A non-linear minimization calculation of the renormalized frequency $\tilde{\omega}$ in dirty d-wave superconductors

This work performs a comparative numerical study of the impurity average self-frequency $\tilde{\omega}$ in an unconventional superconducting alloy with non-magnetic impurities. Two methods are used: the Levenberg-Marquardt algorithm as a non-linear minimization problem, and a fixed-point iteration procedure. The unconventional superconducting renormalized by impurities \bw is a self-consistent complex non-linear equation with two varying parameters: the impurity concentration $\Gamma^+$ and the strength of the impurities c, for which its numerical solution is a computational challenge. Throughout this study $\tilde{\omega}$ is the renormalized frequency, \bg represents the inverse of the residual average lifetime $\tau$ at zero frequency, and $N/N_0$ is the normalized superconducting density of states (DOS). This study uses an order parameter that corresponds to the high-temperature superconducting ceramics (HTS) with a well-established gap symmetry. The results reveal the computational efficiency of the non-linear minimization technique by improving the calculations of the $\tilde{\omega}$ computation when using a 2D parameter space ($\Gamma^+$, c), particularly in the unitary regime, where the imaginary part of $\tilde{\omega}$ is a complicated expression of those parameters; this allows to enhance the study of the universal behavior of this particular quantum mechanical state.