In this work, we examine how charge is transported in a theory where momentum is relaxed by spatially dependent, massless scalars. We analyze the possible IR phases in terms of various scaling exponents and the (ir)relevance of operators in the IR effective holographic theory with a dilaton. We compute the (finite) resistivity and encounter broad families of (in)coherent metals and insulators, characterized by universal scaling behaviour. The optical conductivity at zero temperature and low frequencies exhibits power tails which can violate scaling symmetries, due to the running of the dilaton. At low temperatures, our model captures features of random-field disorder.
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0$ in~\eqref{ACscaling2}. These last two cases are expected to correspond to strong momentum relaxation effects in the IR\@. We shall see whether this is borne out when the running scalar is included. Finally, in section~\ref{section:ACconductivity} we analyze the zero temperature, small frequency behaviour of the real part of the AC conductivity. Metals are expected to develop a delta function, a signal that dissipation turns off at exactly zero temperature. On top of that, a power tail exists, which a priori can come both with a positive or negative exponent. The first case is encountered for gapless, translation-invariant systems~\cite{cgkkm,gk,g2013} and there all the spectral weight is transfered from the Drude peak to the delta function as the temperature is lowered. If this persists after breaking translations invariance, the scale-invariant predictions~\eqref{ACscaling1},~\eqref{ACscaling2} are necessarily violated. On the other hand, when the tail blows up at low frequencies, some spectral weight remains which swamps out the delta function. It should not blow up faster than $1/\omega$ though, in order not to violate sum rules on the conductivity. For insulators, there is no delta function (the DC conductivity vanishes) and consequently the power tail should decay as well. We conclude in section~\ref{section:ccl}, and comment on how our model captures certain features of random-field disorder at low temperatures. ]]>
0$; \item it has only irrelevant deformations, except for the temperature deformation which should be relevant. \end{enumerate} Within this parameter space, they all obey the NEC and the $tt$ and $x^ix^i$ elements of the metric scale the same way with $r$, so the IR is unambiguous. We can work out the spectrum of deformations along the lines of~\cite{gk2012,g2013}: the conjugate modes always sum to $z+d_\theta$ as expected on dimensional grounds, with a temperature deformation associated to (marginal) time rescalings. Consequently, a blackness function can be turned on as \be f(r) = 1-{\bigg(\frac{r}{r_h}\bigg)\!}^{z+d_\theta} , \ee when the other deformations are turned off. The parameter spaces in appendix~\ref{app:B} always take into account the fact that all other deformations should be irrelevant. Whenever $z\neq1$ (so for classes I, II and III), a semi-locally critical limit can be taken (possibly also involving $\zeta$) \be\label{DefSL} \theta \to +\infty \,, \qquad z \to +\infty \,, \qquad \frac{\theta}{z} = -\eta \,. \ee For classes I and II, this imposes $\lambda=0$, so a constant IR axion-dilaton coupling. In this limit, the entropy scales like $T^{\eta}$, so a linear specific heat is obtained when $\eta=1$. ]]>
0$. As discussed at the end of section~\ref{section:classII}, one can take a semi-locally critical limit in this expression, upon which \be\label{resistivitySL2} \rho \sim T^{-\tilde\zeta } \ee which always vanishes, hence the system is still metallic. \pagebreak As the dissipative term will be parametrically smaller than the pair creation term at low temperatures, the metallic phases do not have a Drude peak but rather an incoherent contribution, which is consistent with strong momentum IR relaxation in the IR (marginally relevant axions). \paragraph{Class III: insulators and coherent metals (marginally relevant current, irrelevant axion).