Comparing the elliptic Ruijsenaars–Schneider model to the finite volume sine-Gordon theory

We compare the spectrum of the elliptic Ruijsenaars-Schneider model with the finite-size spectrum of the sine-Gordon model, highlighting both their similarities and differences. Our analysis focuses on the two-particle sector in the center-of-mass frame. At the free point, we carry out an analytic comparison, while at generic couplings we employ non-perturbative numerical calculations based on the truncated Hilbert space method adapted to difference operators. To benchmark this numerical approach, we first study the trigonometric limit, where analytic results are available. We then examine in detail the non-relativistic limit, which encompasses the rational, trigonometric, hyperbolic, and elliptic Calogero-Moser-Sutherland models. Finally, we compare the Bethe-Yang momentum quantization conditions, derived from infinite-volume scattering phases, with the exact finite-volume solutions. We find that these conditions hold exactly in the rational and trigonometric cases, but acquire finite-size corrections in the hyperbolic and elliptic cases, both for the relativistic model and its non-relativistic limit, however, these corrections are different in quantum field theories and in many body systems.