obtained at this highly symmetric point clearly contain non-trivial excitations, consistent with what one would expect from general principles.

By examining the ‘building blocks’ (4.32) and (4.33) from which all Green’s functions are constructed, it can be seen that they do not contain any poles (nor zeroes, in the case of $G_A$) that lie in the upper half of the complex frequency plane and that would indicate perturbative instabilities of the state.

Finally, we can define a ‘viscosity’ for this state using the usual Kubo formula

$$ \eta = - \lim_{\omega \to 0} \frac{1}{\omega} \mathrm{Im} G_{T^{xy}T^{xy}}^{R}(\omega, k=0) \approx 0.325 \times \frac{s}{4\pi}. \qquad (4.43) $$

This violates the KSS viscosity bound [54] (it avoids the general proof [16] because, due to the translational symmetry breaking, the equation of motion for $h_y^x$ is no longer that of a minimally coupled massless scalar field). However, this is not a conventional viscosity controlling, for example, the rate of transverse momentum diffusion, since our system is no longer described by hydrodynamics in this incoherent regime. A violation of the viscosity bound was found in a related holographic model in [55].

# 5 Charge transport in non-Maxwell theories

In this section, we will address some holographic models of strongly interacting field theory states with a global U(1) symmetry. We will primarily be interested in their electrical conductivity: the linear response of the U(1) current $\vec{J}$ to an external source. In a strongly interacting system of this kind, there is no reason to expect that the current is almost-conserved (there is no symmetry protecting it) and thus one would expect the charge transport to be incoherent. Coherent charge transport arises naturally in holographic models in which the current overlaps with the momentum, which can be almost-conserved [3]. Here, we will instead focus on states in which the current and momentum are not coupled, i.e. perturbations of the gauge field in the gravitational theory decouple from perturbations of the metric. We find that there are coherent regimes of charge transport, despite the fact there is no obvious symmetry producing an almost-conserved quantity.

The two kinds of holographic theory we study — probe brane systems and neutral systems with non-minimal gauge field actions — are deformations of the Maxwell action. The Maxwell theory is incoherent: it has a constant optical conductivity $\sigma(\omega) = \sigma_{DC}$, and charge transport occurs via diffusion [37]. Upon deforming this theory — with a probe brane charge density or a higher-derivative gauge field term in the action — coherent peaks of the form (1.2) can emerge.

## 5.1 Probe brane theories at non-zero density

Firstly, we will address simple probe brane theories with a trivial embedding, for which the action may schematically be written

$$ S = N_c^2 \int d^{d+2}x \sqrt{-g} (\mathcal{R} + \dots) + N_c \int d^{p+1}x \sqrt{-\det(g_{ab} + F_{ab})}, \qquad (5.1) $$
obtained at this highly symmetric point clearly contain non-trivial excitations, consistent with what one would expect from general principles.

By examining the ‘building blocks’ (4.32) and (4.33) from which all Green’s functions are constructed, it can be seen that they do not contain any poles (nor zeroes, in the case of $G_A$) that lie in the upper half of the complex frequency plane and that would indicate perturbative instabilities of the state.

Finally, we can define a ‘viscosity’ for this state using the usual Kubo formula

$$ \eta = - \lim_{\omega \to 0} \frac{1}{\omega} \mathrm{Im} G_{T^{xy}T^{xy}}^{R}(\omega, k=0) \approx 0.325 \times \frac{s}{4\pi}. \qquad (4.43) $$

This violates the KSS viscosity bound [54] (it avoids the general proof [16] because, due to the translational symmetry breaking, the equation of motion for $h_y^x$ is no longer that of a minimally coupled massless scalar field). However, this is not a conventional viscosity controlling, for example, the rate of transverse momentum diffusion, since our system is no longer described by hydrodynamics in this incoherent regime. A violation of the viscosity bound was found in a related holographic model in [55].

# 5 Charge transport in non-Maxwell theories

In this section, we will address some holographic models of strongly interacting field theory states with a global U(1) symmetry. We will primarily be interested in their electrical conductivity: the linear response of the U(1) current $\vec{J}$ to an external source. In a strongly interacting system of this kind, there is no reason to expect that the current is almost-conserved (there is no symmetry protecting it) and thus one would expect the charge transport to be incoherent. Coherent charge transport arises naturally in holographic models in which the current overlaps with the momentum, which can be almost-conserved [3]. Here, we will instead focus on states in which the current and momentum are not coupled, i.e. perturbations of the gauge field in the gravitational theory decouple from perturbations of the metric. We find that there are coherent regimes of charge transport, despite the fact there is no obvious symmetry producing an almost-conserved quantity.

The two kinds of holographic theory we study — probe brane systems and neutral systems with non-minimal gauge field actions — are deformations of the Maxwell action. The Maxwell theory is incoherent: it has a constant optical conductivity $\sigma(\omega) = \sigma_{DC}$, and charge transport occurs via diffusion [37]. Upon deforming this theory — with a probe brane charge density or a higher-derivative gauge field term in the action — coherent peaks of the form (1.2) can emerge.

## 5.1 Probe brane theories at non-zero density

Firstly, we will address simple probe brane theories with a trivial embedding, for which the action may schematically be written

$$ S = N_c^2 \int d^{d+2}x \sqrt{-g} (\mathcal{R} + \dots) + N_c \int d^{p+1}x \sqrt{-\det(g_{ab} + F_{ab})}, \qquad (5.1) $$
where $N_c \to \infty$ and $\{a, b\}$ go from 0...p. These theories have a long and rich history and we refer the reader to [56-58] and their references for more details. The $O(N_c^2)$ terms, which dominate in this limit, determine the metric of the background solution. The subleading $O(N_c)$ term is important because it controls the charge transport properties of the theory: it is the leading order at which the gauge field $F_{ab}$ appears. Physically, one can think of this as a system with $O(N_c)$ charged degrees of freedom living in $p-1$ spatial dimensions coupled to a bath of $O(N_c^2)$ neutral degrees of freedom living in $d$ spatial dimensions. In this probe limit, linear perturbations of the metric and the gauge field decouple. This means that the cross correlator of $\vec{J}$ with $\vec{P}$ vanishes in the dual field theory state: the dynamics of the current do not affect the dynamics of the momentum since the contribution of the charged degrees of freedom to $\vec{P}$ is suppressed by $1/N_c$ with respect to the contribution of the neutral degrees of freedom.

In a zero density state (i.e. when the background solution has $A_t = 0$), the gauge field perturbations obey the Maxwell equations and thus there is incoherent transport and charge diffusion. At non-zero densities ($A_t \neq 0$), non-linear effects in the action (5.1) become important and as the ratio of the U(1) chemical potential to the temperature $\mu/T$ increases, these effects are more pronounced and qualitatively change the system's electrical transport properties. For a probe brane in a geometry corresponding to a neutral sector with dynamical critical exponent $z$ and hyperscaling violation exponent $\theta$, there is a delta function in the $T/\mu = 0$ optical conductivity when [59-62]

$$z < 2(1 - \theta/d), \qquad (5.2)$$

assuming that $z > 0$ and $\theta \le 0$. For $z$ larger than this upper bound, there is no delta function the $T = 0$ optical conductivity, which now grows as a power law at small $\omega$.

