The “noncanonicality” of the underlying operators is the fundamental difficulty in formulating theories for most strong interaction systems such as the t-J model, the Hubbard-type model under the large U limit, and quantum spin etc. But B.S. Shastry proposed Schwinger-Dyson equations of motion to study the t-J model, which can rigorously solve this noncanonicality. In this article, we use the perturbation Schwinger-Dyson equation of motion method to study two spin 1/2 transverse field Ising models perturbed by longitudinal fields, and discuss potential generalizations of the corresponding lattice models.
We develop a nonperturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation functions and dimensional regularization. Therefore, the nonperturbative spectral renormalization setup here respects all symmetries of the theories at hand. In particular, this includes space-time symmetries, as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalization can be applied within general functional approaches such as the functional renormalization group, Dyson-Schwinger equations, and two- or
n-particle irreducible approaches. As an application, we compute the full, nonperturbative, spectral function of the scalar field in the ${\ensuremath{\phi}}^{4}$-theory in
2+1 dimensions from spectral Dyson-Schwinger equations. We also compute the
s-channel spectral function of the full ${\ensuremath{\phi}}^{4}$-vertex in this theory.