Self-consistent approach to quenched impurity effects on quantum phase transitions

Abstract

Double expansion renormalization group treatments are still employed in literature to describe the effects of symmetry-conserving quenched disorder on quantum phase transitions as an alternative to more or less phenomenological models. If one assumes ϵ = 4 – d and the dimensionality ϵτ of the imaginary-time space involved in a generalized quantum action as simultaneously small parameters, a stable random fixed point is found which governs a new disorder-induced quantum criticality usually considered meaningful also after extrapolation to the physical time-like dimension ϵτ = 1. In contrast, by considering ϵτ = 1 at the beginning, a runaway takes place in the parameter space for dimensionalities d < 4. To give some insight into these contrasting results, we employ here a simple self-consistent approach for case of short-range correlated quenched disorder which allows us to analyze analytically what happens by continuous variation of the fictitious time-like dimension ϵτ We find that a ϵτ*(n) < 1 exists, depending on the symmetry index n, above which the disorder inhibits the occurrence of a conventional quantum phase transition. This suggests that the usual procedure to extrapolate the small ϵτ-predictions to the value of interest ϵτ = 1 may have no real physical meaning.