In systems that obey the inequality (5.2), there is a very clear agreement with the effective theory of coherent transport outlined in the introduction. For simplicity we will focus on the case $z = 1$, $\theta = 0$ (e.g. the supersymmetric D3/D5 and D3/D7 systems) which has been studied the most thoroughly. The characteristic dissipation rate in this state is $\Gamma \sim T^2/\mu$, and may be read off from the coherent peak observed in the non-zero temperature optical conductivity [58]. This is consistent with the spatially resolved charge transport: at very long distances $k \ll \Gamma$, charge is transported via diffusion while at shorter distances $\Gamma \ll k$, the longest lived excitation that carries charge is the “holographic zero sound” mode [63-65]. A similar transition at intermediate distance scales is found in the Sakai-Sugimoto model, where $\Gamma \sim T^3/\mu^2$ [66, 67].

A simple explanation of this phenomenology would be that the system has an almost-conserved quantity, with dissipation rate $\Gamma \sim T^2/\mu$, that overlaps with the current. This would-be explanation begs the questions of what is conserved when $T = 0$ and why? In our hydrodynamic model of heat conductivity in section 2, the conserved quantity is the total momentum, and it is conserved due to a symmetry: translational invariance. This also makes it clear that $\Gamma$ in that case should increase with $m$ — the parameter controlling the translational symmetry breaking — and that energy should be transported at short distances by sound-like waves — the excitations characteristic of a system with both conserved momentum and energy. One possible answer for our probe brane theories is that the current
where $N_c \to \infty$ and $\{a, b\}$ go from 0...p. These theories have a long and rich history and we refer the reader to [56-58] and their references for more details. The $O(N_c^2)$ terms, which dominate in this limit, determine the metric of the background solution. The subleading $O(N_c)$ term is important because it controls the charge transport properties of the theory: it is the leading order at which the gauge field $F_{ab}$ appears. Physically, one can think of this as a system with $O(N_c)$ charged degrees of freedom living in $p-1$ spatial dimensions coupled to a bath of $O(N_c^2)$ neutral degrees of freedom living in $d$ spatial dimensions. In this probe limit, linear perturbations of the metric and the gauge field decouple. This means that the cross correlator of $\vec{J}$ with $\vec{P}$ vanishes in the dual field theory state: the dynamics of the current do not affect the dynamics of the momentum since the contribution of the charged degrees of freedom to $\vec{P}$ is suppressed by $1/N_c$ with respect to the contribution of the neutral degrees of freedom.

In a zero density state (i.e. when the background solution has $A_t = 0$), the gauge field perturbations obey the Maxwell equations and thus there is incoherent transport and charge diffusion. At non-zero densities ($A_t \neq 0$), non-linear effects in the action (5.1) become important and as the ratio of the U(1) chemical potential to the temperature $\mu/T$ increases, these effects are more pronounced and qualitatively change the system's electrical transport properties. For a probe brane in a geometry corresponding to a neutral sector with dynamical critical exponent $z$ and hyperscaling violation exponent $\theta$, there is a delta function in the $T/\mu = 0$ optical conductivity when [59-62]

$$z < 2(1 - \theta/d), \qquad (5.2)$$

assuming that $z > 0$ and $\theta \le 0$. For $z$ larger than this upper bound, there is no delta function the $T = 0$ optical conductivity, which now grows as a power law at small $\omega$.

In systems that obey the inequality (5.2), there is a very clear agreement with the effective theory of coherent transport outlined in the introduction. For simplicity we will focus on the case $z = 1$, $\theta = 0$ (e.g. the supersymmetric D3/D5 and D3/D7 systems) which has been studied the most thoroughly. The characteristic dissipation rate in this state is $\Gamma \sim T^2/\mu$, and may be read off from the coherent peak observed in the non-zero temperature optical conductivity [58]. This is consistent with the spatially resolved charge transport: at very long distances $k \ll \Gamma$, charge is transported via diffusion while at shorter distances $\Gamma \ll k$, the longest lived excitation that carries charge is the “holographic zero sound” mode [63-65]. A similar transition at intermediate distance scales is found in the Sakai-Sugimoto model, where $\Gamma \sim T^3/\mu^2$ [66, 67].

A simple explanation of this phenomenology would be that the system has an almost-conserved quantity, with dissipation rate $\Gamma \sim T^2/\mu$, that overlaps with the current. This would-be explanation begs the questions of what is conserved when $T = 0$ and why? In our hydrodynamic model of heat conductivity in section 2, the conserved quantity is the total momentum, and it is conserved due to a symmetry: translational invariance. This also makes it clear that $\Gamma$ in that case should increase with $m$ — the parameter controlling the translational symmetry breaking — and that energy should be transported at short distances by sound-like waves — the excitations characteristic of a system with both conserved momentum and energy. One possible answer for our probe brane theories is that the current
itself becomes conserved in the $T=0$ limit, but it is not clear why this would be the case in a strongly interacting system, why it should depend on $T$, and why the characteristic excitation associated with conservation of both charge and current would be the holographic zero sound mode. We note that the $z=1$, $\theta=0$ solution has a boost symmetry in the limit $T=0$ [68], and it would be interesting to see if this generalises to other values of $z$, $\theta$ satisfying the inequality (5.2). On a related note, the existence of zero temperature diffusive excitations in these states was found to be a result of the existence of moduli spaces for certain scalar operators [69]. It could be the case that the probe limit is singular and it would also be worthwhile to go beyond this limit in these kind of systems (e.g. [70]): this may qualitatively change the low energy transport when even a small amount of backreaction is taken into account, as the current will then couple to the total momentum, which is conserved.

For values of $z$ larger than the upper bound in the inequality (5.2), less information is available. At least when $\theta=0$, there appears to be a coherent peak in the optical conductivity at small, non-zero $T/\mu$, for all values of $z$ [58]. Unlike in the cases satisfying the inequality (5.2), there is no delta function peak in the $T=0$ optical conductivity, but a power law divergence as $\omega \to 0$, indicating that the $\omega \to 0$ and $T \to 0$ limits do not commute. Furthermore, the $k \neq 0$ excitation at zero temperature is no longer the holographic zero sound mode [59–62]. It would be worthwhile exploring these type of states more carefully to determine how exactly they transport charge, but this is beyond the scope of this paper.