} These geometries are deformations of those studied in~\cite{cgkkm,gk}. The DC conductivity~\eqref{resistivity} reads at leading order in temperature \be\label{DC3} \sigma_{\rm DC} = T^{-(2+d_\theta)/z} +\frac{q^2}{k^2}T^{(-d_\theta +\kappa \lambda)/z} . \ee It is always dissipation-dominated at low temperatures, with the leading small-$T$ behaviour of the resistivity given by \be\label{resistivity3} \rho \sim T^{-\frac{\kappa \lambda -d_\theta}{z}} . \ee As the momentum dissipation term dominates, we can naively expect to find no insulators but metals with a coherent Drude peak. However, the parameter space allows for both insulators or metals, i.e.\ the resistivity can blow up or vanish. Insulators are found when the axion-dilaton coupling and the conduction exponent are both bounded: \be\label{InsCriterion3} \text{Insulators: } -2 < \kappa \lambda < \zeta_I < 0 \,. \ee The metals are all expected to be coherent, since the dissipative term is parametrically larger than the pair creation at low temperatures. What is perhaps counter-intuitive is that the dissipative term can actually give rise to insulating behaviour. \paragraph{Class IV: coherent metals (irrelevant current and axion).} The DC conductivity~\eqref{resistivity} reads at leading order in temperature \be\label{DC4} \sigma_{\rm DC} = T^{\zeta -2} +\frac{q^2}{k^2}T^{-d_\theta+\kappa \lambda} . \ee It is dissipation-dominated so that the resistivity reads at low temperatures: \be\label{resistivity4} \rho \sim T^{d_\theta -\kappa\lambda} \ee which means that its scaling is not set by the conduction exponent but by the dilaton-axion coupling.Within the parameter space~\eqref{ParSpaceIV}, the resistivity vanishes, which indicates the system always behaves as a (coherent) metal. \paragraph{Discussion.} In this section, we have seen how the DC conductivity could be dominated either by the pair creation term or the dissipation term. Their generic contribution is given~by \be\label{DCgen} \sigma_{\rm DC,pc} \sim T^{(\zeta-2)/z} , \qquad \sigma_{\rm DC,diss} \sim T^{(\kappa \lambda-d_\theta)/z} \ee which reduce to the correct values for each of the classes. On physical grounds, we might expect to find coherent metallic behaviour when the two terms are of the same order, or when the dissipation term dominates. This is the case for the solutions in class III and IV, which is perhaps not suprising since the effects of momentum dissipation are irrelevant in the IR (like in~\cite{Hartnoll-ml-2012rj,Horowitz-ml-2012ky,Blake-ml-2013owa}). Remarkably, insulators can be found in class III in a certain range where both the conduction exponent and the axion-dilaton coupling are bounded by the other scaling exponents. When the effects of momentum dissipation are strong, one may expect to find incoherent metals and insulators. This is partly verified by the solutions in class I, and fully in class II\@. In class I however, the two terms in the resistivity have the same temperature scaling, and thus they can be of the same magnitude temperature-wise and generate a sharp Drude peak for small enough $k$, similarly to what happens in~\cite{Davison-ml-2013jba,Blake-ml-2013bqa,Andrade-ml-2013gsa}. When $k$ increases, the peak should shrink down and broaden out, effectively transferring spectral weight to higher frequencies. In class I and II, insulators also appear whenever the conduction exponent is bounded by a certain range. How does this compare to previous scaling arguments given to predict the behaviour of the conductivity~\cite{Donos-ml-2012ra} when momentum dissipation is relevant? The real part of the conductivity is given by the retarded current-current correlator \be\label{CondGenEq} \sigma_{\rm DC,I}(T) \sim \lim_{\omega\to 0} \frac1\omega \Im \big[\mathcal G^R_{\mathcal J^x\mathcal J^x} (\omega,T)\big] \sim T^{2\Delta_{\mathcal J^x}-1-(z+d_\theta)/z} \ee where $\Delta_{\mathcal J^x}$ is the real space dual dimension of the dual current $\mathcal J^x$ and the scaling takes into account the Fourier transform to frequency space in $d_{\theta}=p-1-\theta$ spatial dimensions. The scaling dimension of $\mathcal J^x$ is related by the current conservation equation to that of the density