Learning Effective Bulk Fields from Fermionic Spectra: Application to an Extended Power-Law Liquid Model for Cuprate Strange Metals

arXiv:2607.02861 · hep-th, cond-mat.str-el, cond-mat.supr-con, gr-qc · Submitted 2026-07-03 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Learning Effective Bulk Fields from Fermionic Spectra".

Mira: The gist: A data-driven framework based on Neural ODEs learns effective bulk metric functions and charge-weighted gauge potential from boundary fermionic spectral functions to probe nodal strange-metal phenomenology in cuprates.

Kai: First, who's behind it and why it matters.

Paper summary: Mira: So looking at the title "Learning Effective Bulk Fields from Fermionic Spectra: Application to an Extended Power-Law Liquid Model for Cuprate Strange Metals," it really points toward a data-driven approach where we learn the bulk geometry directly from experimental spectral data.

Kai: That's right, and it connects those microscopic properties of cuprates, like linear-temperature resistivity or the Hall angle cotangent theta H proportional to T squared, to macroscopic gravitational models like black holes <ref:2607.02861#pg1>.

Lev: It means we’re essentially using gravity as a language to describe things that are notoriously hard to pin down microscopically in these strongly correlated electron systems.

Mira: The paper shows that at low temperatures and near-optimal doping, the results align well with geometries close to the conformal-to- AdS two times R squared black hole class with a nearly vanishing qAt around ten-four electron volts.

Kai: That vanishing charge-weighted gauge potential is a specific finding that simplifies things, suggesting a very simple structure in the effective bulk for those conditions.

Lev: But they also pointed out that the scaled Hawking temperature isn't fixed by the fermionic spectra, and you still need independent experimental input to fix (one T) at the horizon <ref:2607.02861#pg2>.

Mira: So what this paper changes for us is showing how holographic methods can be used to provide an effective bulk description of specific cuprate targets under certain conditions.

Kai: It gives us a concrete tool for testing whether these geometric ideas are applicable to other strongly correlated electron systems beyond just cuprates, like heavy-fermion compounds.

Lev: And the caveat is that at higher temperatures, both the loss and qAt increase substantially, and they can’t yet reproduce the thermal spectral shape while keeping frequency symmetry.

Mira: So we aren't getting a final picture of these geometries yet; we need to keep looking at those other observables they mentioned, like thermodynamic quantities.

Kai: This whole effort with "Learning Effective Bulk Fields from Fermionic Spectra: Application to an Extended Power-Law Liquid Model for Cuprate Strange Metals" is about using data-driven learning to uncover the hidden geometric structure of strange metals.

Lev: It’s about showing that we can build a framework where boundary fermionic spectra actually guide us toward an effective bulk metric and gauge potential, even if the probe fermion is massless.

Mira: And it opens up avenues for connecting condensed matter phenomena with gravitational theory in a way that's non-perturbative and handles strong correlations better than traditional methods.

Conclusion: Kai: So, we're wrapping up this look at that paper, "Learning Effective Bulk Fields from Fermionic Spectra: Application to an Extended Power-Law Liquid Model for Cuprate Strange Metals." It basically shows how you can use neural networks and these neural ordinary differential equations to figure out the underlying geometry—the effective bulk metric—just by looking at the fermionic spectra you measure on your boundary.

Mira: Right, and what's interesting is that they used this framework to test a specific physical model, an extended power-law liquid, which is meant to mimic what we see in cuprates called strange metal behavior. They found that for low temperatures and near optimal doping, the results fit a black hole geometry class very well.

Lev: From an error correction standpoint, I’m interested in that vanishing charge-weighted gauge potential they found, qAt being around ten to the power of negative four electron volts. That means for those specific conditions, the complexity of the effective bulk simplifies significantly.

Kai: So what does that simplification mean for us? It suggests there’s a very specific, simple structure hidden in that material when it's behaving that way.

Mira: Exactly. But they also highlighted some limitations, like how the scaled Hawking temperature doesn't get fixed by the fermionic spectra alone. They need independent data—like macroscopic thermodynamics—to pin down the conformal factor at the horizon of the black hole model.

Lev: And for me, that means if you want to build a real error correction code based on this idea, you can’t just rely on what’s derived from the single-particle spectrum. You need to feed in that extra thermodynamic input to constrain the model properly.

Kai: It sounds like it's not a complete picture yet for those high-temperature regimes where things get messy and the loss function starts increasing.

Mira: That’s true, and they noted that at higher temperatures, both the loss value and that qAt start getting much bigger. The reconstructed spectrum also develops a component that is odd in frequency when you look at unoccupied states above zero energy.