## 5.2 Neutral theories with higher-derivative couplings

The second class of systems that we look at are neutral states of theories in which the gauge field action of the gravitational theory includes not only the Maxwell term, but higher-derivative corrections to this arising due to couplings of the gauge field to the Weyl tensor. In particular, consider the action

$$S = \int d^4 x \sqrt{-g} \left( \mathcal{R} + 6 - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \gamma_1 C_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta} - \gamma_2 C_{\alpha\beta\rho\sigma} C^{\rho\sigma\alpha\beta} F_{\mu\nu} F^{\mu\nu} \right), \quad (5.3)$$

where $C_{\alpha\beta\rho\sigma}$ is the Weyl tensor. The relevant neutral solution to this action is the neutral AdS$_4$-Schwarzschild black brane. When $\gamma_1 = \gamma_2 = 0$, the action (5.3) reduces to the Maxwell action and so the optical conductivity $\sigma(\omega)$ is constant due to self-duality, and charge diffuses at non-zero wavenumbers [37]. There is incoherent charge transport. When either of $\gamma_i$ are non-zero, this self-duality is broken [71–77], and Drude-like peaks can begin to appear. As in the probe brane examples, we do not know of any reason (such as an approximate conservation law) why this is the case.

When $\gamma_1 > 0$ (and $\gamma_2 = 0$), a low frequency peak appears in the optical conductivity which becomes sharper and taller as $\gamma_1$ is increased. This peak arises due to a pole at $\omega = -i\Gamma$ where $\Gamma \sim T/\gamma_1^{0.66}$, which becomes long-lived with respect to the other excitations of the theory (with decay rates $\Lambda \sim T$) in the limit $\gamma_1 \gg 1$. Thus there is coherent charge transport in the limit $\gamma_1 \gg 1$, with a transition to incoherent charge transport when $\gamma_1 \sim 1$. However, $\gamma_1$ is bounded by causality, and must satisfy $-1/12 \le \gamma_1 \le 1/12$ [71, 78]. When $\gamma_1 < 0$, there is no long-lived excitation in the conductivity (in fact, the closest feature to the origin is a zero), and thus the transport is incoherent.
itself becomes conserved in the $T = 0$ limit, but it is not clear why this would be the case in a strongly interacting system, why it should depend on $T$, and why the characteristic excitation associated with conservation of both charge and current would be the holographic zero sound mode. We note that the $z = 1$, $\theta = 0$ solution has a boost symmetry in the limit $T = 0$ [68], and it would be interesting to see if this generalises to other values of $z$, $\theta$ satisfying the inequality (5.2). On a related note, the existence of zero temperature diffusive excitations in these states was found to be a result of the existence of moduli spaces for certain scalar operators [69]. It could be the case that the probe limit is singular and it would also be worthwhile to go beyond this limit in these kind of systems (e.g. [70]): this may qualitatively change the low energy transport when even a small amount of backreaction is taken into account, as the current will then couple to the total momentum, which is conserved.

For values of $z$ larger than the upper bound in the inequality (5.2), less information is available. At least when $\theta = 0$, there appears to be a coherent peak in the optical conductivity at small, non-zero $T/\mu$, for all values of $z$ [58]. Unlike in the cases satisfying the inequality (5.2), there is no delta function peak in the $T = 0$ optical conductivity, but a power law divergence as $\omega \to 0$, indicating that the $\omega \to 0$ and $T \to 0$ limits do not commute. Furthermore, the $k \neq 0$ excitation at zero temperature is no longer the holographic zero sound mode [59–62]. It would be worthwhile exploring these type of states more carefully to determine how exactly they transport charge, but this is beyond the scope of this paper.

## 5.2 Neutral theories with higher-derivative couplings

The second class of systems that we look at are neutral states of theories in which the gauge field action of the gravitational theory includes not only the Maxwell term, but higher-derivative corrections to this arising due to couplings of the gauge field to the Weyl tensor. In particular, consider the action

$$S = \int d^4 x \sqrt{-g} \left( \mathcal{R} + 6 - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \gamma_1 C_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta} - \gamma_2 C_{\alpha\beta\rho\sigma} C^{\rho\sigma\alpha\beta} F_{\mu\nu} F^{\mu\nu} \right), \quad (5.3)$$

where $C_{\alpha\beta\rho\sigma}$ is the Weyl tensor. The relevant neutral solution to this action is the neutral AdS$_4$-Schwarzschild black brane. When $\gamma_1 = \gamma_2 = 0$, the action (5.3) reduces to the Maxwell action and so the optical conductivity $\sigma(\omega)$ is constant due to self-duality, and charge diffuses at non-zero wavenumbers [37]. There is incoherent charge transport. When either of $\gamma_i$ are non-zero, this self-duality is broken [71–77], and Drude-like peaks can begin to appear. As in the probe brane examples, we do not know of any reason (such as an approximate conservation law) why this is the case.

When $\gamma_1 > 0$ (and $\gamma_2 = 0$), a low frequency peak appears in the optical conductivity which becomes sharper and taller as $\gamma_1$ is increased. This peak arises due to a pole at $\omega = -i\Gamma$ where $\Gamma \sim T/\gamma_1^{0.66}$, which becomes long-lived with respect to the other excitations of the theory (with decay rates $\Lambda \sim T$) in the limit $\gamma_1 \gg 1$. Thus there is coherent charge transport in the limit $\gamma_1 \gg 1$, with a transition to incoherent charge transport when $\gamma_1 \sim 1$. However, $\gamma_1$ is bounded by causality, and must satisfy $-1/12 \le \gamma_1 \le 1/12$ [71, 78]. When $\gamma_1 < 0$, there is no long-lived excitation in the conductivity (in fact, the closest feature to the origin is a zero), and thus the transport is incoherent.
When $\gamma_2 > 0$ (and $\gamma_1 = 0$), there are similar results. A low frequency peak appears in the optical conductivity which becomes sharper and taller at larger $\gamma_2$. This is due to a pole in the conductivity at $\omega = -i\Gamma$ where $\Gamma \sim T/\gamma_2^{0.83}$ [76], which is well-separated from the other excitations with imaginary parts $\Lambda \sim T$. Again, there is coherent charge transport when $\gamma_2 \gg 1$, with a transition to incoherent charge transport when $\gamma_2 \sim 1$. However, unlike in the previous example, the coefficient $\gamma_2$ is *not* bounded from above and so there is a coherent regime of charge transport in this theory. However, this coherent regime exists only when the six-derivative term in the action (5.3) is more important than the two- and four-derivative terms. This is totally at odds with the effective field theory approach taken in writing down the action (5.3), and so it would be unwise to take this coherent regime too seriously. The $\gamma_2$ term in the action (5.3) plays the role of an $r$-dependent coupling constant for the U(1) gauge field, and it would be worthwhile to investigate if the same effect (i.e. coherent charge transport) could be obtained in a neutral Einstein-Maxwell-dilaton system with exponential coupling, in which the $r$-dependent coupling constant is generated by the dilaton rather than by a higher derivative term. Indeed, it is known that electro-magnetic duality can be restored in this case by also flipping the sign of the coupling between the dilaton and the Maxwell term in the action and field equations [79, 80].

