Momentum Relaxation and Strange-Metal Transport

The main idea

At finite density, electric current can carry momentum. If the theory is translationally invariant, momentum cannot decay. A constant electric field then accelerates the charged fluid forever, so the DC conductivity is infinite even when the microscopic dynamics is strongly interacting.

This is the first lesson of finite-density transport:

$$\text{finite charge density} + \text{conserved momentum} \quad\Longrightarrow\quad \sigma_{\mathrm{DC}}=\infty$$

unless the electric current has no overlap with momentum. To model an ordinary material, one must give momentum somewhere to go: a lattice, impurities, disorder, umklapp, a pinned density wave, or an explicit translation-breaking source. Holographically, this means that the bulk theory must break boundary spatial translations.

At finite density, current overlaps with momentum. If $P^i$ is conserved, the optical conductivity contains a protected zero-frequency contribution. Translation-breaking sources introduce a relaxation rate $\Gamma$ and replace the pole by a Drude-like factor. Homogeneous holographic models such as linear axions and Q-lattices keep the bulk geometry solvable by ODEs while producing finite DC conductivities determined by horizon data.

The central hydrodynamic formula is

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

Here $\rho=\langle J^t\rangle$ is the charge density, $\chi_{PP}$ is the static momentum susceptibility, and $\Gamma$ is the momentum relaxation rate. For a relativistic CFT,

$$\chi_{PP}=\epsilon+p=sT+\mu\rho.$$

The term proportional to $1/(\Gamma-i\omega)$ is the part of the electric current that moves by dragging the total momentum. The term $\sigma_Q$ is the incoherent conductivity: the part of charge transport that does not require momentum flow.

This page explains three related ideas:

1. why translation-invariant finite-density theories have infinite DC conductivity;
2. how holography breaks translations while keeping calculations controlled;
3. how these models are used, carefully, in discussions of strange-metal transport.

Why conserved momentum makes $\sigma_{\mathrm{DC}}$ infinite

The Kubo formula for the electrical conductivity is

$$\sigma^{ij}(\omega) = \frac{1}{i\omega} G^R_{J^iJ^j}(\omega,\vec k=0),$$

up to contact terms and convention-dependent diamagnetic pieces. The important point is not the precise contact convention. It is the existence of a protected low-frequency contribution when $J^i$ overlaps with a conserved operator.

At finite density, a boost of the fluid produces both momentum and electric current. In relativistic hydrodynamics,

$$T^{ti}=(\epsilon+p)v^i, \qquad J^i=\rho v^i+J_{\mathrm{inc}}^i,$$

where $v^i$ is the local velocity and $J_{\mathrm{inc}}^i$ is the part of the current not tied to the fluid velocity. Thus

$$J^i = \frac{\rho}{\epsilon+p}P^i+J_{\mathrm{inc}}^i, \qquad P^i\equiv T^{ti}.$$

If $P^i$ is exactly conserved, an electric field accelerates the fluid:

$$\dot P^i=\rho E^i.$$

In frequency space,

$$P^i(\omega)=\frac{\rho}{-i\omega}E^i(\omega),$$

so the convective contribution to the current is

$$J^i_{\mathrm{conv}}(\omega) = \frac{\rho}{\epsilon+p}P^i(\omega) = \frac{\rho^2}{\epsilon+p}\frac{1}{-i\omega}E^i(\omega).$$

Therefore

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

Using

$$\frac{i}{\omega+i0^+} = \pi\delta(\omega)+i\,\mathcal P\frac{1}{\omega},$$

the real part contains a delta function:

$$\operatorname{Re}\sigma(\omega) = \pi\frac{\rho^2}{\epsilon+p}\delta(\omega)+\operatorname{Re}\sigma_Q(\omega).$$

This is not a weak-coupling statement. It is a consequence of symmetry and finite density. Interactions can rapidly degrade quasiparticles, but if the total momentum is conserved and the current overlaps with momentum, the DC conductivity is still infinite.

This is why translation breaking is not a decorative complication in finite-density holography. It is necessary if the goal is to model ordinary finite resistivity.

