is, under certain general conditions, probably the only non-trivial example of NRIM may not be accidental and the universal ubiquity of this model has been investigated in a series of paper, listed below. The only other known
NRIM are, essentially, variations of the same theme, alias systems made of coupled Lieb-Liniger models relative to different species of particles. Notice that the integrability of these coupled Lieb-Liniger models is only realised under special conditions: when the particle species have the same mass and all interactions -- both among the particles of the same species or interspecies -- have the same coupling constant.
Possible new NRIM can be identified by making use of the richness of the relativistic ones. Namely, I have explored the possibility of identifying a new class of NRIM in terms of the non-relativistic limit (NR) of known integrable QFT, both with purely boson or fermionic excitation. The first important example has been the non-relativistic limit of the Sinh-Gordon model which was indeed shown to be the Lieb-Liniger model. This identification, in particular, was the key point that permitted to compute previously unknown correlation functions of the Lieb-Liniger model in terms of Form Factors and Thermodynamics Bethe Ansatz of the Sinh-Gordon model. Later this analysis has been generalised to all Toda Field Theories, to O(N) models and supersymmetric version thereof.
Main References
M. Kormos, G. Mussardo, A. Trombettoni, Expectation Values in the Lieb-Liniger Bose Gas
Selected for the December 2009 issue of Virtual Journal of Atomic Quantum Fluids,
Eds. W. Ketterle, M. Greiter, P. Zoller, Phys. Rev. Lett. 103, 210404 (2009);
M. Kormos, G. Mussardo, A. Trombettoni, 1D Lieb-Liniger Model as Non-relativistic limit of the Sinh-Gordon Model, Phys. Rev. A 81 (2010), 043606.
M. Kormos, G. Mussardo, A. Trombettoni, Local Correlations in the Super Tonks-Girardeau Gas, Phys.Rev.A83:013617,2011
M. Kormos, G. Mussardo, B. Pozsgay, Bethe Ansatz Matrix Elements as Non-relativistic limit of Form Factors of Quantum Field Theories, JSTAT 105 (2010), P05014 arXiv:0912.3502 [cond-mat.stat-mech]
A. Bastianello, A. De Luca and G. Mussardo, Non-relativistic limit of integrable QFT and Lieb-Liniger Models, JSTAT 1612 (2016), n. 12, 123104
A. Bastianello, A. De Luca and G. Mussardo, Non-relativistic limit of integrable QFT with fermionic excitations, invited paper on Journal of Physics, arXiv:1701.06542