## 6 Discussion

In this paper, we have studied how heat is transported in neutral systems in which momentum is not conserved. Firstly, we examined a hydrodynamic system in which the conservation equations were modified to include momentum dissipation. We then compared its transport properties to those of a strongly coupled, holographic system, in which momentum dissipates due to the breaking of translational invariance by linearly dependent scalar sources controlled by a parameter $m$, and found excellent agreement. At sufficiently small $m$, momentum dissipates slowly at the rate $\Gamma \sim m^2/T$. The low energy, spatially uniform transport properties of these systems are controlled by the interplay of two scales: the momentum dissipation rate $\Gamma$, and the thermal scale $\Lambda \sim T$. The systems can be broadly classified into two regimes: the *coherent regime* when $\Gamma \ll \Lambda$, and the *incoherent regime* when $\Gamma \gtrsim \Lambda$. Within each of these regimes, the spatially resolved transport properties depend upon the distance scale one examines.

In the coherent regime, the low frequency AC conductivity at zero $k$ has a Drude-like form (figure 2). The late time transport properties are dominated by a single pole in the complex frequency plane with dispersion relation $\omega = -i\Gamma$, which is parametrically separated from the other poles with decay rates $\sim \Lambda$ (figure 3). At non-zero $k$, the heat transport mechanism depends upon the distance scale. Over the longest distance scales $k \ll \Lambda \ll \Gamma$, momentum significantly dissipates and the heat transport is dominated by diffusion. Over shorter distances $\Lambda \ll k \ll \Gamma$, momentum is long-lived and sound-like modes carry heat. As $k$ is varied, the crossover between these is realised by a collision of poles of the conductivity in the complex frequency plane (figure 4).

In the incoherent regime $\Gamma \gtrsim \Lambda$, the low energy AC conductivity no longer has a simple Drude-like form, and there is no longer a Drude-like collective excitation that is well-
When $\gamma_2 > 0$ (and $\gamma_1 = 0$), there are similar results. A low frequency peak appears in the optical conductivity which becomes sharper and taller at larger $\gamma_2$. This is due to a pole in the conductivity at $\omega = -i\Gamma$ where $\Gamma \sim T/\gamma_2^{0.83}$ [76], which is well-separated from the other excitations with imaginary parts $\Lambda \sim T$. Again, there is coherent charge transport when $\gamma_2 \gg 1$, with a transition to incoherent charge transport when $\gamma_2 \sim 1$. However, unlike in the previous example, the coefficient $\gamma_2$ is *not* bounded from above and so there is a coherent regime of charge transport in this theory. However, this coherent regime exists only when the six-derivative term in the action (5.3) is more important than the two- and four-derivative terms. This is totally at odds with the effective field theory approach taken in writing down the action (5.3), and so it would be unwise to take this coherent regime too seriously. The $\gamma_2$ term in the action (5.3) plays the role of an $r$-dependent coupling constant for the U(1) gauge field, and it would be worthwhile to investigate if the same effect (i.e. coherent charge transport) could be obtained in a neutral Einstein-Maxwell-dilaton system with exponential coupling, in which the $r$-dependent coupling constant is generated by the dilaton rather than by a higher derivative term. Indeed, it is known that electro-magnetic duality can be restored in this case by also flipping the sign of the coupling between the dilaton and the Maxwell term in the action and field equations [79, 80].

## 6 Discussion

In this paper, we have studied how heat is transported in neutral systems in which momentum is not conserved. Firstly, we examined a hydrodynamic system in which the conservation equations were modified to include momentum dissipation. We then compared its transport properties to those of a strongly coupled, holographic system, in which momentum dissipates due to the breaking of translational invariance by linearly dependent scalar sources controlled by a parameter $m$, and found excellent agreement. At sufficiently small $m$, momentum dissipates slowly at the rate $\Gamma \sim m^2/T$. The low energy, spatially uniform transport properties of these systems are controlled by the interplay of two scales: the momentum dissipation rate $\Gamma$, and the thermal scale $\Lambda \sim T$. The systems can be broadly classified into two regimes: the *coherent regime* when $\Gamma \ll \Lambda$, and the *incoherent regime* when $\Gamma \gtrsim \Lambda$. Within each of these regimes, the spatially resolved transport properties depend upon the distance scale one examines.

In the coherent regime, the low frequency AC conductivity at zero $k$ has a Drude-like form (figure 2). The late time transport properties are dominated by a single pole in the complex frequency plane with dispersion relation $\omega = -i\Gamma$, which is parametrically separated from the other poles with decay rates $\sim \Lambda$ (figure 3). At non-zero $k$, the heat transport mechanism depends upon the distance scale. Over the longest distance scales $k \ll \Lambda \ll \Gamma$, momentum significantly dissipates and the heat transport is dominated by diffusion. Over shorter distances $\Lambda \ll k \ll \Gamma$, momentum is long-lived and sound-like modes carry heat. As $k$ is varied, the crossover between these is realised by a collision of poles of the conductivity in the complex frequency plane (figure 4).

In the incoherent regime $\Gamma \gtrsim \Lambda$, the low energy AC conductivity no longer has a simple Drude-like form, and there is no longer a Drude-like collective excitation that is well-
separated from the other collective excitations of the theory. In our holographic model, we
have shown that as $\Gamma$ is increased, the heat conductivity changes from a sharp Drude-like
peak to a constant, and then to a valley (figure 2). The longitudinal heat transport is always
diffusive in this regime, as momentum is never approximately conserved. Although this is
in agreement with the statement of [31] that incoherent transport is mediated by diffusive
processes, we emphasise that long distance transport in the coherent regime is also mediated
by diffusive processes. The difference between them lies at shorter distance scales, where
in the coherent case a new excitation, associated with the approximate conservation law,
takes over. At low temperatures $T \ll m$ (which coincides with the incoherent regime), the
diffusion constant does not depend on temperature, $D_{\parallel} = \sqrt{3/2}/m$, and the dimensionless
combination $D_{\parallel}T$ can be made arbitrarily small. It would be interesting to re-examine it
at non-zero density in light of the bound on energy/charge diffusion constants proposed
in [31] (see also [81]).

In the coherent regime, the approximate conservation of momentum means that heat
is transported in the transverse directions by a pseudo-diffusive excitation (figure 7). At
shorter distances than this, and in the incoherent regime, there is no approximate conservation law and thus no long-lived collective excitations.

In this paper, we have focused mainly on the heat conductivity and its poles in the com-
plex frequency plane. There are other potentially important features in the conductivity besides its poles, such as its zeroes [72, 74], and other analytic contributions at low frequencies.
It would be interesting to study these more carefully in our model. It would also be interesting to extend our analysis of low energy transport $\omega \ll T$ to study processes at higher energies $T \ll \omega \ll m$. We have focused on a holographic model where the zero temperature IR fixed point is given by an $AdS_2 \times R^2$ geometry. It is easy to generalise this geometry to other values of the exponents $z$ and $\theta$, by introducing couplings to a neutral scalar [21, 41, 42], and it would be interesting to ascertain the effects of this on the transport properties. Including a constant background magnetic field would also help bridge the results of [2, 17, 82, 83].