The incoherent current

The convective part of the current is not the whole story. Define the current component orthogonal to momentum by

$$J_{\mathrm{inc}}^i = J^i-\frac{\rho}{\epsilon+p}P^i.$$

It has no static overlap with momentum:

$$\chi_{J_{\mathrm{inc}}P}=0.$$

The conductivity associated with this current can remain finite even in a perfectly clean system. This is the quantity often denoted $\sigma_Q$ in relativistic hydrodynamics. It measures charge diffusion in the local rest frame of the fluid.

A particularly clean example occurs at particle-hole symmetry. If

$$\rho=0,$$

then the electric current does not overlap with momentum, and the conductivity can be finite even without momentum relaxation. At nonzero density, however, the convective part normally dominates the low-frequency DC response unless momentum relaxes.

This separation is conceptually important:

Quantity	Physical meaning
$\rho^2/(\chi_{PP}\Gamma)$	coherent momentum-drag contribution
$\sigma_Q$	incoherent charge diffusion contribution
$\Gamma$	rate at which total momentum relaxes
$\chi_{PP}$	inertia of the charged fluid
$J_{\mathrm{inc}}^i$	current combination orthogonal to momentum

In a clean strongly coupled finite-density CFT, $\sigma_Q$ is a genuine transport coefficient, but it is not the full DC conductivity. The full DC conductivity is infinite because the fluid can move without friction.

Slow momentum relaxation: hydrodynamics with $\Gamma$

Suppose translations are weakly broken. Momentum is no longer exactly conserved, but it is long-lived. The simplest effective equation is

$$\dot P^i+ \Gamma P^i = \rho E^i.$$

Solving in frequency space gives

$$P^i(\omega) = \frac{\rho}{\Gamma-i\omega}E^i(\omega).$$

Thus

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

At zero frequency,

$$\sigma_{\mathrm{DC}} = \sigma_Q+\frac{\rho^2}{\chi_{PP}\Gamma}.$$

This is a Drude formula, but it is not necessarily a quasiparticle formula. The long-lived object is total momentum, not an electron quasiparticle. A strongly interacting fluid with no quasiparticles can still have a narrow Drude peak if translations are only weakly broken.

The distinction is useful:

Regime	Condition	Optical conductivity
clean finite-density fluid	$\Gamma=0$	delta function at $\omega=0$
coherent metal	$\Gamma\ll T$	narrow Drude peak
incoherent metal	$\Gamma\sim T$ or larger	no sharp momentum peak
particle-hole symmetric plasma	$\rho=0$	finite $\sigma_Q$ even when clean

The word coherent here means that momentum is a long-lived transport carrier. The word incoherent means that charge and heat flow mainly by diffusion of locally equilibrated densities rather than by the slow decay of momentum.

Translation breaking from Ward identities

In QFT, translations are generated by the stress tensor. If the action is deformed by a spacetime-dependent source,

$$S = S_{\mathrm{CFT}} + \int d^d x\,J_I(x)\mathcal O_I(x),$$

then the stress tensor Ward identity becomes

$$\partial_\mu T^{\mu}{}_{\nu} = \sum_I \langle\mathcal O_I\rangle\,\partial_\nu J_I +F_{\nu\lambda}\langle J^\lambda\rangle.$$

For spatially homogeneous sources $J_I=\mathrm{constant}$, translations remain unbroken. For spatially dependent sources $J_I(x)$, momentum is not conserved. The external sources can absorb momentum.

A periodic lattice source has the schematic form

$$J(x)=J_0\cos(kx).$$

A disordered source is a random function $J(x)$. Umklapp in a microscopic lattice also relaxes continuum momentum because momentum is conserved only modulo a reciprocal lattice vector. In holography, the cleanest implementation depends on how hard a gravitational boundary value problem one is willing to solve.

Holographic mechanisms for momentum relaxation

There are several common holographic ways to break translations. They differ in physical realism, calculational complexity, and microscopic interpretation.