operator $\mathcal J^t$, which can be worked out from the mode analysis in appendix~\ref{app:B}: \be\label{ScalDim1} \Delta_{\mathcal J^x} = \Delta_{\mathcal J^t} +1-\frac1z \,, \ee where \be\label{ScalDim2} \Delta_{\mathcal J^t} = \frac{d_\theta+\zeta}{2z} \,, \ee keeping in mind that the modes are quadratic in the irrelevant current/axion and that the above expressions are in units of frequency. From~\eqref{ScalDim2}, it is clear that $\zeta$ characterizes deviation from the dimension of a conserved current in a scale invariant theory in $d_\theta$ spatial dimensions. Plugging~\eqref{ScalDim1} and~\eqref{ScalDim2} in~\eqref{CondGenEq}, we recover the pair creation term of the DC conductivity~\eqref{DCgen}. As we have already commented in the main text, for classes I and II where momentum dissipation is relevant, pair creation is always dominant and sets the scaling of the resistivity at low temperatures. On the other hand, when translation breaking and momentum dissipation are irrelevant,~\cite{Hartnoll-ml-2012rj} predicted that the relaxation rate $\Gamma$ (and hence the contribution to the resistivity) should be given by \be\label{DClattice} \rho_{\rm diss} \sim \Gamma = \frac{g^2 k_L^2}{\chi_{\vec P\vec P}} \lim_{\omega\to 0} \frac1\omega \Im \big[\mathcal G^R_{\mathcal O\mathcal O} (\omega,T)\big] \sim T^{2\Delta+\frac2z-1-\frac{z+d_\theta}{z}} \ee where $g$ is the coupling constant of the translation-breaking deformation, $k_L$ is the lattice wavevector, $\chi_{\vec P\vec P}$ the static susceptibility of the momentum operator $\vec P$, $\mathcal O$ the operator dual to the lattice deformation and $\Delta$ its scaling dimension in real space and units of frequency. The term $+2/z$ comes from the dimension of $k^2_L$ in units of frequency, and the last term from the Fourier transform to frequency space \emph{in $d_\theta$ spatial dimensions} to take into account hyperscaling violation. We do not have a lattice in this work, but we can still work out the scaling dimension of the irrelevant operator dual to the axion in the IR, for classes III and IV\@. From our analysis in appendix~\ref{app:B}, a deformation $\psi=k x$ of the translation-invariant ground states generates a mode (at quadratic order) $1+\# k^2r^{2+\kappa\lambda}$ where $\#$ is a dimensionless number. So we can identify (in $d_\theta$ spatial dimensions) \be \Delta = \frac{z+d_\theta}{z} -\frac{2+\kappa\lambda}{2z} \ee which yields a relaxation rate consistent with $\sigma_{\rm DC,diss}$ in~\eqref{DCgen}. This confirms the prediction in~\cite{Hartnoll-ml-2012rj} (see also~\cite{Blake-ml-2013owa} for irrelevant lattice deformations). It is also instructive to compare our results with the case of probe brane charge carriers studied in~\cite{cgkkm,gk}, where the DBI action is used to model the dynamics of the charge carriers. This gives rise to a finite DC conductivity since there is a parametrically small number of charge carriers diluted in a neutral bath: this allows them to dissipate their momentum. The following expression was obtained \be\label{DCprobe} \sigma_{\rm DC,DBI} = \frac{e^{-k\phi_\star}}{C_\star} \sqrt{q^2+C_\star^{p-1}Z_\star^2 e^{2k\phi_\star}} \ee where all quantities are evaluated at the turning point of the brane $r=r_\star$ and here $k$ labels the frame dependence of the metric as well as the origin of the neutral scalar (see~\cite{cgkkm} for details). The important point to note is that~\eqref{DCprobe} also displays two terms: the first is the contribution of the charge carriers to the DC conductivity, while the second is the pair creation term. The first is expected to dominate at high densities for massive carriers, while the other does for massless carriers. When the electric field on the boundary is small, the turning point $r_\star$ is well approximated by the horizon $r_h$. This means that the resistivity obtained from~\eqref{DCprobe} bears a close relation to the thermal entropy,~\cite{cgkkm}, just as in the formula~\eqref{resistivity}. We can now compare the temperature dependence of the pair creation term with that of~\eqref{resistivity}: \be \begin{aligned} &\text{Relevant