Kubo Formulae and Holographic Transport

A Kubo formula identifies a transport coefficient with a particular low-frequency, low-momentum limit of a retarded correlator. The operator channel, source normalization, order of limits, equilibrium ensemble, and contact or magnetization subtractions are part of the definition. Holography computes the correlator by solving an ingoing bulk boundary-value problem; horizon data often determine the dissipative coefficient because the canonical response becomes radially conserved in the hydrodynamic limit.

Required background. Gauge Theory and String-Theory Preparation supplies conserved currents, stress tensors, and operator/source normalization used below.

Helpful background. Quantum Gravity frames the classical bulk limit, Quantum Information distinguishes dissipative response from information recovery, Black-Hole Perturbation Theory supplies the ingoing spectral problem, and Gravitational Radiation and Waveforms provides complementary linear-response conventions.

Shear viscosity from the tensor channel

Perturb the boundary metric by $h_{xy}^{(0)}(t)$, which sources $T^{xy}$. For an isotropic translation-invariant state, the shear viscosity is

$$\eta =- \lim_{\omega\to0} \frac{1}{\omega} \operatorname{Im}G_R^{T^{xy}T^{xy}}(\omega,\mathbf 0).$$

The zero-momentum limit is taken before extracting the coefficient of $-i\omega$. With the Fourier convention $e^{-i\omega t}$, a positive viscosity corresponds to the displayed minus sign. The real part may contain pressure and contact contributions; these are not dissipative viscosity.

The order of limits is physical. Taking frequency to zero at nonzero momentum probes a static susceptibility, while taking momentum to zero first probes homogeneous transport. Hydrodynamic poles make the two operations noncommuting. A complete Kubo statement therefore names the path to the origin in frequency-momentum space rather than writing only “the zero-frequency correlator.”

For a two-derivative Einstein black brane, the transverse graviton behaves as a minimally coupled massless scalar. Its ingoing flux at the horizon yields

$$\eta=\frac{A_h}{16\pi G_{d+1}V}, \qquad s=\frac{A_h}{4G_{d+1}V}, \qquad \frac{\eta}{s}=\frac{1}{4\pi}.$$

The original absorption and Kubo calculation is given in Kovtun, Son, and Starinets 2005, complete article, pp. 1–4. The result assumes the Einstein kinetic term, isotropy, translation invariance, and a regular nonextremal horizon; later corrections belong to the viscosity comparison.

Conductivity and diffusion require the conserved density

For a current $J^i$ sourced by a boundary gauge field,

$$\sigma =- \lim_{\omega\to0} \frac{1}{\omega} \operatorname{Im}G_R^{J^xJ^x}(\omega,\mathbf 0).$$

At finite density, electric and heat currents mix, and a translation-invariant system can contain a delta function at zero frequency because momentum cannot relax. The “DC conductivity” must then be separated into coherent and incoherent pieces or evaluated after a declared momentum-relaxing deformation. In a magnetic field, magnetization currents must be subtracted before identifying transport current.

For a single conserved charge whose fluctuations decouple from energy and momentum—for example, a charge-conjugation-symmetric state at zero density—charge diffusion appears in the density correlator through

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

The Einstein relation is an independent check: compute the finite regular $\sigma$ from the current-current Kubo limit, $\chi$ from equilibrium thermodynamics, and verify that their ratio matches the hydrodynamic pole. At finite density, heat and charge generally diffuse together; the diffusion constants are eigenvalues of a transport matrix times an inverse susceptibility matrix. The full electric conductivity, with its momentum delta function, cannot be inserted into the single-charge formula above. A mismatch often means the ensemble, current normalization, or order of limits differs between calculations.

Why horizon data can suffice

Let $\Pi(r,\omega)$ be the radial canonical momentum conjugate to a gauge-invariant perturbation and define the response

$$\mathcal R(r,\omega)=\frac{\Pi(r,\omega)}{i\omega\phi(r,\omega)}.$$

At zero spatial momentum and leading order in frequency, the radial equation often gives $\partial_r\mathcal R=O(\omega)$. The boundary Kubo coefficient can therefore be evaluated at the horizon, where ingoing regularity fixes $\mathcal R$ algebraically. This membrane argument and its domain are developed in Iqbal and Liu 2009, §§III–IV, pp. 7–15.

The simplification fails when fields mix nontrivially, higher derivatives introduce additional radial data, translation breaking changes the zero mode, or an extremal throat makes the frequency expansion nonuniform. In those cases one evolves a response matrix or solves the full fluctuation system. Horizon regularity remains necessary but no longer determines the coefficient alone.

Ward identities and positivity checks

Stress-tensor conservation relates shear and sound correlators; current conservation relates density and longitudinal-current response. Contracting a computed correlator with its momentum must reproduce the appropriate Ward identity, including equilibrium contact terms. Failure diagnoses an unconstrained gauge component or omitted counterterm.

Spectral positivity supplies another check. For a Hermitian operator in a thermal state, the dissipative spectral weight has the sign required to make entropy production nonnegative. A numerical result with negative shear viscosity or negative regular conductivity in a stable passive state usually indicates a Fourier-sign, horizon-branch, or source-normalization error.

Dimensions offer a quick diagnostic. In a d-dimensional conformal theory, shear viscosity has dimension $d-1$, charge conductivity dimension $d-3$ for a conventionally normalized current, and diffusion dimension minus one. Restoring temperature should produce the corresponding powers unless another density, chemical potential, or explicit scale is present. This check catches hidden choices of horizon radius or AdS length set to unity.

Exercises

Derive the diffusion constant from the constitutive relation and verify the Einstein relation using the static susceptibility.

Solution

In the decoupled charge sector use $J^i=-D\partial_i n+\sigma E^i$ and $\partial_t n+\partial_iJ^i=0$. With no external field, a plane wave gives $-i\omega+Dk^2=0$, hence $\omega=-iDk^2$. Write $n=\chi\mu_{\rm int}$ and denote the physical potential energy per unit charge by $U$, so $E=-\nabla U$. Electrochemical equilibrium requires $\mu_{\rm int}+U$ to be constant, hence $\nabla\mu_{\rm int}=E$. Then $J=(-D\chi+\sigma)E=0$, so $D=\sigma/\chi$. This argument fixes the physical sign without confusing $U$ with a convention-dependent covariant component $A_t$.

References

• Iqbal, Nabil, and Hong Liu. “Universality of the Hydrodynamic Limit in AdS/CFT and the Membrane Paradigm.” Physical Review D 79 (2009): 025023. DOI.
• Kovtun, Pavel, Dam T. Son, and Andrei O. Starinets. “Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics.” Physical Review Letters 94 (2005): 111601. DOI.