Mechanism	Boundary interpretation	Bulk advantage	Main warning
explicit lattice	periodic source $J(x)$	most literal	usually PDEs
disorder	random spatial source	closer to impurities	hard numerically and statistically
Q-lattice	spatial phase of a complex scalar	ODEs from internal symmetry	homogeneous trick, not a literal ionic lattice
linear axions	scalar sources $\psi_I=kx_I$	analytic homogeneous black branes	very coarse effective translation breaking
massive gravity	graviton mass/Stückelberg fields	simple effective momentum relaxation	microscopic dual may be obscure
probe-brane disorder or impurities	flavor-sector momentum loss	useful in probe limits	often neglects full backreaction

The fundamental problem is that a generic lattice makes the bulk solution inhomogeneous. For example, if a boundary source depends on $x$, the metric, gauge field, and matter fields typically depend on both the radial coordinate and $x$:

$$g_{ab}=g_{ab}(r,x), \qquad A_a=A_a(r,x), \qquad \phi=\phi(r,x).$$

One then must solve coupled nonlinear PDEs. This is possible and important, but it is not the simplest way to build intuition.

Homogeneous translation-breaking models exploit a loophole. They break translations in the boundary theory, but preserve enough combined symmetry that the bulk stress tensor is homogeneous. The background equations reduce to ODEs in the radial coordinate.

Linear axion models

The simplest homogeneous model uses massless neutral scalar fields. In a bulk theory with $d-1$ boundary spatial directions, introduce scalars

$$\psi_I, \qquad I=1,\ldots,d-1,$$

with the background profile

$$\psi_I=k\,x_I.$$

The source is spatially dependent, so translations are explicitly broken. But the stress tensor built from gradients is homogeneous:

$$\partial_i\psi_I=k\,\delta_{iI}, \qquad (\partial\psi)^2=\text{constant on boundary spatial slices}.$$

A common Einstein-Maxwell-axion model is

$$S = \frac{1}{16\pi G_{d+1}} \int d^{d+1}x\sqrt{-g} \left[ R-\frac12(\partial\phi)^2+V(\phi) -\frac14 Z(\phi)F_{ab}F^{ab} -\frac12Y(\phi)\sum_{I=1}^{d-1}(\partial\psi_I)^2 \right] +S_{\partial}.$$

Here $\phi$ is often a neutral dilaton. The simplest case has constant $Z$ and $Y$ and no running scalar. A standard homogeneous ansatz is

$$ds^2 = -g_{tt}(r)dt^2+g_{rr}(r)dr^2+g_{xx}(r)d\vec x^{\,2},$$ $$A=A_t(r)dt, \qquad \phi=\phi(r), \qquad \psi_I=kx_I.$$

The parameter $k$ controls the strength of explicit translation breaking. It is not always identical to the momentum relaxation rate $\Gamma$. In weak breaking one often finds

$$\Gamma\propto k^2,$$

but at strong breaking the notion of a single relaxation rate may cease to be useful.

Why the axion model is so useful

The linear axion model is popular because it combines three features that rarely coexist:

1. it explicitly breaks translations;
2. it keeps the background homogeneous and isotropic;
3. it often gives analytic DC transport formulas.

For a large class of Einstein-Maxwell-dilaton-axion theories, the DC conductivities are determined entirely by horizon data. In a common convention for isotropic $d=3$ boundary theories,

$$\sigma_{\mathrm{DC}} = Z_h+\frac{4\pi\rho^2}{s k^2Y_h},$$ $$\alpha_{\mathrm{DC}} = \frac{4\pi\rho}{k^2Y_h},$$ $$\bar\kappa_{\mathrm{DC}} = \frac{4\pi sT}{k^2Y_h}.$$

Here

$$Z_h=Z(\phi_h), \qquad Y_h=Y(\phi_h),$$

and $\phi_h$ is the scalar value at the horizon. The entropy density $s$ is the horizon area density divided by $4G_{d+1}$.

The thermal conductivity at zero electric current is

$$\kappa_{\mathrm{DC}} = \bar\kappa_{\mathrm{DC}} -\frac{T\alpha_{\mathrm{DC}}^2}{\sigma_{\mathrm{DC}}}.$$

The two terms in $\sigma_{\mathrm{DC}}$ have distinct meanings:

$$\sigma_{\mathrm{DC}} = \underbrace{Z_h}_{\text{incoherent pair-creation/horizon term}} + \underbrace{\frac{4\pi\rho^2}{s k^2Y_h}}_{\text{momentum-drag term}}.$$

In the simplest four-dimensional bulk model with $Z=Y=1$, this reduces schematically to

$$\sigma_{\mathrm{DC}}=1+\frac{\mu^2}{k^2},$$

in standard units. The constant term is the particle-hole symmetric contribution. The second term becomes large when momentum relaxation is weak.