Lev: So it’s a viable model for low-T, near-optimal doping strange metals, but it’s not yet robust enough to handle the full thermal behavior across the whole doping range without more constraints.

Kai: It opens up a way to connect these microscopic measurements directly to gravitational theories in a semi-holographic setting.

Mira: It definitely shows that these geometric ideas aren't just abstract math; they can actually give us a concrete, data-driven description of what might be happening inside those complex materials.

Lev: The next step for this kind of research is looking at how to relax those strict assumptions they made—like the conformal boundary condition—and see if we can incorporate more observables like Hall angle scaling to tighten up the constraints on that geometry.

Department of Physics, Shanghai University · Center for Gravitation and Cosmology, Yangzhou University · Department of Physics, Freie Universite Berlin

hep-th, cond-mat.str-el, cond-mat.supr-con, gr-qc

Submitted: 2026-07-03

Updated: 2026-10-19

Comments: 46 pages, 10 figures, 3 tables, accepted by JHEP

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 80/100

The gist: The gist: A data-driven framework based on Neural ODEs learns effective bulk metric functions and charge-weighted gauge potential from boundary fermionic spectral functions to probe nodal

Key concepts

Neural-ODE Framework: This is a machine learning method that uses ordinary differential equations (ODEs) to model complex physical processes. It learns how a system evolves by fitting data points, allowing researchers to infer underlying physical properties like the bulk metric from boundary measurements.
Effective Bulk Metric and Gauge Potential: These are mathematical descriptions of the spacetime geometry (metric) and electromagnetic fields (gauge potential) within the black hole's interior. The framework attempts to learn these functions directly from how electrons behave near the boundary, rather than calculating them from first principles.
Extended Power-Law Liquid Model: This is a specific theoretical model used as a target for testing. It describes how particles behave in strongly correlated systems like cuprates at low temperatures. The framework was trained to match the spectral features predicted by this specific liquid model.
Spectral Extraction Formula: A mathematical tool derived from the conserved radial flux of a massless probe field. This formula allows researchers to extract crucial information about the spectral function (how electrons behave) directly from measurable data in a semi-holographic setting.

Terminology

Accuracy — 1× in this paper · explained in 1 episode
Degeneracy — 1× in this paper · explained in 4 episodes
Redundancy — 1× in this paper · explained in 2 episodes

Summary

The gist: A data-driven framework based on Neural ODEs learns effective bulk metric functions and charge-weighted gauge potential from boundary fermionic spectral functions to probe nodal strange-metal phenomenology in cuprates.

How it works

The framework develops a data-driven approach using Neural ODEs to learn the effective bulk metric and the charge-weighted gauge potential qAt of a static, planar-symmetric black hole from boundary fermionic spectral functions. This is achieved by coupling a weakly coupled UV electron sector to a strongly coupled IR sector represented by a composite operator. The physical electron's Green’s function is then given by GΨΨ(ω, k) = ħ−ħω + ϵ(k) − µ + Σ(ω, k), from which the electron self-energy reads as Σ(ω, k) = −g2kGk(ω).

The training protocol involves a four-stage process to screen multiple initializations and refine the lowest-loss candidates. The loss function adopted is the grid-averaged Itakura–Saito divergence, L = 1/2NDX(ω,k)∈DχNODE(ω, k)χDATA(ω, k)− 1 − ln χNODE(ω, k)χDATA(ω, k)>. The training is performed by integrating the complex-valued master equations into Neural-ODE form using an explicit adaptive Runge–Kutta solver.

Key Structural Observations and Findings

The study identifies two distinct sources of nonuniqueness in the inverse problem: the massless Dirac probe is insensitive to the conformal factor omega(z), leaving a coordinate/Weyl redundancy in the learned metric functions. Spectral normalization introduces a degeneracy in the scaled Hawking temperature. 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-AdS2 × R2 black hole class, with a nearly vanishing qAt (∼ 10−4 eV).

The learned effective model remains viable throughout the studied doping range at low temperatures, with only a mild increase in loss toward the overdoped side. At higher temperatures, both the loss and qAt increase substantially. The reconstructed spectrum develops a sizable ω-odd component when extrapolated to unoccupied states (ω > 0) at high temperatures.

Geometric Equivalence and Temperature Degeneracy

The massless probe is insensitive to the conformal factor, leading to a coordinate/Weyl redundancy in the learned metric functions. The gauge-invariant information that the low-temperature spectra support reduces to the spatial factor h0 ≈ (kF /α)2 and the near-vanishing of qAt, while the specific member of that class (its TH) and the conformal factor omega remain undetermined. The successful fit supports using the framework as an effective bulk description of the normalized extended-PLL target for the UD32K sample at low temperatures.

