Phase Diagram from Nonlinear Interaction between Superconducting Order and Density: Toward Data-Based Holographic Superconductor

TL;DR

The paper tackles an inverse problem in holographic superconductivity by learning a nonlinear bulk interaction $M(F^2)$ that governs the coupling between the superconducting order and charge density, aiming to reproduce experimental superconducting domes. It introduces a physics-informed neural network with positional embedding to infer $M(F^2)$ from critical-temperature data, validating the approach on artificial phase borders and applying it to real data from YBCO and transition-metal dichalcogenides. Key findings show that the learned $M(F^2)$ shapes the phase boundary and, below $T_c$, hairy black-brane solutions are thermodynamically preferred with a mean-field condensation profile, supporting a data-driven holographic route to high-$T_c$ phenomenology. The work provides a robust methodology for constructing data-based holographic quantum matter, with future directions including strange-metal normal phases, quantum criticality, and broader material classes.

Abstract

We address an inverse problem in modeling holographic superconductors. We focus our research on the critical temperature behavior depicted by experiments. We use a physics-informed neural network method to find a mass function $M(F^2)$, which is necessary to understand phase transition behavior. This mass function describes a nonlinear interaction between superconducting order and charge carrier density. We introduce positional embedding layers to improve the learning process in our algorithm, and the Adam optimization is used to predict the critical temperature data via holographic calculation with appropriate accuracy. Consideration of the positional embedding layers is motivated by the transformer model of natural-language processing in the artificial intelligence (AI) field. We obtain holographic models that reproduce borderlines of the normal and superconducting phases provided by actual data. Our work is the first holographic attempt to match phase transition data quantitatively obtained from experiments. Also, the present work offers a new methodology for data-based holographic models.

Figures (8)

• Figure 1: This figure illustrates the structured layout of our neural network. The process showcases how the network learns to approximate the mass function $M$ from critical temperature data, employing physics-informed strategies for an enhanced understanding of holographic superconductivity.
• Figure 2: (a) The decrease in the loss function value of our method based on the PINN as training progresses (as epochs increase). (b) Predicting of the mass function $M(F^2)$ as epochs increase as 50, 200, 400, 600, and 1000. (C) The superconducting phase diagram provided by the predicted mass function $M(F^2)$.
• Figure 3: Mass functions for (a) Gaussian Shape, (b) Double Gaussian shape, (c) Semi-circle shape, (d) Triangle shape phase diagram (Inset figure in each figure).
• Figure 4: The phase diagram and the mass function $M(F^2)$ for (a) a Gaussian shape, (b) two overlapping Gaussians, (c) a semicircular shape, and (d) a triangular shape superconductor dome. The achieved loss values (\ref{['loss']}) are $7.53\times 10^{-5}, 4.81\times 10^{-4}, 2.51\times 10^{-4}$, and $1.07\times 10^{-3}$ for (a), (b), (c), and (d), respectively. The red curves denote the artificial phase borderlines and the blue curves show the borderlines obtained using our PINN method.
• Figure 5: (a) Temperature dependence $\Delta\mathcal{F}$ of the free energy difference for $M(F^2)= -1 +\frac{3}{2}F^2 + \frac{3}{8} (F^2)^2$(red) and $M(F^2) = -4 e^{-(F^2+2)^2/2}$ (blue) to the RN black hole. (b) The condensation curves of the scalar operators for (a).