How horizon DC formulas work

DC transport in black-hole backgrounds has a special simplification. To compute $\sigma_{\mathrm{DC}}$, one applies a constant boundary electric field and studies time-independent linearized perturbations that are regular at the future horizon. The bulk Maxwell equation implies the existence of a radially conserved electric flux,

$$\mathcal J^i(r) = -\sqrt{-g}\,Z(\phi)F^{ri}+\cdots,$$

where the dots denote possible mixing with metric and matter perturbations. Since

$$\partial_r\mathcal J^i=0$$

at zero frequency and zero spatial momentum, the boundary current can be evaluated at the horizon:

$$\langle J^i\rangle=\mathcal J^i(r\to\infty)=\mathcal J^i(r_h).$$

The horizon regularity condition then relates $\mathcal J^i(r_h)$ to the applied electric field $E_i$ and temperature gradient $\nabla_iT$. This yields DC conductivities without solving the full radial perturbation problem.

For fully inhomogeneous holographic lattices, the same idea generalizes: the DC conductivities are found by solving forced Stokes equations on the black-hole horizon. The homogeneous axion formula is the simplest avatar of a broader principle:

$$\text{DC transport} \quad\longleftrightarrow\quad \text{horizon fluid problem}.$$

This does not mean AC transport is trivial. Optical conductivity at finite $\omega$ requires solving the radial fluctuation equations with infalling boundary conditions, exactly as in the previous module.

Q-lattices

A Q-lattice uses a complex scalar field with a spatially dependent phase:

$$\Phi(r,x)=\varphi(r)e^{ikx}.$$

If the bulk action has a global $U(1)$ symmetry

$$\Phi\to e^{i\theta}\Phi,$$

then the explicit $x$-dependence can be absorbed into the internal phase. The stress tensor depends on $|\Phi|$ and derivatives in a way that keeps the background homogeneous. Again, the gravitational problem reduces to ODEs.

The boundary interpretation is a spatially modulated source for a scalar operator. The modulation wave number is $k$, and the amplitude is controlled by the leading near-boundary coefficient of $\varphi(r)$. Q-lattices are useful for constructing holographic metallic and insulating phases because the lattice deformation can become relevant or irrelevant in the IR.

The essential lesson is not that real crystals are Q-lattices. It is that Q-lattices give a controlled way to study finite-density transport without a protected momentum delta function.

Massive gravity and Stückelberg language

Another effective approach gives the graviton a mass in the spatial directions. In the boundary theory, a massless bulk graviton is tied to conservation of the boundary stress tensor. If the graviton acquires suitable spatial mass terms, the dual stress tensor no longer has conserved momentum.

A more transparent version introduces Stückelberg scalar fields,

$$\psi_I=kx_I,$$

which is closely related to the linear axion construction. In this language, the graviton mass comes from spatial scalar profiles. The lesson is that massive-gravity models are often effective descriptions of homogeneous translation breaking.

One should be cautious: not every massive-gravity action has a clean microscopic CFT dual. But as low-energy effective holographic models, they are valuable because they isolate the role of momentum relaxation.

Weak lattices and the memory-matrix formula

When translations are weakly broken by a small periodic source,

$$\delta H = \lambda\int d^{d-1}x\,\cos(kx)\mathcal O(x),$$

momentum relaxation can be computed in perturbation theory around the translation-invariant theory. A memory-matrix estimate gives the schematic scaling

$$\Gamma \sim \frac{\lambda^2 k^2}{\chi_{PP}} \lim_{\omega\to0} \frac{1}{\omega} \operatorname{Im}G^R_{\mathcal O\mathcal O}(\omega,k).$$

This formula is extremely useful conceptually. It says that momentum relaxation is efficient only if the clean theory has low-energy spectral weight at the lattice wavevector $k$.

For a conventional Fermi liquid, finite-momentum low-energy spectral weight is controlled by the Fermi surface and leads to familiar powers of temperature. For a locally critical holographic IR region, such as an $\mathrm{AdS}_2\times\mathbb R^{d-1}$ throat, momenta do not scale but frequencies do. The finite-$k$ spectral density can then remain important at low energies, leading to nontrivial power laws for resistivity.

