Fast holographic inversion of superconducting domes

Sejin Kim

hep-thOct 7, 2026

Fast holographic inversion of superconducting domes

Authors: Sejin Kim

Organizations: Center for AI and Natural Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea

Abstract

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 wherever the horizons of the dome reach, including the minima of $M$ , and differ only on that interval. A rule that keeps the reconstruction with the simplest closed form recovers both the scalar mass and the mass function of a test dome. On Gaussian and double-Gaussian domes and on the measured phase diagrams of YBa $_{2}$ Cu $_{3}$ O $_{y}$ and 2M-WS $_{2}$ , however, the pinned mass it keeps rests on ties or on narrow margins, so for these targets the scalar mass is left open. The dome thus constrains $M$ where its horizons reach, and fixing the dimension of the dual operator needs a second observable.

Figures & tables

parameters	method	integrations	seconds
$25$	earlier	$728$	$12.6$
	this	$56$	$1.27$
$50$	earlier	$1{,}428$	$25.6$
	this	$56$	$1.33$
$101$	earlier	$2{,}856$	$53.9$
	this	$56$	$1.35$

Table 1: One update in each method, on the phase boundary of Sec. 5.1 at

28

points, with the same basis and the same integration accuracy. The earlier method is one shooting solution per point,

\mathcal{J}

from a fit of the near-boundary expansion, and a finite-difference gradient for Adam Kim et al. (2025) . Ours is the two solutions, each integrated across the whole bulk, with

\hat{W}

evaluated at the matching point and the Jacobian of Eq. ( 1 ).

phase boundary	$m_{0}$	$\Delta$	$(n_{c},n_{f})$	$k$	spline	closed
generated by Eq. ( 18 )	$-1/2$	$2.823$	$(2,0)$	$2$	$0.247\%$	$0.538\%$
Gaussian	$-1/2$	$2.823$	$(2,3)$	$11$	$0.172\%$	$2.286\%$
double Gaussian	$-3/4$	$2.725$	$(2,4)$	$14$	$0.512\%$	$1.003\%$
YBCO	$-1/4$	$2.914$	$(2,3)$	$11$	$0.306\%$	$0.662\%$
2M-WS 2	$-3/4$	$2.725$	$(3,5)$	$18$	$2.398\%$	$2.424\%$

Table 2: The pinned mass each phase boundary keeps under the two criteria, with the dimension it implies, under the conventions of Sec. 4.3 . The form

(n_{c},n_{f})

gives the polynomial degree and the number of Gaussian functions of Eq. ( 17 ), and

k=n_{c}+3n_{f}

its symbolic parameters. The last two columns are the mean error of the phase boundary regenerated from the

101

-parameter reconstruction and from the closed form. The first row is the test of Sec. 5.1 . Eqs. ( 19 ) and ( 21 ) to ( 24 ) write the forms out.

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