current:}\ &\sigma_{\rm DC,DBI} &\sim T^{-(2+d_\theta)/z} \\ &\text{Irrelevant current:}\ &\sigma_{\rm DC,DBI} &\sim T^{\zeta-2} \end{aligned} \ee where the relevant solutions are the class III and class IV solutions without axions, studied in~\cite{cgkkm,gk}. This precisely matches the scaling of~\eqref{DCgen}, hinting that there is some universality behind how the pair creation contribution to the DC conductivity scales with temperature in various setups. The charge contribution in~\eqref{DCprobe} is however quite different from~\eqref{resistivity}, but this should not surprise us as the two terms have very different origins. We anticipate similar scalings would be found in massive gravity~\cite{Blake-ml-2013bqa} if the same IR analysis were performed. ]]>
0$, then we find that the IR in the Schr\"odinger coordinate $\rho$ is always located at $\rho\to+\infty$. The various Schr\"odinger potentials we will find will always scale like $1/\rho^2$ in the IR, and so will vanish there, indicating a gapless spectrum irrespective of the UV behaviour. So generically we obtain a Schr\"odinger equation for a generic perturbation $\Psi_I$ with a dual operator $\mathcal O_I$ \be\label{SchrEq} \Psi_I''(\rho) +\omega^2\Psi_I(\rho) -V_I(\rho)\Psi_I(\rho) = 0 \,, \qquad V_I(\rho) = \frac{c_I}{\rho^2}+\cdots \ee where the dots denote subleading contributions to the Schr\"odinger potential in the IR\@. From this, we can extract the scaling of the imaginary part of the Green's function of $\Psi_I$ \be\label{ImGreenScaling} \Im \big[\mathcal G^R_{\mathcal{O}_I\mathcal{O}_I} (\omega\ll\mu,T=0)\big] \sim \omega^{\sqrt{4c_I+1}} . \ee We then have to compare the various contributions from the different perturbations at small $\omega$ in~\eqref{Im}, from which the real part of the optical conductivity reads \be \Re \big[\sigma (\omega\ll\mu,T=0)\big] = \frac1\omega \Im \big[G^R_{\mathcal J^x\mathcal J^x} (\omega\ll\mu,T=0)\big] \,. \ee \paragraph{Class I (marginally relevant current and axion).} The two equations~\eqref{PerEq2} can be decoupled using the linear combinations \be \lambda_1 = a-\frac{i q \bar\chi}{2 k^2} \,, \qquad \lambda_2 = a-\frac{i (k^2+2 q^2)\bar\chi}{2 k^2 q} \ee and take the form of two Schr\"odinger equations~\eqref{SchrEq}, from which we can extract the scalings\footnote{We have simplified an absolute value in the $\lambda_2$ scaling which always encloses a positive expression within the parameter space.} \be\label{ACI} \Im \big[\mathcal G^R_{\lambda_1\lambda_1}\big] \sim \omega^{|1-(2+d_\theta)/z|} , \qquad \Im \big[\mathcal G^R_{\lambda_2\lambda_2}\big] \sim \omega^{|3+(d_\theta-2)/z|} \sim \omega^{3+(d_\theta-2)/z} . \ee Within the allowed parameter space, $\lambda_1$ is always the most relevant of the two in the IR, so that the optical conductivity scales like \be\label{ACcond1} \Re [\sigma] \sim \omega^{|1-(2+d_\theta )/z|-1} . \ee The next question is the sign of the exponent, as well as the sign of the expression within the absolute value. We find that the expression in the absolute value is positive whenever the gauge coupling is bounded\footnote{We thank A.~Donos for pointing this out to us.} by \be\label{ABsCrit1} \begin{aligned} -\frac{(p-3)}{p-1} +\frac{(p-2)}{p-1}\,z &< \frac{\kappa \gamma}2 < 1 \,, \quad &z &< 0 \,, \\ \text{or}\quad 1 &< \frac{\kappa \gamma}2 < -\frac{(p-3)}{p-1} +\frac{(p-2)}{p-1}\,z \,, \quad &z &> 2 \,. \end{aligned} \ee This range contains both insulating solutions~\eqref{InsCriterion1} as well as metallic ones. Then, the scaling of the optical conductivity in~\eqref{ACcond1} agrees with the scaling of the resisitivity we derived in~\eqref{resistivity1}, as expected from the scaling argument in~\cite{Donos-ml-2012ra}. There is however a region of the allowed parameter space, where the gauge coupling is not bounded and the resistivity vanishes at zero temperatures, such that the absolute value takes the opposite sign. In this case, we generically have a metal (vanishing resistivity) with a positive power tail in the optical conductivity, which always differs from the DC scaling. We will come back to this in the discussion. In the semi-locally critical limit $\theta=-\eta z$, $\z\to+\infty$, the optical conductivity becomes \be\label{ACcondSL1} \Re [\sigma] \sim \omega^{|1-\eta|-1} , \ee which associates a $1/\omega$ power tail to the linear resistivity case $\eta=1$, in agreement with the argument of~\cite{Donos-ml-2012ra}. It is worth noting that in this limit, the resistivity vanishes at zero temperature and the system always describes a metal. \paragraph{Class II (irrelevant current, marginally relevant axion).