This is one route by which holography can produce strange-metal-like power-law transport. The logic is:

$$\text{translation breaking} + \text{strong finite-$k$ low-energy spectral weight} \quad\Longrightarrow\quad \text{efficient momentum relaxation}.$$

But the exponent is model-dependent. Holography gives controlled examples, not a universal explanation of all strange metals.

Coherent versus incoherent metals

A clean finite-density fluid has a delta function in $\operatorname{Re}\sigma$. Weak translation breaking broadens it into a narrow peak:

$$\operatorname{Re}\sigma(\omega) \approx \sigma_Q+\frac{\rho^2}{\chi_{PP}}\frac{\Gamma}{\Gamma^2+\omega^2}.$$

This is the coherent regime. The low-frequency current is mostly the motion of the conserved densities.

As translation breaking becomes strong, the peak may disappear. Then there is no useful separation between a narrow momentum mode and the rest of the spectrum. Transport is governed by diffusion of charge and energy, by IR critical scaling, and by horizon data. This is the incoherent regime.

A useful diagnostic is the optical conductivity:

Optical feature	Interpretation
sharp Drude peak	momentum is long-lived
broad low-frequency bump	intermediate relaxation
featureless low-frequency response	strongly incoherent transport
$\omega/T$ scaling	possible quantum critical control
hard gap or suppressed low-$\omega$ weight	insulating or ordered response, depending on context

The absence of a Drude peak does not automatically imply a strange metal. It only says that momentum is not a long-lived carrier of current. One must still identify the charge carriers, scaling regime, entropy, susceptibilities, and relaxation mechanisms.

Strange metals: what holography tries to capture

The phrase strange metal is used for metallic states whose transport is hard to explain using long-lived Landau quasiparticles. Common phenomenological features include large resistivity, broad optical conductivity, short relaxation times, and resistivity approximately linear in temperature over broad ranges.

Holography is attractive here because classical black holes naturally describe strongly interacting thermal states without quasiparticles. Horizons absorb perturbations. Transport is dissipative. Many holographic models produce relaxation times of order

$$\tau\sim\frac{1}{T}$$

in units with $\hbar=k_B=1$. Restoring units, this is the so-called Planckian scale

$$\tau_{\mathrm{P}} \sim \frac{\hbar}{k_B T}.$$

But this slogan needs discipline. A holographic model does not prove that every strange metal has a universal Planckian bound. Rather, it gives examples in which strong interactions and horizons make the only available local equilibration scale of order $T$.

A modern holographic strange-metal model usually combines:

1. finite charge density;
2. an IR scaling geometry, often with dynamical exponent $z$ and hyperscaling violation exponent $\theta$;
3. translation breaking by axions, a Q-lattice, disorder, or umklapp;
4. horizon formulas for DC transport;
5. careful matching between model parameters and field-theory observables.

In Einstein-Maxwell-dilaton-axion models, the couplings $Z(\phi)$ and $Y(\phi)$ can run in the IR. The DC conductivity

$$\sigma_{\mathrm{DC}} = Z_h+\frac{4\pi\rho^2}{s k^2Y_h}$$

then acquires temperature dependence through the horizon value $\phi_h(T)$ and entropy density $s(T)$. By choosing IR exponents and couplings, one can engineer many power laws:

$$\rho_{\mathrm{dc}}(T)=\frac{1}{\sigma_{\mathrm{DC}}(T)} \sim T^\alpha.$$

The valuable part is not that any desired exponent can be engineered. That is also a warning. The valuable part is that holography gives a controlled strongly coupled framework in which one can ask which scaling laws follow from which IR fixed points and which translation-breaking deformations.

Metallic, insulating, and bad-metal behavior

A holographic metal is often diagnosed by the low-temperature behavior of its DC conductivity. Very roughly,

$$\frac{d\sigma_{\mathrm{DC}}}{dT}<0$$

is metallic-like, while

$$\frac{d\sigma_{\mathrm{DC}}}{dT}>0$$

is insulating-like. But this diagnostic is crude. A better analysis examines charge susceptibility, heat transport, entropy, optical spectral weight, and the IR geometry.

In holographic scaling models, an insulating phase may arise because the charge sector decouples in the IR, the lattice deformation becomes relevant, the horizon conductivity $Z_h$ vanishes, or charge transport is suppressed by the geometry. A metallic phase may arise because charge remains mobile at the horizon, because an incoherent conductivity stays finite, or because momentum relaxation is weak.

