Neural Ordinary Differential Equations for Mapping the Magnetic QCD Phase Diagram via Holography

TL;DR

This work addresses mapping the QCD phase diagram in the space of temperature $T$, baryon chemical potential $\mu_B$, and magnetic field $B$ by learning the holographic action from lattice QCD data using neural ordinary differential equations. The authors construct a (2+1)-flavor holographic QCD model with a neural ODE-based determination of the magnetic coupling $\hat{Z}(\phi)$, enabling a precise fit to lattice data and predictive mapping of the phase structure under finite $B$. They discover a nontrivial phase structure at strong magnetic fields, including multiple critical endpoints in the $T$-$\mu_B$ plane and a turning-point in $T_C(B)$, with critical exponents that depend on the CEP location and even violate conventional scaling relations at high $B$. These results provide concrete, testable predictions for upcoming facilities such as FAIR, JPARC-HI, and NICA, and illustrate a powerful integration of holography and machine learning for nonperturbative QCD phenomenology under extreme conditions.

Abstract

The QCD phase diagram is crucial for understanding strongly interacting matter under extreme conditions, with major implications for cosmology, neutron stars, and heavy-ion collisions. We present a novel holographic QCD model utilizing neural ordinary differential equations (ODEs) to map the QCD phase diagram under magnetic field $B$, baryon chemical potential $μ_B$, and temperature $T$. By solving the inverse problem of constructing a gravitational theory from Lattice QCD data, we reveal an unprecedentedly rich phase structure at finite $B$, including multiple critical endpoints (CEPs) in strong magnetic fields. Specifically, for {$B = 1.618 \, \mathrm{GeV}^2=2.592 \times 10^{19}$ Gauss}, we identify two distinct CEPs at $(T_C = 87.3 \, \mathrm{MeV}, \, μ_C = 115.9 \, \mathrm{MeV})$ and $(T_C = 78.9 \, \mathrm{MeV}, \, μ_C = 244.0 \, \mathrm{MeV})$. Notably, the critical exponents vary depending on the CEP's location, and the conventional scaling relations can be violated in the presence of strong magnetic fields. These findings significantly advance our understanding of the QCD phase structure and provide concrete predictions for experimental validation at upcoming facilities such as FAIR, JPARC-HI, and NICA.

Figures (11)

• Figure 1: Schematic of the neural ODE approach for solving the inverse problem in QCD phase diagram analysis. Initial conditions from the black hole (BH) horizon are connected to QCD boundary data via EoMs. The neural network initialized as a trial function for the coupling $\hat{Z}(\phi)$ is optimized via backpropagation to minimize the loss function $\text{L}$, representing the difference from lattice QCD data. Parameters $\xi$ are optimized using gradient descent. Technical details are provided in Appendix B.
• Figure 2: The magnetic coupling $\hat{Z}(\phi)$ as a function of $\phi$ from machine learning. The black solid curve is the one obtained from our neural ODEs architecture, and the red dotted one from the fitting function \ref{['myfit']} in Appendix B.
• Figure 3: Thermodynamic Quantities from Holographic QCD Model vs. Lattice Data. Temperature dependence of (a) magnetic susceptibility $\chi_B$, (b) magnetization $M$, (c) entropy density $s/T^3$, and (d) longitudinal pressure $\Delta p_z = p_z|_B - p_z|_{B=0}$ across magnetic fields. The shaded regions show lattice QCD estimates Bali:2014kia; the solid lines indicate model predictions. Here, $e = 1$, giving $B = 1 \, \text{GeV}^2 = 1.602 \times 10^{19}$ Gauss.
• Figure 4: The phase diagram on the $B$-$T$ plane at vanishing $\mu_B$. The blue dot denotes the CEP, and the blue line corresponds to the first-order line.
• Figure 5: QCD phase diagram at finite magnetic field $B$. Phase structure in temperature $T$, baryon chemical potential $\mu_B$, and magnetic field $B$ from our holographic model. The light blue surface denotes the first-order transition boundary, separating the hadronic phase from the quark-gluon plasma. The dark blue line traces the CEP trajectory, marking where the first-order transition ends in a crossover.