Chemical Potential and Charged Black Branes

In holography, the boundary chemical potential is the gauge-invariant potential difference between the AdS boundary and a regular horizon, while the charge density is the conserved radial electric flux with a normalization fixed by the bulk Maxwell term. Dirichlet and fixed-flux boundary conditions define grand-canonical and canonical ensembles; using the same metric with different gauge boundary terms changes the thermodynamic question. A broad derivation of these finite-density conventions is given by Hartnoll 2009.

Required background. Chemical Potentials and Finite-Density Ensembles fixes the thermodynamics, and AdS Black Branes and Holographic Thermodynamics supplies the neutral saddle.

Helpful background. Thermodynamic Limits, Phases, and Ensemble Equivalence distinguishes ensemble stability; Currents, Stress Tensor, and Bulk Gauge and Metric Fields fixes the current dictionary.

Maxwell flux and horizon regularity

Take

$$S=\frac{1}{2\kappa^2} \int d^{d+1}x\sqrt{\lvert g\rvert} \left[ R+\frac{d(d-1)}{L^2} -\frac{L^2}{4g_F^2}F_{MN}F^{MN} \right].$$

For a homogeneous electric ansatz, Maxwell’s equation gives the radially conserved quantity

$$\mathcal Q =-\frac{L^2}{2\kappa^2g_F^2} \sqrt{\lvert g\rvert}\,F^{rt}, \qquad \partial_r\mathcal Q=0.$$

With this sign convention, the renormalized boundary charge density is $\rho=\mathcal Q$. For a planar black brane,

$$A_t(r)=\mu-\frac{Q}{r^{d-2}}, \qquad A_t(r_h)=0 \quad\Longrightarrow\quad \mu=\frac{Q}{r_h^{d-2}}.$$

The last condition is regularity of the Euclidean one-form at the contractible thermal circle, or equivalently regularity in a horizon gauge. A constant shift of $A_t$ changes neither $F$ nor physics, so $\mu$ is the potential difference, not the unqualified boundary value.

Charged equation of state

The Einstein equation yields

$$f(r)=\frac{r^2}{L^2} -\frac{m}{r^{d-2}} +\alpha_d\frac{Q^2}{r^{2d-4}},$$

where $\alpha_d$ is fixed by $\kappa/g_F$ and the definition of $Q$. The horizon condition $f(r_h)=0$ eliminates $m$, and

$$T=\frac{f'(r_h)}{4\pi}, \qquad s=\frac{2\pi}{\kappa^2} \left(\frac{r_h}{L}\right)^{d-1}.$$

Holographic renormalization gives the pressure $p=-\Omega/V$, energy density, and charge. They must satisfy

$$d p=s\,dT+\rho\,d\mu, \qquad \epsilon+p=Ts+\mu\rho$$

for a conformal homogeneous state. These identities are stronger checks than fitting $f(r)$ alone.

For an explicit first application, set $d=3$, $L=g_F=1$, and use $z=1/r$ with boundary at $z=0$:

$$ds^2=\frac{1}{z^2}\left[f(z)dt^2-\frac{dz^2}{f(z)}-dx^2-dy^2\right], \qquad A_t=\mu\left(1-\frac{z}{z_h}\right),$$ $$f(z)=1-\left(1+q^2\right)\left(\frac{z}{z_h}\right)^3 +q^2\left(\frac{z}{z_h}\right)^4, \qquad q^2=\frac{\mu^2z_h^2}{4}.$$

With $C_J=1/(2\kappa^2)$ in these units,

$$T=\frac{3-q^2}{4\pi z_h}, \qquad \rho=C_J\frac{\mu}{z_h}.$$

The fixed-temperature susceptibility must include the implicit motion of the horizon:

$$\chi_T =\left(\frac{\partial\rho}{\partial\mu}\right)_T =\left(\frac{\partial\rho}{\partial\mu}\right)_{r_h} +\left(\frac{\partial\rho}{\partial r_h}\right)_\mu \left(\frac{\partial r_h}{\partial\mu}\right)_T.$$

Equivalently differentiating with $z_h$ gives

$$\left(\frac{\partial z_h}{\partial\mu}\right)_T =-\frac{\mu z_h^3/2}{3+\mu^2z_h^2/4}, \qquad \chi_T=\frac{C_J}{z_h} \frac{3+3\mu^2z_h^2/4}{3+\mu^2z_h^2/4}>0.$$

Thus the horizon gauge $A_t(z_h)=0$ and the equation of state together determine a positive grand-canonical susceptibility on this branch. In a different Maxwell normalization, both $q(\mu)$ and $C_J$ must be changed consistently.

The implicit horizon response is essential. Charged AdS thermodynamics and ensemble-dependent stability were analyzed systematically by Chamblin et al. 1999.

Canonical versus grand-canonical adversary

Dirichlet boundary data $\delta A_t^{(0)}=0$ define fixed $\mu$. Fixed electric flux requires a Legendre boundary term and defines fixed $\rho$. The appropriate Hessians are different:

$$\chi_T=\left(\frac{\partial\rho}{\partial\mu}\right)_T>0 \quad\text{versus}\quad \left(\frac{\partial\mu}{\partial\rho}\right)_T>0.$$

A branch can be stable or dominant in one ensemble and not the other. Reusing the Euclidean on-shell action without the Legendre transform is the adversarial error.

Evidence ceiling and handoff

The calculation establishes finite-density thermodynamics for the specified bulk action, gauge normalization, boundary condition, and saddle. It does not identify the boundary charge with electrons, baryon number, or any material current without an independent microscopic dictionary.

Thermal and Nonequilibrium QFT owns finite-density thermodynamics; the thermal black-hole chapter owns phase structure; later pages add charged matter and transport.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Chamblin, Andrew; Emparan, Roberto; Johnson, Clifford V.; and Myers, Robert C. “Charged AdS Black Holes and Catastrophic Holography.” Physical Review D 60, 064018 (1999). doi:10.1103/PhysRevD.60.064018.
• Hartnoll, Sean A. “Lectures on Holographic Methods for Condensed Matter Physics.” Classical and Quantum Gravity 26, 224002 (2009). doi:10.1088/0264-9381/26/22/224002.