The term bad metal refers to a metal whose resistivity exceeds what would be expected from quasiparticle mean-free-path reasoning. Since holographic models often lack quasiparticles from the outset, they naturally produce bad-metal-like transport. However, matching a holographic conductivity to a real material requires a choice of units, normalization, number of degrees of freedom, and microscopic current.

Momentum relaxation versus charge relaxation

Momentum relaxation does not mean charge relaxation. Charge is still conserved:

$$\partial_\mu J^\mu=0.$$

Energy is usually also conserved in a time-independent background:

$$\partial_\mu T^{\mu t}=0.$$

What is broken is spatial momentum conservation:

$$\partial_\mu T^{\mu i}\neq0.$$

This distinction matters because hydrodynamic modes change. In a clean charged fluid, one has sound modes controlled by energy and momentum conservation. With momentum relaxation, sound can cross over to diffusion or become a damped pseudo-hydrodynamic mode. At strong translation breaking, the late-time dynamics may be governed by charge and energy diffusion alone.

A minimal hydrodynamic pole structure is:

Conservation law	Clean system	With momentum relaxation
charge	diffusion or coupled sound/charge modes	diffusion remains
energy	sound/heat modes	energy diffusion can dominate
momentum	hydrodynamic sound/shear	pole shifts by $-i\Gamma$

In holography, these poles are quasinormal modes. Momentum relaxation moves the hydrodynamic poles in the complex frequency plane and can create a crossover from propagating to diffusive behavior.

Spontaneous versus explicit translation breaking

Not all translation breaking produces finite DC resistivity by itself. If translations are broken spontaneously, as in a clean charge-density wave or crystal, total momentum may still be conserved. The system has phonons, and a uniform electric current can still drag the whole crystal unless there is also pinning, impurities, a lattice, or another explicit relaxation mechanism.

The distinction is:

Breaking pattern	Momentum conservation	Transport consequence
explicit translations broken	$P^i$ not conserved	finite DC possible
spontaneous translations broken	$P^i$ conserved	phonons, but DC may remain infinite
weak explicit + spontaneous	pseudo-phonons	pinned collective mode
strong explicit breaking	no long-lived momentum	incoherent transport

Holographic models of pinned density waves, helical lattices, and spatially modulated phases enrich this story, but the basic lesson is simple: a finite DC resistivity requires an explicit sink for momentum or a situation where the electric current has no momentum overlap.

What is universal?

Some statements are robust:

• At finite density, conserved momentum produces an infinite DC conductivity if $J^i$ overlaps with $P^i$.
• Weak momentum relaxation broadens the protected pole into a Drude-like peak.
• The incoherent current orthogonal to momentum can have finite conductivity even in a clean theory.
• In many two-derivative holographic theories, DC conductivities can be expressed in terms of horizon data.

Other statements are model-dependent:

• whether the resistivity is linear in $T$;
• whether the phase is metallic or insulating;
• whether optical conductivity shows a power law;
• whether a proposed Planckian time controls the measured conductivity;
• whether a bottom-up translation-breaking model has a UV-complete string embedding.

This distinction is crucial for using holography honestly. The strongest conclusions are symmetry and hydrodynamic statements. The most phenomenological conclusions require explicit model assumptions.

A practical workflow

When computing a holographic finite-density conductivity with momentum relaxation, use the following checklist.

Step 1: Identify the conserved quantities

Ask which of the following are conserved:

$$J^\mu, \qquad T^{\mu t}, \qquad T^{\mu i}.$$

If momentum is conserved and $\rho\neq0$, expect a delta function in $\operatorname{Re}\sigma$.

Step 2: Identify the translation-breaking source

Is the source explicit or spontaneous? Is it homogeneous in the bulk? Is it periodic, random, linear, helical, or Q-lattice-like?

For axions,

$$\psi_I=kx_I.$$

For a Q-lattice,

$$\Phi=e^{ikx}\varphi(r).$$

For a literal lattice,

$$J(x)=J_0\cos(kx),$$

and the bulk solution is usually inhomogeneous.