} As mentioned above, the two perturbations $a$ and $\bar\chi$ decouple in the IR and can be shown to obey Schr\"odinger equations. From this, we derive the scalings \be\label{ACII} \Im \big[\mathcal G^R_{aa}\big] \sim \omega^{|1+(\zeta -2)/z|} , \qquad \Im \big[\mathcal G^R_{\tilde\chi\tilde\chi}\big] \sim \omega^{|3+(d_\theta-2)/z|} \sim \omega^{3+(d_\theta-2)/z} . \ee The $\bar\chi$ scaling\footnote{We have simplified an absolute value which always encloses a positive expression within the parameter space.} is identical to the $\lambda_2$ perturbation of class I, while the $a$ scaling reduces to the $\lambda_1$ one upon taking $\zeta=-d_\theta$. The $a$ perturbation is the most IR-relevant if the conduction exponent is bounded by \be \operatorname{Min} [4(1-z) ,0] < \zeta-\zeta_I < \operatorname{Max} [4(1-z),0] \,. \ee This range has to be further restricted to \be \operatorname{Min} [2-z ,-d_\theta] < \zeta+d_\theta < \operatorname{Max} [2-z,-d_\theta] \ee in order for the expression within the absolute value to have the right sign to match the resistivity scaling~\eqref{resistivity2}. Outside of that range, the optical conductivity scaling differs from the resistivity scaling~\eqref{resistivity2}. When the optical conductivity is given by the $a$ perturbation, then the system can behave both as a metal or as an insulator. When it is metallic, the power tail can decay or blow up towards $\omega\to0$, while it always decays for insulators. When the optical conductivity scaling is given by the $\bar\chi$ perturbation, it reads \be \sigma_{\bar\chi} \sim \omega^{2+(d_\theta-2)/z} \ee and the exponent is always positive, so this power tail decays towards $\omega\to0$. The resisitivity~\eqref{resistivity2} always vanishes, so we have a metal. This is consistent with the fact that the conduction exponent is not bounded, so the system does not localize. \paragraph{Class III (marginally relevant current, irrelevant axion).} We find the two following contributions to the imaginary part of the UV Green's function: \be\label{ACIII} \Im \big[\mathcal G^R_{aa}\big] \sim \omega^{|3+(d_\theta-2 )/z|} \sim \omega^{3+(d_\theta-2)/z} , \qquad \Im \big[\mathcal G^R_{\tilde\chi\tilde\chi}\big] \sim \omega^{|1+(\kappa \lambda-d_\theta)/z|} . \ee Both can dominate the low-frequency behaviour. When the $a$ contribution does, the system is always metallic and the frequency-dependent power tail at zero temperature is always decaying. When the $\tilde\chi$ contribution dominates, the system can be both metallic and insulating. The frequency-dependent power tail at zero temperature can both vanish or blow up at zero frequency in the metallic case, and it always vanishes in the insulating case. Moreover, the expression in the absolute value matches the resistivity scaling~\eqref{resistivity3} when the axion-dilaton coupling is bounded \be \operatorname{Min}(-2,d_\theta-z) < \kappa \lambda < \operatorname{Max} (-2,d_\theta-z) \ee in terms of the conduction exponent. \paragraph{Class IV (irrelevant current, irrelevant axion).} We find the two following contributions to the imaginary part of the UV Green's function: \be\label{ACIV} \Im \big[\mathcal G^R_{aa}\big] \sim \omega^{|\zeta-1|} \sim \omega^{\zeta-1} , \qquad \Im \big[\mathcal G^R_{\tilde\chi\tilde\chi}\big] \sim \omega^{|1-d_\theta +\kappa \lambda|} \sim \omega^{-(1-d_\theta +\kappa \lambda)} . \ee In both cases we have removed the absolute value, as allowed by the parameter space~\eqref{ParSpaceIV}. Both can dominate the low-frequency behaviour. The $\tilde\chi$ perturbation dominates when \be -\zeta+d_\theta < \kappa \lambda < -2 \ee in terms of the conduction exponent. Irrespectively of which perturbation is the most relevant at low frequencies, the power tails always decay. Because of the sign inversion of the $\tilde\chi$ perturbation, the scaling of the optical conductivity at low frequencies can never match that of the resistivity~\eqref{resistivity4}. \paragraph{Discussion.