A more pressing question is to generalise our results to systems with a charge density,
such that the system’s momentum and current are coupled. The DC conductivities of the
two charges have been investigated in holographic systems of this type [11, 15, 17–27] but, as
we have demonstrated, this is not sufficient to infer the nature (i.e. coherent or incoherent)
of the conduction of each charge in a system. For very weak translational symmetry
breaking, the DC electrical conductivity in these systems is not directly proportional to the
momentum dissipation rate, indicating that there is a richer structure in the low energy
physics of the system. This is presumably due to the existence of an extra conserved
charge (the electrical U(1) charge) and its associated diffusive excitation. Calculations to
determine the precise nature of the transport in these systems are underway, and we hope
to report on this soon.

Besides this pressing question, there are further interesting questions raised by our
results. We found that for a specific value of the parameter *m* of our holographic model,
there is a constant AC heat conductivity due to a self-duality of the gravitational perturbations analogous to that of Maxwell theory in (Schwarzschild-)AdS₄. In the Maxwell
case, small deviations from the self-dual action turn this self-duality into duality relating
separated from the other collective excitations of the theory. In our holographic model, we
have shown that as $\Gamma$ is increased, the heat conductivity changes from a sharp Drude-like
peak to a constant, and then to a valley (figure 2). The longitudinal heat transport is always
diffusive in this regime, as momentum is never approximately conserved. Although this is
in agreement with the statement of [31] that incoherent transport is mediated by diffusive
processes, we emphasise that long distance transport in the coherent regime is also mediated
by diffusive processes. The difference between them lies at shorter distance scales, where
in the coherent case a new excitation, associated with the approximate conservation law,
takes over. At low temperatures $T \ll m$ (which coincides with the incoherent regime), the
diffusion constant does not depend on temperature, $D_{\parallel} = \sqrt{3/2}/m$, and the dimensionless
combination $D_{\parallel}T$ can be made arbitrarily small. It would be interesting to re-examine it
at non-zero density in light of the bound on energy/charge diffusion constants proposed
in [31] (see also [81]).

In the coherent regime, the approximate conservation of momentum means that heat
is transported in the transverse directions by a pseudo-diffusive excitation (figure 7). At
shorter distances than this, and in the incoherent regime, there is no approximate conservation law and thus no long-lived collective excitations.

In this paper, we have focused mainly on the heat conductivity and its poles in the com-
plex frequency plane. There are other potentially important features in the conductivity be-
sides its poles, such as its zeroes [72, 74], and other analytic contributions at low frequencies.
It would be interesting to study these more carefully in our model. It would also be interest-
ing to extend our analysis of low energy transport $\omega \ll T$ to study processes at higher ener-
gies $T \ll \omega \ll m$. We have focused on a holographic model where the zero temperature IR
fixed point is given by an $\text{AdS}_2 \times \mathbb{R}^2$ geometry. It is easy to generalise this geometry to other
values of the exponents $z$ and $\theta$, by introducing couplings to a neutral scalar [21, 41, 42], and
it would be interesting to ascertain the effects of this on the transport properties. Including
a constant background magnetic field would also help bridge the results of [2, 17, 82, 83].

A more pressing question is to generalise our results to systems with a charge density,
such that the system’s momentum and current are coupled. The DC conductivities of the
two charges have been investigated in holographic systems of this type [11, 15, 17–27] but, as
we have demonstrated, this is not sufficient to infer the nature (i.e. coherent or incoherent)
of the conduction of each charge in a system. For very weak translational symmetry
breaking, the DC electrical conductivity in these systems is not directly proportional to the
momentum dissipation rate, indicating that there is a richer structure in the low energy
physics of the system. This is presumably due to the existence of an extra conserved
charge (the electrical U(1) charge) and its associated diffusive excitation. Calculations to
determine the precise nature of the transport in these systems are underway, and we hope
to report on this soon.

Besides this pressing question, there are further interesting questions raised by our
results. We found that for a specific value of the parameter *m* of our holographic model,
there is a constant AC heat conductivity due to a self-duality of the gravitational perturbations analogous to that of Maxwell theory in (Schwarzschild-)AdS<sub>4</sub>. In the Maxwell
case, small deviations from the self-dual action turn this self-duality into duality relating
two different theories: one with “particle-like” excitations carrying charge and one with “vortex-like” excitations carrying charge (i.e. one with a low energy pole in the conductivity and one with a low energy zero) [37, 71], and we expect an analogous duality to hold for our neutral theory close to the self-dual point $m = \sqrt{8\pi T}$. Near the self-dual point (where the thermodynamic energy vanishes), our system has either a small negative or positive thermodynamic energy. To get a better understanding of the physics of this neutral state, it would be very interesting to find a (non-holographic) field theory with heat conduction properties of this type, along the lines of [37]. Moreover, at the self-dual point, an extra $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$ is present and allows to solve exactly for the Green’s functions for all $\omega$ and $k$. Assuming it is still given by the usual Kubo formula, a ‘shear viscosity’ can be calculated exactly $\eta \approx 0.325 \times s/4\pi$, and violates the KSS bound. Such violations are due to translational symmetry breaking and will also occur in states with different values of $m$.

On a related note, the similarities between our holographic model and massive gravity (where the parameter $m$ now becomes related to the graviton mass) have been discussed in [18, 26]. It is known that linearized massive gravity in de Sitter backgrounds becomes partially massless at a specific value of the graviton mass, with a corresponding emergent self-duality [50, 51]. It would be interesting to understand better the links between the two theories, and possibly the emergence of self-duality in massive gravity.

Finally, in light of recent memory matrix studies of transport properties, we highlighted some interesting properties of probe brane theories and neutral theories with higher-derivative couplings between the gauge field and the Weyl tensor. In particular, probe brane theories at low temperatures typically exhibit coherent peaks in their low energy transport, as can the higher-derivative theories (although in limits where effective field theory breaks down or causality is violated). According to memory matrix calculations, an almost-conserved quantity overlapping with the charge current will produce a coherent peak in the low energy electrical conductivity [1–4]. But is the converse true? Does the existence of coherent peaks, e.g. in probe brane systems, indicate an approximate conservation law at work? If so, this conservation law should also explain the nature of the holographic zero sound excitation in these systems.

## Acknowledgments

We are grateful to Sean Hartnoll, Diego Hofman, Elias Kiritsis, Andrei Parnachev, Anastasios Petkou, Koenraad Schalm, David Tong and Jan Zaanen for helpful discussions. We also thank Sean Hartnoll, Chris Herzog and Andy Lucas for comments on the manuscript. The work of R.D. is supported by a VIDI grant from NWO, the Netherlands Organisation for Scientific Research. The work of B.G. is supported by the Marie Curie International Outgoing Fellowship nr 624054 within the 7th European Community Framework Programme FP7/2007-2013.
two different theories: one with “particle-like” excitations carrying charge and one with “vortex-like” excitations carrying charge (i.e. one with a low energy pole in the conductivity and one with a low energy zero) [37, 71], and we expect an analogous duality to hold for our neutral theory close to the self-dual point $m = \sqrt{8\pi T}$. Near the self-dual point (where the thermodynamic energy vanishes), our system has either a small negative or positive thermodynamic energy. To get a better understanding of the physics of this neutral state, it would be very interesting to find a (non-holographic) field theory with heat conduction properties of this type, along the lines of [37]. Moreover, at the self-dual point, an extra $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$ is present and allows to solve exactly for the Green’s functions for all $\omega$ and $k$. Assuming it is still given by the usual Kubo formula, a ‘shear viscosity’ can be calculated exactly $\eta \approx 0.325 \times s/4\pi$, and violates the KSS bound. Such violations are due to translational symmetry breaking and will also occur in states with different values of $m$.