Step 3: Choose the ensemble

Finite-density transport depends on whether one fixes charge density or chemical potential. For thermoelectric transport, state the convention for

$$\begin{pmatrix} J^i\\ Q^i \end{pmatrix} = \begin{pmatrix} \sigma & T\alpha\\ T\alpha & T\bar\kappa \end{pmatrix} \begin{pmatrix} E_i\\ -\nabla_iT/T \end{pmatrix}.$$

The physical thermal conductivity at zero electric current is

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

Step 4: For DC transport, look for radially conserved fluxes

At $\omega=0$, the Maxwell and Einstein equations often imply conserved radial quantities. Evaluate them at the horizon using regularity.

Step 5: For AC transport, solve the fluctuation equations

At finite $\omega$, impose infalling boundary conditions at the horizon and extract the source and response at the boundary. Momentum relaxation couples gauge, metric, and scalar perturbations, so gauge-invariant variables are often helpful.

Step 6: Interpret the result hydrodynamically

Ask whether the low-frequency response is coherent:

$$\Gamma\ll T,$$

or incoherent:

$$\Gamma\gtrsim T.$$

Do not call every finite conductivity a strange metal. First identify the scaling regime and the mechanism that controls the relaxation.

Common mistakes

Mistake 1: forgetting the momentum delta function

A finite-density translation-invariant holographic black brane usually has infinite DC conductivity. A finite value obtained by numerics may be an artifact of taking limits incorrectly, omitting the metric perturbation, or confusing $\sigma_Q$ with the full conductivity.

Mistake 2: treating $k$ and $\Gamma$ as identical

In linear axion models, $k$ is the source strength or wave number of translation breaking. The relaxation rate $\Gamma$ is a physical pole location. For weak breaking, $\Gamma$ is often proportional to $k^2$, but at strong breaking there may be no single narrow relaxation pole.

Mistake 3: calling homogeneous axions a literal lattice

The source $\psi_I=kx_I$ breaks translations, but it is not a periodic ionic lattice. It is a homogeneous effective way to relax momentum.

Mistake 4: ignoring thermoelectric mixing

At finite density, electric, heat, and momentum transport mix. The measured thermal conductivity depends on whether the electric current is set to zero.

Mistake 5: overclaiming Planckian universality

Holography often produces timescales of order $1/T$, but a model-specific relaxation time is not automatically a universal bound for all materials.

Exercises

Exercise 1: The conductivity delta function

Assume a clean relativistic finite-density fluid with

$$J^i=\frac{\rho}{\epsilon+p}P^i+J_{\mathrm{inc}}^i$$

and

$$\dot P^i=\rho E^i.$$

Derive the pole in $\sigma(\omega)$ and show that $\operatorname{Re}\sigma(\omega)$ contains a delta function.

Solution

Fourier transforming the momentum equation gives

$$-i\omega P^i(\omega)=\rho E^i(\omega),$$

so

$$P^i(\omega)=\frac{\rho}{-i\omega}E^i(\omega).$$

The convective current is therefore

$$J_{\mathrm{conv}}^i(\omega) = \frac{\rho}{\epsilon+p}P^i(\omega) = \frac{\rho^2}{\epsilon+p}\frac{1}{-i\omega}E^i(\omega).$$

Thus

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

Using

$$\operatorname{Re}\frac{i}{\omega+i0^+}=\pi\delta(\omega),$$

one obtains

$$\operatorname{Re}\sigma(\omega) = \operatorname{Re}\sigma_Q(\omega) + \pi\frac{\rho^2}{\epsilon+p}\delta(\omega).$$

The delta function is protected by momentum conservation.

Exercise 2: Drude peak from momentum relaxation

Now replace momentum conservation by

$$\dot P^i+\Gamma P^i=\rho E^i.$$

Show that

$$\sigma_{\mathrm{DC}} = \sigma_Q+\frac{\rho^2}{(\epsilon+p)\Gamma}.$$

Solution

In frequency space,

$$(\Gamma-i\omega)P^i(\omega)=\rho E^i(\omega),$$

so

$$P^i(\omega)=\frac{\rho}{\Gamma-i\omega}E^i(\omega).$$

The current is

$$J^i(\omega)=J^i_{\mathrm{inc}}(\omega)+\frac{\rho}{\epsilon+p}P^i(\omega).$$

If the incoherent part contributes $\sigma_QE^i$, then

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

Taking $\omega\to0$ gives

$$\sigma_{\mathrm{DC}}=\sigma_Q+\frac{\rho^2}{(\epsilon+p)\Gamma}.$$

Exercise 3: Why linear axions are homogeneous

Consider two scalar fields in a $2+1$-dimensional boundary theory with sources

$$\psi_1=kx, \qquad \psi_2=ky.$$

Show why translations are broken, but the scalar contribution to the bulk stress tensor can remain homogeneous and isotropic.

