Holographic Diffusion, Conductivity, and Momentum Relaxation

At finite density, electric current overlaps with momentum. Exact translations then produce an infinite dc conductivity even when an incoherent current dissipates. A holographic transport calculation must distinguish those contributions, declare how translations are relaxed, and use the longitudinal fluctuation problem—not the optical limit alone—to establish diffusion.

Required background. Kubo Formulae and Horizon Response supplies the retarded and horizon-flux prescriptions. Chemical Potential and Charged Black Branes supplies charge density and ensemble normalization.

Helpful background. Transport Extraction, Inverse Problems, and Error Budgets supplies fit and uncertainty controls. Diffusion, Conductivity, and Susceptibility supplies the field-theory Einstein relations.

Momentum, current, and the Drude pole

At nonzero charge density $\rho$, hydrodynamics gives an electric current of the form

$$J^i=\rho u^i+\sigma_Q\left(E^i-T\nabla^i\frac{\mu}{T}\right)+\cdots .$$

If momentum is conserved, a homogeneous electric field accelerates the fluid. The optical conductivity contains

$$\sigma(\omega)=\sigma_Q+\frac{\rho^2}{\epsilon+p}\frac{i}{\omega+i0^+}+\cdots,$$

so $\operatorname{Re}\sigma$ has a delta function. With weak momentum relaxation rate $\Gamma$,

$$\sigma(\omega)=\sigma_Q+\frac{\rho^2}{\epsilon+p}\frac{1}{\Gamma-i\omega}+\cdots .$$

The incoherent term and the momentum-drag term are physically distinct even when a single Drude fit approximates both over a narrow interval.

A homogeneous axion model

For a concrete first application, take four-dimensional bulk units $L=1$, $2\kappa^2=1$, and Maxwell coupling $g_F=1$:

$$S=\int d^4x\sqrt{\lvert g\rvert}\left[R+6-\frac14F^2 +\frac12\sum_{I=1}^{2}(\partial\phi_I)^2\right], \qquad \phi_1=kx,\quad\phi_2=ky .$$

The scalar sources break translations while preserving a homogeneous stress tensor. Solving the coupled, zero-frequency perturbations and constructing radially conserved electric and heat currents gives Donos and Gauntlett 2014

$$\sigma_{\mathrm{dc}}=1+\frac{4\pi\rho^2}{k^2s}, \qquad \alpha_{\mathrm{dc}}=\frac{4\pi\rho}{k^2}, \qquad \bar\kappa_{\mathrm{dc}}=\frac{4\pi sT}{k^2}.$$

Here $J=\sigma E-\alpha\nabla T$ and $Q=T\alpha E-\bar\kappa\nabla T$ define the thermoelectric signs, and $\bar\kappa$ is measured at zero electric field. The open-circuit thermal conductivity is

$$\kappa=\bar\kappa-\frac{T\alpha^2}{\sigma}.$$

Different bulk dimensions, scalar normalizations, gauge couplings, or nonlinear axion actions change the formula. The parameters $(T,\mu,k)$ and these normalizations must accompany any numerical value.

The background and its optical response were constructed explicitly by Andrade and Withers 2014. The horizon formula is a dc statement; the Drude width and non-Drude modes require solving the frequency-dependent coupled system with infalling boundary conditions.

Diffusion and susceptibility

At zero density, charge fluctuations decouple from momentum and the longitudinal pole obeys

$$\omega(k)=-iDk^2+O(k^4), \qquad D=\frac{\sigma_{\mathrm{dc}}}{\chi}, \qquad \chi=\left(\frac{\partial\rho}{\partial\mu}\right)_T .$$

At finite density with energy mixing, diffusion is matrix-valued. The eigenvalues of the conductivity matrix times the inverse susceptibility matrix give the diffusive constants only after momentum is either relaxed or projected out. Susceptibilities must be computed in the same ensemble and current normalization as the conductivities.

The limit orders are different:

• optical transport sets $k=0$ and then takes $\omega\to0^+$;
• static susceptibility sets $\omega=0$ and then takes $k\to0$;
• diffusion follows a pole with $\omega/k^2$ fixed as both vanish.

Conflating them can produce a numerically plausible but physically incorrect Einstein relation.

Translation-restoring adversary

Send $k\to0$ at fixed nonzero $\rho$. The formulas give

$$\sigma_{\mathrm{dc}}\sim\frac{4\pi\rho^2}{k^2s}\to\infty,$$

while the first term remains finite. This divergence is required by momentum conservation. If a computation at finite density instead approaches a finite total dc conductivity, it has either projected onto an incoherent current, retained another relaxation mechanism, or mishandled the zero-frequency limit.

A second check compares the pole extracted from optical conductivity with the lowest vector-channel quasinormal mode. Agreement at weak $k$ supports the Drude interpretation; disagreement signals additional slow modes or an invalid single-pole fit.

Exercises

Using the three horizon coefficients above, evaluate $\kappa_{\mathrm{dc}}$.

Solution

Substitution gives

$$\kappa_{\mathrm{dc}} =\frac{4\pi sT}{k^2} -\frac{T(4\pi\rho/k^2)^2}{1+4\pi\rho^2/(k^2s)} =\frac{4\pi s^2T}{k^2s+4\pi\rho^2}.$$

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

• Andrade, Tomás, and Benjamin Withers. “A Simple Holographic Model of Momentum Relaxation.” Journal of High Energy Physics 2014, 101 (2014). DOI.
• Donos, Aristomenis, and Jerome P. Gauntlett. “Thermoelectric DC Conductivities from Black Hole Horizons.” Journal of High Energy Physics 2014, 081 (2014). DOI.