On a related note, the similarities between our holographic model and massive gravity (where the parameter $m$ now becomes related to the graviton mass) have been discussed in [18, 26]. It is known that linearized massive gravity in de Sitter backgrounds becomes partially massless at a specific value of the graviton mass, with a corresponding emergent self-duality [50, 51]. It would be interesting to understand better the links between the two theories, and possibly the emergence of self-duality in massive gravity.

Finally, in light of recent memory matrix studies of transport properties, we highlighted some interesting properties of probe brane theories and neutral theories with higher-derivative couplings between the gauge field and the Weyl tensor. In particular, probe brane theories at low temperatures typically exhibit coherent peaks in their low energy transport, as can the higher-derivative theories (although in limits where effective field theory breaks down or causality is violated). According to memory matrix calculations, an almost-conserved quantity overlapping with the charge current will produce a coherent peak in the low energy electrical conductivity [1–4]. But is the converse true? Does the existence of coherent peaks, e.g. in probe brane systems, indicate an approximate conservation law at work? If so, this conservation law should also explain the nature of the holographic zero sound excitation in these systems.

## Acknowledgments

We are grateful to Sean Hartnoll, Diego Hofman, Elias Kiritsis, Andrei Parnachev, Anastasios Petkou, Koenraad Schalm, David Tong and Jan Zaanen for helpful discussions. We also thank Sean Hartnoll, Chris Herzog and Andy Lucas for comments on the manuscript. The work of R.D. is supported by a VIDI grant from NWO, the Netherlands Organisation for Scientific Research. The work of B.G. is supported by the Marie Curie International Outgoing Fellowship nr 624054 within the 7th European Community Framework Programme FP7/2007-2013.
## A Details of numerical calculations

The linear response properties of quantum field theory states are encoded in their two-point retarded Green's functions. To calculate these from the gravitational dual, we must solve the equations of motion for the linear perturbations around the corresponding black brane solution. The two-point retarded Green's function of an operator $O$, dual to a bulk field $\phi_O$, is then given by the value of the on-shell gravitational action at quadratic order in perturbations, after imposing the boundary conditions that the 'source' term of each field is zero, except the source term of $\phi_O$ which should be equal to one, and that all field perturbations are ingoing at the black brane horizon. We refer the reader to [84, 85] for more technical details.

In the following subsection we will give the relevant equations of motion for the fluctuations of each field, which are typically coupled to one another. After determining the appropriate decoupled field perturbations, we solve their equations of motion by numerical integration from the black brane horizon to the spacetime boundary and use these solutions to compute the retarded Green's functions, and their poles, that we present in the main body of the paper.

### A.1 Neutral axionic theory

In the neutral axionic theory with action (3.1), we are primarily concerned with the linear perturbations of $\delta g_{tx^i}$, which are dual to the momentum density operators $T^{tx^i}$ of the dual field theory. In general, we are interested in these perturbations at non-zero frequency $\omega$ and wavenumber $\vec{k}$. We choose our co-ordinates such that the wavenumber $\vec{k}$ points in the $x$-direction. The linear perturbations of the metric $\delta g_{\mu\nu} \equiv h_{\mu\nu}$ and of the scalar fields $\delta\phi_i$ form two decoupled sets: those related to longitudinal transport ($h_{tt}$, $h_{xx}$, $h_{yy}$, $h_{rr}$, $h_{xt}$, $h_{xr}$, $h_{tr}$, $\delta\phi_1$), and those related to transverse transport ($h_{yt}$, $h_{yx}$, $h_{yr}$, $\delta\phi_2$). The longitudinal fields obey the coupled equations of motion

$$0 = \frac{d}{dr} [r^4 f (\delta\phi_1' - mh_r^x)] + \frac{1}{f} (\omega^2 - k^2 f) \delta\phi_1 - \frac{i\omega m}{f} h_r^x + \frac{ikm}{2} (h_t^t - h_x^x + h_y^y + h_r^r), \quad (A.1)$$

$$0 = \frac{d}{dr} [r^4 f (h_x^{x'} - h_y^{y'} - 2ikh_r^x)] + 2ikm\delta\phi_1 + \frac{1}{f} (\omega^2 - m^2 f) (h_x^x - h_y^y) + \frac{2\omega k}{f} h_r^x - k^2 (h_t^t + h_r^r), \quad (A.2)$$

$$0 = \frac{d}{dr} [r^4 (h_t^{x'} + i\omega h_r^x)] - \frac{ik}{f} h_r^{t'} + \frac{\omega k}{f} (h_y^y + h_r^r) - \frac{i\omega m}{f} \delta\phi_1 - \frac{m^2}{f} h_t^x, \quad (A.3)$$

$$0 = i\omega (h_x^{x'} + h_y^{y'}) + ikh_t^{x'} + \omega kh_r^x + \frac{k^2}{r^4 f} h_t^r - \frac{i\omega f'}{2f} (h_x^x + h_y^y) - \frac{2i\omega}{r} h_r^r - ik \frac{f'}{f} h_t^x, \quad (A.4)$$

$$0 = ik (h_t^{t'} + h_y^{y'}) - \frac{i\omega}{f} h_t^{x'} + \frac{\omega^2}{f} h_r^x + \frac{\omega k}{r^4 f^2} h_t^r + \frac{ik f'}{2f} h_t^t - ik \left(\frac{f'}{2f} + \frac{2}{r}\right) h_r^r - m^2 h_r^x + m\delta\phi_1', \quad (A.5)$$

$$0 = 2r^3 f^2 h_t^{t'} + 2r^4 f^2 \left(\frac{f'}{4f} + \frac{1}{r}\right) (h_x^{x'} + h_y^{y'}) + ikmf\delta\phi_1 - 2ikr^4 f^2 \left(\frac{f'}{2f} + \frac{2}{r}\right) h_r^x - \frac{4i\omega}{r} h_t^r + \omega^2 (h_x^x + h_y^y) - k^2 f (h_t^t + h_y^y) + 2\omega k h_t^x - \frac{1}{2} m^2 f (h_x^x + h_y^y) + f (m^2 - 6r^2) h_r^r, \quad (A.6)$$
## A Details of numerical calculations

The linear response properties of quantum field theory states are encoded in their two-point retarded Green's functions. To calculate these from the gravitational dual, we must solve the equations of motion for the linear perturbations around the corresponding black brane solution. The two-point retarded Green's function of an operator $O$, dual to a bulk field $\phi_O$, is then given by the value of the on-shell gravitational action at quadratic order in perturbations, after imposing the boundary conditions that the 'source' term of each field is zero, except the source term of $\phi_O$ which should be equal to one, and that all field perturbations are ingoing at the black brane horizon. We refer the reader to [84, 85] for more technical details.

