Finite Temperature Large N Gauge Theory with Quarks in an External Magnetic Field

TL;DR

This work probes finite-temperature, large-N gauge theories with fundamental quarks in a constant external magnetic field using holography, implementing a D7-brane probe in the AdS5-Schwarzschild × S5 background. By analyzing brane embeddings (Minkowski vs black-hole) and solving the D7 equations of motion both analytically at large mass and numerically, the authors map a magnetic-field–dependent phase diagram, identifying a critical η_cr ≈ 7.89 at which the meson-melting transition disappears and chiral symmetry is spontaneously broken at m=0. They compute thermodynamic quantities (free energy, entropy, magnetization, speed of sound) and study the low-lying meson spectra, including a GMOR-type Goldstone mode and Zeeman-like splittings, finding that a strong magnetic field enforces a discrete meson spectrum in the melted phase and enhances magnetization and sound speed. The results corroborate magnetic catalysis in a strongly coupled plasma and provide quantitative, holographically derived benchmarks relevant to QCD-like dynamics in magnetic environments, with potential connections to heavy-ion phenomenology and lattice studies. All findings are expressed in a controlled holographic setup with large-N, small-Nf, and explicit η-dependence, offering insight into the non-perturbative interplay between temperature, magnetic fields, and fundamental flavors.

Abstract

Using a ten dimensional dual string background, we study aspects of the physics of finite temperature large N four dimensional SU(N) gauge theory, focusing on the dynamics of fundamental quarks in the presence of a background magnetic field. At vanishing temperature and magnetic field, the theory has N=2 supersymmetry, and the quarks are in hypermultiplet representations. In a previous study, similar techniques were used to show that the quark dynamics exhibit spontaneous chiral symmetry breaking. In the present work we begin by establishing the non-trivial phase structure that results from finite temperature. We observe, for example, that above the critical value of the field that generates a chiral condensate spontaneously, the meson melting transition disappears, leaving only a discrete spectrum of mesons at any temperature. We also compute several thermodynamic properties of the plasma.

Figures (16)

• Figure 1: The solid curve starting far left (red) represents solutions falling into the black hole, the dotted (blue) curve represents solutions with shrinking $S^3$. The vertical dashed line corresponds to the critical value of $\tilde{m}$ at which the first order phase transition takes place. The solid black curve dropping sharply from above is the function derived in equation (\ref{['weakcond']}), corresponding to the large mass limit.
• Figure 2: The area below the $(-{\tilde{c}}, {\tilde{m}})$ curve has the interpretation of the free energy of the D7--brane; thus the phase transition pattern obeys the "equal area law"--- the area of the shaded regions is equal.
• Figure 3: The effect of the weak magnetic field is to decrease the values of $\tilde{m}_{\rm cr}$ and the condensate. Equation (\ref{['weakcond']}) is still a good approximation for $\tilde{m}>\tilde{m}_{\rm cr}$.
• Figure 4: For strong magnetic field the condensate is negative. The value of $\tilde{m}_{\rm cr}$ continues to drop as we increase $\eta$.
• Figure 5: For sufficiently high values of $\eta$ there are states with negative $\tilde{m}$, which are considered non-physical. However the equal area law is still valid as long as $m_{\rm cr}>0$.