} In~\cite{g2013}, we found that there was a single operator in the IR, giving rise to the following scaling of the optical conductivity at zero temperature and low frequency \be\label{ACscalingsTI} \begin{aligned} &\text{Relevant current:}\ &\Re \big[\sigma (\omega\ll\mu,T=0)\big] &\sim \delta(\omega)+\omega^{|3+(d_\theta-2)/z|-1} \\ &\text{Irrelevant current:}\ &\Re \big[\sigma (\omega\ll\mu,T=0)\big] &\sim \delta(\omega)+\omega^{|1-\zeta|-1} . \end{aligned} \ee Moreover, in the allowed parameter space, the expression within the absolute value was always positive and could be simplified. In the setup of this paper, the analysis above shows that there is now an extra propagating mode due to the presence of the axions. The scalings of~\eqref{ACscalingsTI} are most obviously compared to those of the class III and IV solutions~\eqref{ACIII} and~\eqref{ACIV}, from which it is clear that indeed the same mode is still present (the $a$ mode). This is because in these classes the axions are treated as irrelevant IR operators. Remarkably however, a mode with the same IR dimension is still present in class I and II, where the IR operators mix the $a$ and $\tilde\chi$ perturbations. Generically, we find that the expression under the square root in~\eqref{ImGreenScaling} is always a perfect square, hence it simplifies into an absolute value. The fact that this absolute value can change sign can be understood in the following way: the procedure we have just described amounts to taking the ratio of the normalisable over the non-normalisable piece of the IR perturbation, which then gives the imaginary part of the IR Green's function. These two pieces now typically come accompanied by a power of the radial coordinate which depends on the set of scaling exponents of the solution. Depending on their value, the two pieces can actually exchange roles, the non-normalisable piece becoming normalisable and vice-versa. This explains the absolute value, which accounts for the uncertainty over which piece is which. This has a dramatic consequence: only one sign for the absolute value (the positive sign in our covention) for only one of the IR perturbations can match the resistivity scalings of the previous section. It turns out that the scaling of the optical conductivity at zero temperature and low frequencies can differ from the scaling of the resistivity, either because the absolute value has the wrong sign, or because the `wrong' perturbation is the most relevant. Ultimately, this can be traced back to the presence of the running scalar and to the violation of scaling symmetries~\eqref{ACscaling1},~\eqref{ACscaling2}. This is confirmed by the fact that the scaling symmetries are violated when the gauge or axion-dilaton couplings are unbounded by other scaling exponents, so that the dilaton running is `strong'. At the transition point where the expression in the absolute value changes sign and scale invariance~\eqref{ACscaling1},~\eqref{ACscaling2} is violated, the resistivity is automatically linear in temperature with a $1/\omega$ tail in the optical conductivity, which is reminescent of the mechanism pointed out in~\cite{Donos-ml-2012ra}. We do however find consistent behaviours. Whenever the system is insulating (so there is no delta function at zero temperature and zero frequency), the diverging of the resistivity at small temperatures is matched with a vanishing of the optical conductivity at zero temperature and small frequencies. On the other hand, when the system is metallic and the resistivity vanishes at low temperatures, one may expect the Drude peak to sharpen into a delta function at exactly zero frequency, that is \be \Re \big[\sigma (\omega\ll\mu,T=0)\big] \sim \delta(\omega)+\omega^{|n|-1} . \ee However, $|n|-1$ can actually be