In the following subsection we will give the relevant equations of motion for the fluctuations of each field, which are typically coupled to one another. After determining the appropriate decoupled field perturbations, we solve their equations of motion by numerical integration from the black brane horizon to the spacetime boundary and use these solutions to compute the retarded Green's functions, and their poles, that we present in the main body of the paper.

### A.1 Neutral axionic theory

In the neutral axionic theory with action (3.1), we are primarily concerned with the linear perturbations of $\delta g_{tx^i}$, which are dual to the momentum density operators $T^{tx^i}$ of the dual field theory. In general, we are interested in these perturbations at non-zero frequency $\omega$ and wavenumber $\vec{k}$. We choose our co-ordinates such that the wavenumber $\vec{k}$ points in the $x$-direction. The linear perturbations of the metric $\delta g_{\mu\nu} \equiv h_{\mu\nu}$ and of the scalar fields $\delta\phi_i$ form two decoupled sets: those related to longitudinal transport ($h_{tt}$, $h_{xx}$, $h_{yy}$, $h_{rr}$, $h_{xt}$, $h_{xr}$, $h_{tr}$, $\delta\phi_1$), and those related to transverse transport ($h_{yt}$, $h_{yx}$, $h_{yr}$, $\delta\phi_2$). The longitudinal fields obey the coupled equations of motion

$$0 = \frac{d}{dr} [r^4 f (\delta\phi_1' - mh_r^x)] + \frac{1}{f} (\omega^2 - k^2 f) \delta\phi_1 - \frac{i\omega m}{f} h_r^x + \frac{ikm}{2} (h_t^t - h_x^x + h_y^y + h_r^r), \quad (A.1)$$

$$0 = \frac{d}{dr} [r^4 f (h_x^{x'} - h_y^{y'} - 2ikh_r^x)] + 2ikm\delta\phi_1 + \frac{1}{f} (\omega^2 - m^2 f) (h_x^x - h_y^y) + \frac{2\omega k}{f} h_t^x - k^2 (h_t^t + h_r^r), \quad (A.2)$$

$$0 = \frac{d}{dr} [r^4 (h_t^{x'} + i\omega h_r^x)] - \frac{ik}{f} h_r^{t'} + \frac{\omega k}{f} (h_y^y + h_r^r) - \frac{i\omega m}{f} \delta\phi_1 - \frac{m^2}{f} h_t^x, \quad (A.3)$$

$$0 = i\omega (h_x^{x'} + h_y^{y'}) + ikh_t^{x'} + \omega kh_r^x + \frac{k^2}{r^4 f} h_t^r - \frac{i\omega f'}{2f} (h_x^x + h_y^y) - \frac{2i\omega}{r} h_r^r - ik \frac{f'}{f} h_t^x, \quad (A.4)$$

$$0 = ik (h_t^{t'} + h_y^{y'}) - \frac{i\omega}{f} h_t^{x'} + \frac{\omega^2}{f} h_r^x + \frac{\omega k}{r^4 f^2} h_t^r + \frac{ik f'}{2f} h_t^t - ik \left(\frac{f'}{2f} + \frac{2}{r}\right) h_r^r - m^2 h_r^x + m\delta\phi_1', \quad (A.5)$$

$$0 = 2r^3 f^2 h_t^{t'} + 2r^4 f^2 \left(\frac{f'}{4f} + \frac{1}{r}\right) (h_x^{x'} + h_y^{y'}) + ikmf\delta\phi_1 - 2ikr^4 f^2 \left(\frac{f'}{2f} + \frac{2}{r}\right) h_r^x - \frac{4i\omega}{r} h_t^r + \omega^2 (h_x^x + h_y^y) - k^2 f (h_t^t + h_y^y) + 2\omega k h_t^x - \frac{1}{2} m^2 f (h_x^x + h_y^y) + f (m^2 - 6r^2) h_r^r, \quad (A.6)$$
$$ 
\begin{aligned}
0 = & r^4 f^2 (h_x^{x''} + h_y^{y''}) + \frac{1}{2} r^3 f (rf' + 8f) (h_x^{x'} + h_y^{y'}) - 2ikr^4 f^2 h_r^{x'} - 2r^3 f^2 h_r^{r'} \\
& - ikr^3 f (rf' + 8f) h_r^x + ikm f \delta\phi_1 - k^2 f (h_y^y + h_r^r) - \frac{1}{2} m^2 f (h_x^x + h_y^y) \\
& + \frac{f}{3} [3m^2 - 18r^2 + 6r^2 (3f + rf')] h_t^t - \frac{r^2 f}{3} (6rf' + 18f) h_r^r,
\end{aligned} 
\quad (A.7) $$

$$ 
\begin{aligned}
0 = & 2r^4 f^2 h_t^{t''} + r^3 f (3rf' + 8f) h_t^{t'} + r^4 f^2 (h_x^{x''} + h_y^{y''}) - 2ikr^4 f^2 h_r^{x'} - 4i\omega h_t^{r'} \\
& + r^3 f (rf' + 4f) (h_x^{x'} + h_y^{y'} - h_r^{r'}) - 2ikr^3 f (rf' + 4f) h_r^x + 2i\omega \frac{f'}{f} h_t^r + 2\omega^2 h_r^r \\
& + \omega^2 (h_x^x + h_y^y) - k^2 f (h_t^t + h_r^r) + 2\omega k h_t^x - \frac{2r^2 f}{3} (18rf' + 18f + 3r^2 f'') h_r^r,
\end{aligned} 
\quad (A.8) $$

and the transverse fields obey the coupled equations of motion

$$ 0 = \frac{d}{dr} [r^4 f (h_x^{y'} - ikh_r^y)] + \frac{\omega}{f} (\omega h_x^y + kh_t^y) - m^2 h_x^y + ikm\delta\phi_2, \quad (A.9) $$

$$ 0 = \frac{d}{dr} \left[ r^4 \left( h_t^{y'} + i\omega h_r^y \right) \right] - \frac{k}{f} (\omega h_x^y + kh_t^y) - \frac{m^2}{f} h_t^y - \frac{i\omega m}{f} \delta\phi_2, \quad (A.10) $$

$$ 0 = \frac{d}{dr} [r^4 f (\delta\phi_2' - m h_r^y)] + \frac{1}{f} (\omega^2 - k^2 f) \delta\phi_2 - \frac{m}{f} (i\omega h_t^y + ik f h_x^y), \quad (A.11) $$

$$ 0 = i\omega h_t^{y'} + ik f h_x^{y'} - (\omega^2 - m^2 f - k^2 f) h_r^y - m f \delta\phi_2', \quad (A.12) $$

where we raise indices using the background metric (3.2) and primes denote derivatives with respect to $r$. Longitudinal and transverse transport in this theory are therefore, in general, independent of one another.

