Fast holographic inversion of superconducting domes

Sejin Kim

A holographic superconductor whose scalar mass depends on the gauge field strength, $M(\Fsq)$, reproduces a superconducting dome for a suitable $M$, and recovering that $M$ from a given dome has so far taken days for a single training run. We propose a new way of training this model, with which an inversion takes from about ten minutes to an hour. Training needs the gradient of the condition that fixes the critical temperature, which the earlier method obtains by finite differences, repeating the bulk integrations for every training parameter. Here that condition is obtained, without any fit, from two integrations started at the horizon and at the boundary, and its derivative with respect to $M$ is an integral over the same two solutions, so the gradient needs no integration of its own. We use the speed to study the part of $M$ that a dome cannot determine, on the interval between the value $\Fsq$ takes at the horizon for the lowest doping and $\Fsq=0$, at which $M$ is the scalar mass $M(0)$ that fixes the dimension of the dual operator. We hold the scalar mass at several values, which we call pinned masses, retrain everything else at each, and find that the reconstructions agree…