Baryon-charge Chemical Potential in AdS/CFT

TL;DR

This work provides a gauge-invariant AdS/CFT framework for finite $U(1)_B$ chemical potential using flavor D7-branes in the D3-D7 system. The chemical potential is defined as the bulk electric flux, $oldsymbol{} = $int d ho F_{ ho t}$, with the grand potential and the Legendre-transformed canonical free energy derived consistently within this formulation. Numerical results demonstrate that both Minkowski and black-hole D7-brane embeddings are required to cover the full thermodynamic parameter space and to ensure stability, with Maxwell constructions enforcing thermodynamic consistency across first-order transitions. The authors interpret Minkowski embeddings as physically essential for low-$T$ and low-$$ physics and discuss an external charged-source picture, as well as possible extensions to include baryon vertices. Altogether, the paper clarifies how to holographically describe baryon-like density while resolving previous incomplete-ness issues in finite-density holography.

Abstract

We present a closed framework of AdS/CFT with finite U(1)B-charge chemical potential. We show how the gauge-invariant identification of the chemical potential with the bulk gauge field emerges from the standard AdS/CFT dictionary. Physical importance and necessity of the Minkowski embeddings within the present framework is also shown numerically in the D3-D7 systems. We point out that the D3-D7 model with only the black-hole embeddings does not have the low-temperature and low-chemical-potential region in the grand-canonical ensemble, hence it is incomplete. A physical interpretation that explains these numerical results is also proposed.

Figures (10)

• Figure 1: (a) An example of $\mu$-$Q$ diagram. The shaded regions indicate the Maxwell construction in the canonical ensemble. (b) Determination of the critical densities from $F$. We observe two phase transitions in this case. (c) The same $\mu$-$Q$ diagram can be used in the grand-canonical ensemble. The shaded regions indicate the Maxwell construction in the grand-canonical ensemble. (d) Determination of the critical value of $\mu$ from $\Omega$.
• Figure 2: ${ \mu}$-$Q$ diagrams at various temperature.
• Figure 3: The structure of the D7-brane solutions at $\mu=0.01$ fixed. Solid line: black-hole embeddings, dashed line: Minkowski embeddings.
• Figure 4: (a) The phase diagram in the canonical ensemble. The shaded region is where the number-density instability presents. (b) The region of two phase transitions exist is zoomed.
• Figure 5: Phase diagram in the grand-canonical ensemble