positive or negative. If it is positive, we have a vanishing power law and a diverging DC conductivity, so all the low energy spectral weight is transferred to the delta function. If it is negative, there is a diverging power tail which seems to wash out the delta function and signals that some spectral weight does remain at non zero energies. It would be interesting to verify this by numerical computations, in particular whether a delta function is still present when there is a negative power tail, which cannot be inferred from the analytical calculations above (but could be on inspection of the imaginary part of the conductivity from the presence of a $1/\omega$ pole). ]]>
0$, which again is not necessarily guaranteed by the parameter space. This reflects the fact that in the IR region, the `source' and the `vev' (i.e.\ the non-normalisable and normalisable pieces) can be exchanged depending on the values of the scaling exponents. However, when the violation of scale invariance is realised in this way, the resistivity becomes linear in temperature at this transition point $n=0$, which is reminescent of the mechanism described in~\cite{Donos-ml-2012ra} (albeit at zero momentum). One interesting consequence of our analysis is the following: whenever the resistivity is linear (like for instance in the semi-locally critical case analogous to~\cite{Davison-ml-2013txa}), the power tail in the optical conductivity goes like $1/\omega$ (as was also pointed out in~\cite{Donos-ml-2012ra}). It would be desirable to understand what consequences this has on the calculation of the sum rule on the real part of the conductivity, which is not integrable at $\omega=0$, and whether these two features can be decoupled. In this work, we have considered spatially-dependent but linear sources for the axions, which are marginal deformations of the UV CFT\@. It would be very interesting to make these deformations relevant while retaining homogeneity, like in~\cite{Donos-ml-2013eha}, and investigate how and if these results change, particularly the various scaling behaviours. This also opens the way for an analysis of phases which are spatially anisotropic. Another interesting setup could involve helical (Bianchi VII) symmetries,~\cite{Donos-ml-2012js}. There, some extra scaling exponents are needed to parameterize the spatial anisotropy, but the conductivity displays similar scaling properties, in particular with negative, frequency-dependent power tails as well as insulating behaviour,~\cite{BianchiVII,BianchiVII.m001}. Of course, efficient, power-law momentum relaxation will not always occur in this setup: lattices at finite $z$ relaxing momentum at the lattice scale should result in Boltzmann-suppressed resistivities~\cite{Hartnoll-ml-2012rj,semilocal,semilocal.m001}. In~\cite{Davison-ml-2013txa}, it was shown that for a specific $\eta=1$ semi-locally critical geometry in holographic massive gravity, the resistivity scaled linearly like the entropy $\rho\sim s$: the explanation put forward was that if the late-time behaviour of the system is controlled by hydrodynamics,\footnote{Assuming a hydrodynamic state can form, which means Umklapp scattering should occur on much longer timescales.} the momentum relaxation rate associated to quenched disorder is set by the shear viscosity, which is famously related to the entropy density via a universal ratio. Hence~\cite{Davison-ml-2013txa} concluded that massive gravity captures leading effects of quenched disorder, while subleading corrections in $1/\log T$ have to be worked out for instance using the memory matrix formalism. When it dominates, the dissipative term in~\eqref{DCgenccl} generates a leading contribution to the resistivity \be\label{rhodissccl} \rho \sim s \,T^{-\frac{\kappa\lambda}{z}} , \qquad s \sim T^{\frac{d_\theta}z} . \ee Obviously, it reproduces the result of~\cite{Davison-ml-2013txa} in the semi-locally critical limit $z\to+\infty$. In the presence of spatially-dependent axions, the universality of the shear viscosity to entropy ratio will be violated, but will only generate subleading corrections~\cite{DiscDavison} so