The temperature degeneracy holds for any regular qAt profile transported by Eq. (E.6), with qAt = 0 as the simplest special case. The learned constant h0 fixes the momentum scaling exponent, while the remaining radial profile is encoded in the gauge-invariant blackening factor F(u); however, the scaled Hawking temperature is not fixed by the fermionic spectra.

Doping and Temperature Dependence

The learned effective model remains viable throughout the studied doping range at low temperatures, with only a mild increase in loss toward the overdoped side. At elevated temperatures, both the loss and qAt increase substantially. The gauge-potential profile develops pronounced oscillations; the final-candidate profiles become more dispersed; and the reconstructed spectrum develops a sizable ω-odd component when extrapolated to unoccupied states (ω > 0) at high temperatures.

Future Directions

A natural next step is to relax the assumptions that define the present inverse problem: the strict conformal AdS2×R2 boundary condition, the minimal massless Dirac dynamics of the probe fermion, and the clean factorization of the non-holographic contributions G0(ω) and gk. Complementary observables, such as macroscopic thermodynamics and Hall-angle scaling, could then be incorporated to constrain the conformal factor and test whether the same effective geometry controls both single-particle spectra and transport. Applying the framework to other strongly correlated systems, including heavy-fermion compounds, and exploring the hidden gravitational dynamics would further clarify the scope and limitations of holographic and semi-holographic methods in condensed matter physics.

Unit Conventions

The dimensionless bulk quantities admit a well-defined mapping to the dimensionful experimental observables in SI units. The scaled Hawking temperature is defined as TH = −f′(1)/4π. The bulk frequency ω is related to the dimensionless boundary ratio ħω/(kBT) via [124].

Flux-Based Extraction

The extraction formula (3.13) for a massless Dirac field on a real static background probes the conserved radial flux, which fixes the spectral function Im Gk without isolating the subleading response coefficient. This construction manifestly respects spectral positivity.

Gauge Fixing

The learned pair is not separately physical, and a change of radial coordinate, z → u(z), accompanied by a Weyl rescaling can trade content between f and h without changing the fermionic observable. The gauge-fixed data is defined by the gauge-invariant content that survives it.

Conclusion

The framework successfully demonstrates 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-AdS2 × R2 black hole class, with a nearly vanishing qAt (∼ 10−4 eV). The learned constant h0 fixes the momentum scaling exponent, while the remaining radial profile is encoded in the gauge-invariant blackening factor F(u); however, the scaled Hawking temperature is not fixed by the fermionic spectra. The conformal-factor ambiguity further implies that predicting macroscopic thermodynamics—such as the entropy density or the electronic specific heat—requires independent experimental input to fix omega(1, T) at the horizon.

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Improvements for AI systems

Bold Header: Data-Driven Inverse Mapping of Bulk Physics to Spectral Functions. This improved system can perform the complementary inverse problem of reconstructing the bulk physics from boundary data, specifically learning the effective metric profiles and the charge-weighted gauge potential qAt of a static, planar-symmetric black hole from boundary fermionic spectra.
Bold Header: Accurate Reconstruction of Strange Metal Phenomenology across Doping and Temperature. The system can identify effective geometries for cuprates like the (Pb, Bi)2Sr2−xLaxCuO6+δ within a semi-holographic setting, successfully mapping the normalized extended-PLL target against the learned models across a range of doping and temperature.
Bold Header: Identification of Geometric Redundancies and Ambiguities. The system can explicitly quantify that the massless Dirac probe is insensitive to the conformal factor, leading to a coordinate/Weyl redundancy, and that spectral normalization introduces a degeneracy in the scaled Hawking temperature, allowing researchers to distinguish between physical constraints (like fixing macroscopic thermodynamics) and model ambiguities.
Bold Header: Constraint-Based Model Selection for Physical Interpretation. The system can utilize a four-stage protocol (Seed Model Selection, Variant Generation, Deep Optimization, Precision Polishing) to select the most robust bulk solutions, leading to learned metrics that are close to the analytical solutions with high precision.
Bold Header: Verification of Temperature Degeneracy and Robustness. The system can test whether the approximate temperature degeneracy is not confined to UD32K by repeating reconstructions with a fixed hyperparameter for TH, demonstrating that the learned effective model remains viable throughout the studied doping range and confirming that "the reconstructed spectrum develops a sizable ω-odd component when extrapolated to unoccupied states (ω > 0)" at elevated temperatures.