Complexity Growth in Flavor-Dependent Systems

TL;DR

This work analyzes holographic complexity growth in a flavor-dependent Einstein-Maxwell-Dilaton model by computing the time derivative of the Nambu-Goto action for a probe string on the Wheeler-DeWitt patch, in the CA framework. The background parameters for different flavor contents are determined via machine learning trained on lattice QCD EoS and baryon susceptibility, yielding EMD solutions that reproduce lattice thermodynamics. The NG action growth is explored as a function of string velocity $v$, chemical potential $\mu$, temperature $T$, and flavor number, showing maximum growth at $v=0$, enhancement with $\mu$ and $T$, and flavor-driven suppression at $\mu=0$; multi-valued behavior near first-order transitions and single-valued behavior in crossovers indicate that complexity growth can diagnose phase transitions in QCD-like systems. The work provides a concrete ML-enabled holographic framework for QCD thermodynamics and offers predictions for how complexity evolves across phase boundaries, with potential extensions to magnetic fields and rotation.

Abstract

In this work, we investigate holographic complexity growth in a flavor-dependent Einstein-Maxwell-Dilaton (EMD) model, where the parameters are determined through machine learning algorithms fitted to lattice QCD equation of state (EoS) and baryon number susceptibility data. Within the Complexity=Action (CA) conjecture, we introduce a probe string into the bulk geometry and evaluate the time derivative of its Nambu-Goto (NG) action on the Wheeler-DeWitt (WDW) patch as the holographic dual of complexity growth. Our analysis explores the dependence of complexity growth on string velocity, chemical potential, temperature, and the number of flavors. Results show maximum complexity growth for stationary strings, decreasing with string velocity. At zero chemical potential, complexity growth is largest in the pure gluon system and reduces with the addition of quark flavors. Increasing temperature and chemical potential consistently enhance complexity growth. Furthermore, complexity growth exhibits multi-valued behavior in regions corresponding to first-order transitions and single-valued behavior in crossover regimes, indicating that complexity can serve as a probe for phase transitions.

Figures (5)

• Figure 1: Holographic EoS (lines) compared with lattice QCD results for systems (symbols with error bar) with different flavors Chen:2024mmd.
• Figure 2: Action growth versus string velocity for different flavors at $T= 0.6GeV$.
• Figure 3: Action growth versus string velocity for different chemical potentials at $T= 0.2GeV$ for the $N_{f} = 2$, $N_{f} = 2+1$, and $N_{f} = 2+1+1$ flavor systems.
• Figure 4: Action growth versus temperature at vanishing chemical potential for different flavors with $\upsilon=0$.
• Figure 5: Action growth versus temperature at finite chemical potential for the $N_{f} = 2$, $N_{f} = 2+1$, and $N_{f} = 2+1+1$ flavor systems with $\upsilon=0$.