we can expect the previous result to still hold. It would be interesting to understand if there is some universality behind the temperature prefactor. At finite $z$,~\cite{Lucas-ml-2014zea} analyzed the effects of random-field disorder on a generic hyperscaling violating but scale invariant theory. They found that for relevant disorder with UV scaling dimension $\Delta$, the contribution to the resistivity was $\rho\sim T^{2(1+\Delta-z)/z}$, which upon saturation of the Harris criterion (meaning that disorder becomes marginally relevant and perturbation theory breaks down), turned into $\rho\sim s\, T^{2/z}$. Remarkably, this scaling coincides with~\eqref{rhodissccl} when the axions are marginally relevant in the IR and $\kappa\lambda=-2$ (classes I and II), bringing further evidence that the axions capture some of the IR physics associated with random-field disorder and can relax momentum efficiently at finite $z$. We have worked at zero momentum throughout this paper: indeed the disorder calculations~\cite{Davison-ml-2013txa,Lucas-ml-2014zea} are also dominated by low momenta modes.~\cite{Hartnoll-ml-2014cua} has shown that (UV) marginal disorder gave rise to Lifshitz IR geometries with $z$ finite: our results for class II solutions, which have IR marginally relevant axions and finite $z$ backgrounds, seem to resonate with the interpretation that our massless scalars capture random disorder physics at low temperatures. \paragraph{Note added.} \cite{Donos-ml-2014uba} appeared simultaneously where a subset of class II solutions as well as anisotropic phases analogous to class I are discussed in four bulk dimensions. The formula for the DC conductivity is also obtained for $D=4$ as well as qualitatively similar results for the scaling of the conductivity. ]]>
0 \,, \quad k^2V_0 < \frac{-2+2 z}{-z+p z-\theta } \,, \quad \text{and} \\ z &< 0 \,, ~~ \theta > p-1 \quad\text{or}\quad 1 < z \leq 2 \,, ~~ \theta < (z-1)(p-1) \quad\text{or}\quad z > 2 \,, ~~ \theta < p-1 \end{split} \ee with a maximum value for $k^2$. \paragraph{Semi-locally critical limit.} In the limit, \be \theta \to +\infty \,, \qquad z \to +\infty \,, \qquad \frac{\theta}{z} = -\eta \ee the solution~\eqref{solClassI} becomes conformal to AdS$_2\times\mathbf R^{p-1}$: \be\label{solClassISL} \begin{aligned} \ud s^2 &= r^{-\frac{2\eta}{p-1}} \bigg[\frac{L^2\ud r^2-\ud t^2}{r^2}+\ud\vec{x}^2\bigg] \,, &L^2 &= \frac{2 (1+\eta )^2}{2 V_0-k^2 (p-2)} \\ A &= \sqrt{\frac{2 \big(2V_0+k^2 (p -1+\eta)\big)}{\big(k^2(p-2)-2 V_0\big) (1+\eta)}} r^{-1-\eta}\ud t \,, \\ e^\phi &= r^\kappa &\kappa^2 &= \frac{2\eta (p-1+\eta )}{p-1} \\ \gamma &= (2-p) \delta \,, \qquad\qquad \lambda = 0 \,, &\kappa\delta &= \frac{-2 \eta }{p-1} \,, \end{aligned} \ee with blackness function \be f(r) = 1-{\bigg(\frac{r}{r_h}\bigg)\!}^{1+\eta} . \ee The allowed parameter space is \be\label{ParSpaceISL} V_0 > 0 \,, \qquad k^2V_0 < \frac{2}{-1+p+\eta } \,, \qquad \eta > 0 \,. \ee ]]>
0 \,, \qquad \theta &\leq 0 \,, &z &> 1 \,, &\zeta &< 1-p+\theta \quad\text{or} \\ 0 < \theta &< -1+p \,, \quad &z &> \frac{-1+p+\theta }{-1+p} \,, \quad &\zeta &< 1-p+\theta \quad\text{or} \\ \theta &> -1+p \,, &z &< 0 \,, &\zeta &> 1-p+\theta \,. \end{aligned} \ee Once all these constraints are taken into account, the Null Energy Condition holds and the $tt$ element of the metric scales together with the spatial elements. \paragraph{Semi-locally critical limit.} A semi-locally critical limit can be taken as well, upon which the axion-dilaton coupling goes to a constant in the IR $\lambda=0$. However, here the limit should also include the conduction exponent $\zeta=\tilde\zeta z$, $z\to+\infty$, in order to allow for full generality in the scaling of the electric potential. The allowed parameter space is \be\label{ParSpaceIISL} V_0 > 0 \,, \qquad \eta > 0 \,, \qquad \tilde\zeta < -\eta \,. \ee ]]>
0$ as well as: \be\label{ParSpaceIII} \begin{aligned} z &< 0 \,, \quad &\theta &> -1+p \,, &\kappa \lambda &> -2 \\ 1 < z &\leq 2 \,, &\theta &< 1-p-z+p z \,, \quad &\kappa \lambda &< -2 \\ z &> 2 \,, &\theta &< -1+p \,, &\kappa \lambda &< -2 \end{aligned} \ee ]]>