A Hybrid Lattice Scalar Field Model with Stability Bounds and Nonzero Truncated Correlation Functions

We introduce a Euclidean lattice model of scaled interacting scalar fields, defined on a finite lattice in spacetime dimension $d\geq 2$, which we refer to as a hybrid model. The coupling parameters of the model are the hopping parameter $\kappa^2 > 0$ and the field interaction coupling $\alpha > 0$. By employing the technique of lattice scaled fields with a noncanonical scaling corresponding to a partial wavefunction renormalization, we show that the partition function of the model satisfies both thermodynamic and ultraviolet stability bounds, with constants that are uniform in the lattice volume $\Lambda_s$ and the lattice spacing $a \in (0, 1]$. As a consequence, the model admits a finite free energy per degree of freedom in the thermodynamic limit $\Lambda_s \nearrow \infty$ and the continuum limit $a\searrow 0$, at least in the subsequential sense. Furthermore, for sufficiently small values of the coupling parameters $\kappa^2$ and $\alpha$, we perturbatively compute the truncated two point correlation function and derive the relevant analyticity criterion in the form of a critical line. Finally, we perturbatively compute the truncated four point correlation function and demonstrate that it is nonzero. In this sense, the model exhibits nontrivial interacting correlations in the weak coupling regime. Using some well established results for the free field theory, we also investigate the existence of the model in the thermodynamic and continuum limits. The underlying physical Hilbert space is also discussed briefly. Together, these results establish the stability bounds of the model and provide perturbative evidence of nontrivial connected correlations, while leaving the full continuum construction and Osterwalder--Schrader positivity for further investigation.