  Investigation of strongly interacting, nonlinear quantum field theories
(QFT-s) remains one of the outstanding challenges of modern physics. Here, we
describe analog quantum simulators for nonlinear QFT-s using mesoscopic
superconducting circuit lattices. Using the Josephson effect as the source of
nonlinear interaction, we investigate generalizations of the quantum
sine-Gordon model. In particular, we consider a two-field generalization, the
double sine-Gordon model. In contrast to the sine-Gordon model, this model can
be purely quantum integrable, when it does not admit a semi-classical
description - a property that is generic to many multi-field QFT-s. The primary
goal of this work is to investigate different thermodynamic properties of the
double sine-Gordon model and propose experiments that can capture its subtle
quantum integrability. First, we analytically compute the mass-spectrum and the
ground state energy in the presence of an external `magnetic' field using Bethe
ansatz and conformal perturbation theory. Second, we calculate the
thermodynamic Bethe ansatz equations for the model and analyze its finite
temperature properties. Third, we propose experiments to verify the theoretical
predictions.
