Finite baryon and isospin chemical potential in AdS/CFT with flavor

TL;DR

The paper extends gauge/gravity duality to a thermal theory with both baryon and isospin chemical potentials by embedding $N_f$ D7-branes in the AdS-Schwarzschild background. It develops a detailed holographic setup, decouples the non-Abelian DBI action into Abelian sectors, and analyzes both canonical and grand canonical thermodynamics, revealing symmetry-driven phase structure and potential instabilities. It also computes vector spectral functions to track mesonic quasiparticles, uncovering a new low-energy excitation at large densities and an instability suggesting a meson-condensed phase. The results illuminate how finite densities shape embeddings, thermodynamics, diffusion, and meson spectra, offering quantitative predictions and a framework for exploring condensed-meson phases in holographic QCD-like theories.

Abstract

We investigate the thermodynamics of a thermal field theory in presence of both a baryon and an isospin chemical potential. For this we consider a probe of several D7-branes embedded in the AdS-Schwarzschild black hole background. We determine the structure of the phase diagram and calculate the relevant thermodynamical quantities both in the canonical and in the grand canonical ensemble. We discuss how accidental symmetries present reflect themselves in the phase diagram: In the case of two flavors, for small densities, there is a rotational symmetry in the plane spanned by the baryon and isospin density which breaks down at large densities, leaving a Z_4-symmetry. Finally, we calculate vector mode spectral functions and determine their dependence on either the baryon or the isospin density. For large densities, a new excitation forms with energy below the known supersymmetric spectrum. Increasing the density further, this excitation becomes unstable. We speculate that this instability indicates a new phase of condensed mesons.

Figures (32)

• Figure 1: Embedding function $L(r)$ of D$7$-branes in the AdS black hole background, with the dimensionless coordinates $L=\rho\cos\theta=\rho\chi$ and $r=\rho\sin\theta=\rho\sqrt{1-\chi^2}$.
• Figure 2: The dimensionless mass parameter $m$ as defined in equation \ref{['dicmc']} versus the horizon value $\chi_0=\lim_{\rho\to 1}\chi$ of the embedding at baryon density $\tilde{d}^B=0.5$ for the case $N_f=2$. The five different curves correspond to isospin density $\tilde{d}^I=0$ (black), $\tilde{d}^I=\frac{1}{4}\tilde{d}^B$ (green), $\tilde{d}^I=\frac{1}{2}\tilde{d}^B$ (blue), $\tilde{d}^I=\frac{3}{4}\tilde{d}^B$ (red) and $\tilde{d}^I=\tilde{d}^B$ (orange).
• Figure 3: The dimensionless chiral condensate $c$ versus the mass parameter $m$ as defined in equation \ref{['dicmc']} at baryon density $\tilde{d}^B=5\cdot 10^{-5}$ (a), $\tilde{d}^B=0.5$ (b) and $\tilde{d}^B=20$ (c) for the case $N_f=2$. The five different curves in each figure correspond to isospin density $\tilde{d}^I=0$ (black), $\tilde{d}^I=\frac{1}{4}\tilde{d}^B$ (green), $\tilde{d}^I=\frac{1}{2}\tilde{d}^B$ (blue), $\tilde{d}^I=\frac{3}{4}\tilde{d}^B$ (red) and $\tilde{d}^I=\tilde{d}^B$ (orange).
• Figure 4: The baryon (left) and isospin (right) chemical potential divided by the bare quark mass $M_q$ versus the mass parameter $m$ as defined in equation \ref{['dicmc']} at baron density $\tilde{d}^B=5\cdot 10^{-5}$ in the upper figures and $\tilde{d}^B=0.5$ in the lower ones for the case $N_f=2$. The five different curves in each figure correspond to isospin density $\tilde{d}^I=0$ (black), $\tilde{d}^I=\frac{1}{4}\tilde{d}^B$ (green), $\tilde{d}^I=\frac{1}{2}\tilde{d}^B$ (blue), $\tilde{d}^I=\frac{3}{4}\tilde{d}^B$ (red) and $\tilde{d}^I=\tilde{d}^B$ (orange).
• Figure 5: The dimensionless mass parameter $m$ as defined in equation \ref{['dicmc']} versus the horizon value of the black hole embeddings $\chi_0=\lim_{\rho\to 1}\chi$ between 0 to 1 and the asymptotic value of the Minkowski embeddings $L_0=\lim_{r\to 0}L$ between 1 and 2 at baryon chemical potential $\mu^B/M_q=0.01$ (a), $\mu^B/M_q=0.1$ (b) and $\mu^B/M_q=0.8$ (c) for the case $N_f=2$. The dotted purple curve corresponds to Minkowski embeddings and the five other curves to black hole embeddings with isospin chemical potential $\mu^I=0$ (black), $\mu^I=\frac{1}{4}\mu^B$ (green), $\mu^I=\frac{1}{2}\mu^B$ (blue), $\mu^I=\frac{3}{4}\mu^B$ (red) and $\mu^I=\mu^B$ (orange).