Solution

The sources depend explicitly on spatial coordinates, so under a boundary translation $x^i\to x^i+a^i$ the source values change:

$$\psi_I(x^i+a^i)=\psi_I(x^i)+k a_I.$$

Thus translations are explicitly broken by the sources.

However, the stress tensor of shift-symmetric massless scalars depends on derivatives, not on the absolute values of $\psi_I$. The gradients are constants:

$$\partial_i\psi_I=k\delta_{iI}.$$

Therefore scalar invariants such as

$$\sum_I\partial_i\psi_I\partial_j\psi_I=k^2\delta_{ij}$$

are independent of $x$ and rotationally invariant. The boundary sources break translations, but the bulk stress tensor sourcing the background geometry is homogeneous and isotropic.

Exercise 4: Interpreting the axion DC formula

For a simple isotropic axion model, suppose

$$\sigma_{\mathrm{DC}}=Z_h+\frac{4\pi\rho^2}{s k^2Y_h}.$$

Explain the behavior at $\rho=0$, at weak translation breaking $k\to0$, and at strong translation breaking.

Solution

At $\rho=0$,

$$\sigma_{\mathrm{DC}}=Z_h.$$

This is the incoherent or particle-hole symmetric contribution. It does not require momentum drag.

At weak translation breaking, $k\to0$ with $\rho\neq0$, the second term diverges:

$$\frac{4\pi\rho^2}{s k^2Y_h}\to\infty.$$

This reproduces the infinite DC conductivity of the translation-invariant finite-density system.

At strong translation breaking, the second term can become small, and the conductivity may be controlled by $Z_h$ and the IR behavior of the couplings. In that regime the response need not have a narrow Drude peak, so interpreting $k$ as a simple relaxation rate can be misleading.

Exercise 5: Source strength versus relaxation rate

A linear axion model has parameter $k$. Explain why $k$ is not always the same as the physical momentum relaxation rate $\Gamma$.

Solution

The parameter $k$ appears in the boundary source:

$$\psi_I^{(0)}=kx_I.$$

It measures the strength or gradient of explicit translation breaking. The physical relaxation rate $\Gamma$ is extracted from a pole of a retarded Green function, for example from the low-frequency conductivity:

$$\sigma(\omega)\approx\sigma_Q+\frac{D}{\Gamma-i\omega}.$$

For weak translation breaking, perturbation theory or hydrodynamics often gives

$$\Gamma\propto k^2.$$

At stronger breaking, there may be no isolated Drude pole, so no unique $\Gamma$ controls the response. Thus $k$ is a Lagrangian/source parameter, while $\Gamma$ is an emergent dynamical quantity.

Further reading

• S. A. Hartnoll, Lectures on holographic methods for condensed matter physics, arXiv:0903.3246.
• S. A. Hartnoll and D. M. Hofman, Locally critical umklapp scattering and holography, arXiv:1201.3917.
• G. T. Horowitz, J. E. Santos, and D. Tong, Optical Conductivity with Holographic Lattices, arXiv:1204.0519.
• D. Vegh, Holography without translational symmetry, arXiv:1301.0537.
• M. Blake and D. Tong, Universal Resistivity from Holographic Massive Gravity, arXiv:1308.4970.
• A. Donos and J. P. Gauntlett, Holographic Q-lattices, arXiv:1311.3292.
• T. Andrade and B. Withers, A simple holographic model of momentum relaxation, arXiv:1311.5157.
• A. Donos and J. P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, arXiv:1406.4742.
• R. A. Davison and B. Goutéraux, Momentum dissipation and effective theories of coherent and incoherent transport, arXiv:1411.1062.
• R. A. Davison, B. Goutéraux, and S. A. Hartnoll, Incoherent transport in clean quantum critical metals, arXiv:1507.07137.