Restoration of a Spontaneously Broken Symmetry in an Euclidean Quantum $\lambda\varphi^{4}_{d+1}$ model with Quenched Disorder

We investigate the low temperature behavior of a system in a spontaneously broken symmetry phase described by an Euclidean quantum $\lambda\varphi^{4}_{d+1}$ model with quenched disorder. Using a series representation for the averaged generating functional of connected correlation functions in terms of the moments of the partition function, we study the effects of the disorder linearly coupled to the scalar field. To deal with the strongly correlated disorder in imaginary time, we employthe equivalence between the model defined in a $d$-dimensional space with imaginary time with the statistical field theory model defined on a space ${\mathbb R}^{d}\times S^{1}$ with anisotropic quenched disorder. Next, using fractional derivatives and stochastic differential equations we obtain at tree-level the Fourier transform of the correlation functions of the disordered system. In one-loop approximation, we prove that there is a denumerable collection of moments of the partition function that can develop critical behavior. Below the critical temperature of the pure system, with the bulk in the ordered phase, there are a large number of critical temperatures that take each of these moments from an ordered to a disordered phase. We show the emergence of generic scale invariance in the system.