Holographic Learning from Fermionic Spectra: Application to Strange Metal Phenomenology

Abstract

We develop a data-driven framework based on Neural ODEs that learns the effective bulk metric functions and the charge-weighted gauge potential $qA_t$ of a static, planar-symmetric black hole from boundary fermionic spectral functions. After validating the framework on the Einstein--Maxwell and Gubser--Rocha models with high accuracy, we apply it to the nodal strange-metal phenomenology of the cuprate $\mathrm{(Pb,Bi)_2Sr_{2-x}La_xCuO_{6+\delta}}$ within a semi-holographic setting, taking as input the normalized target generated from the extended power-law liquid (PLL) model calibrated by angle-resolved photoemission measurements. A key structural observation is that our probe fermion is massless and therefore insensitive to the conformal factor, leading to a coordinate/Weyl redundancy, while spectral normalization introduces a degeneracy in the scaled Hawking temperature. After identifying these sources of nonuniqueness, we find that, at low temperatures and near-optimal doping, the normalized extended-PLL target can be well described by a family of effective geometries close to the conformal-to-$\mathrm{AdS}_{2}\times\mathbb{R}^{2}$ black-hole class, with a nearly vanishing $qA_t$ ($\sim10^{-4}\,\mathrm{eV}$). The conformal-factor ambiguity further implies that fixing macroscopic thermodynamics such as the electronic specific heat requires independent input beyond the fermionic spectra. We also examine the applicability of our framework across doping and temperature: at low temperatures, the learned effective model remains viable throughout the studied doping range, with only a mild increase in loss toward the overdoped side; at higher temperatures, however, both the loss and $qA_t$ increase substantially.