Despite the large number of fields and equations, there are only four independent degrees of freedom in the system: two in the longitudinal sector and two in the transverse sector. The multitude of field equations are a redundancy due to the diffeomorphism symmetry of the theory, that can be removed by working directly with combinations of the fields that are invariant under linearised diffeomorphisms $h_{\mu\nu} \to h_{\mu\nu} + \nabla_\mu \xi_\nu + \nabla_\nu \xi_\mu$ and $\delta\phi_i \to \delta\phi_i + m\xi^i$. By choosing these variables carefully, we can identify a decoupled, gauge-invariant field perturbation in each sector that determines the two-point function of the appropriate component of the momentum density $P_i = T^{tx^i}$. This decoupling is extremely advantageous when it comes to solving these equations. In the longitudinal sector, the field

$$ 
\begin{aligned}
\psi_{\parallel} \equiv & \frac{r}{m^2 + k^2} [m (h_x^x - h_y^y) - 2ik\delta\phi_1] && (A.13) \\
& - \frac{2r^4 f}{m (k^2 + r^3 f')} \left[ h_x^{x'} + h_y^{y'} - \frac{2}{r} h_r^r - 2ik h_r^x - \frac{k^2 + r^3 f'}{r^3 f} h_y^y \right], &&
\end{aligned}
$$

obeys the equation of motion

$$ \frac{d}{dr} [r^2 f \psi_{\parallel}'] + \left( \frac{\omega^2 - k^2 f}{r^2 f} + V(r) \right) \psi_{\parallel} = 0, \quad (A.14) $$

where

$$ V(r) = - \frac{3r_0 (m^2 - 2r_0^2)}{2r^3 [2k^2 r + m^2 (2r - 3r_0) + 6r_0^3]^2} \left[ 4k^4 r^2 + m^4 (-4r^2 + 6rr_0 - 3r_0^2) \right. \\
\left. \qquad + 12m^2 r_0 (r^3 - rr_0^2 + r_0^3) - 12r_0^3 (2r^3 + r_0^3) \right], \quad (A.15) $$
$$ 
\begin{aligned}
0 ={}& r^4 f^2 (h_x^{x\prime\prime} + h_y^{y\prime\prime}) + \frac{1}{2} r^3 f (rf' + 8f) (h_x^{x\prime} + h_y^{y\prime}) - 2ikr^4 f^2 h_r^{x\prime} - 2r^3 f^2 h_r^{r\prime} \\
& - ikr^3 f (rf' + 8f) h_r^x + ikmf\delta\phi_1 - k^2 f (h_y^y + h_r^r) - \frac{1}{2}m^2 f (h_x^x + h_y^y) \\
& + \frac{f}{3} [3m^2 - 18r^2 + 6r^2 (3f + rf')] h_t^t - \frac{r^2 f}{3} (6rf' + 18f) h_r^r,
\end{aligned} 
\quad (A.7) $$

$$ 
\begin{aligned}
0 ={}& 2r^4 f^2 h_t^{t\prime\prime} + r^3 f (3rf' + 8f) h_t^{t\prime} + r^4 f^2 (h_x^{x\prime\prime} + h_y^{y\prime\prime}) - 2ikr^4 f^2 h_r^{x\prime} - 4i\omega h_t^{r\prime} \\
& + r^3 f (rf' + 4f) (h_x^{x\prime} + h_y^{y\prime} - h_r^{r\prime}) - 2ikr^3 f (rf' + 4f) h_r^x + 2i\omega \frac{f'}{f} h_t^r + 2\omega^2 h_r^r \\
& + \omega^2 (h_x^x + h_y^y) - k^2 f (h_t^t + h_r^r) + 2\omega k h_t^x - \frac{2r^2 f}{3} (18rf' + 18f + 3r^2 f'') h_r^r,
\end{aligned} 
\quad (A.8) $$

and the transverse fields obey the coupled equations of motion

$$ 0 = \frac{d}{dr} [r^4 f (h_x^{y\prime} - ikh_r^y)] + \frac{\omega}{f} (\omega h_x^y + kh_t^y) - m^2 h_x^y + ikm\delta\phi_2, \quad (A.9) $$

$$ 0 = \frac{d}{dr} \left[ r^4 \left( h_t^{y\prime} + i\omega h_r^y \right) \right] - \frac{k}{f} (\omega h_x^y + kh_t^y) - \frac{m^2}{f} h_t^y - \frac{i\omega m}{f} \delta\phi_2, \quad (A.10) $$

$$ 0 = \frac{d}{dr} [r^4 f (\delta\phi_2' - m h_r^y)] + \frac{1}{f} (\omega^2 - k^2 f) \delta\phi_2 - \frac{m}{f} (i\omega h_t^y + ik f h_x^y), \quad (A.11) $$

$$ 0 = i\omega h_t^{y\prime} + ik f h_x^{y\prime} - (\omega^2 - m^2 f - k^2 f) h_r^y - m f \delta\phi_2', \quad (A.12) $$

where we raise indices using the background metric (3.2) and primes denote derivatives with respect to $r$. Longitudinal and transverse transport in this theory are therefore, in general, independent of one another.

Despite the large number of fields and equations, there are only four independent degrees of freedom in the system: two in the longitudinal sector and two in the transverse sector. The multitude of field equations are a redundancy due to the diffeomorphism symmetry of the theory, that can be removed by working directly with combinations of the fields that are invariant under linearised diffeomorphisms $h_{\mu\nu} \rightarrow h_{\mu\nu} + \nabla_{\mu}\xi_{\nu} + \nabla_{\nu}\xi_{\mu}$ and $\delta\phi_i \rightarrow \delta\phi_i + m\xi^i$. By choosing these variables carefully, we can identify a decoupled, gauge-invariant field perturbation in each sector that determines the two-point function of the appropriate component of the momentum density $P_i = T^{tx^i}$. This decoupling is extremely advantageous when it comes to solving these equations. In the longitudinal sector, the field

$$ 
\begin{aligned}
\psi_{\parallel} ={}& \frac{r}{m^2 + k^2} [m (h_x^x - h_y^y) - 2ik\delta\phi_1] \\
& - \frac{2r^4 f}{m (k^2 + r^3 f')} \left[ h_x^{x\prime} + h_y^{y\prime} - \frac{2}{r} h_r^r - 2ik h_r^x - \frac{k^2 + r^3 f'}{r^3 f} h_y^y \right],
\end{aligned} 
\quad (A.13) $$

obeys the equation of motion

$$ \frac{d}{dr} [r^2 f \psi_{\parallel}'] + \left( \frac{\omega^2 - k^2 f}{r^2 f} + V(r) \right) \psi_{\parallel} = 0, \quad (A.14) $$

where

$$ V(r) = - \frac{3r_0 (m^2 - 2r_0^2)}{2r^3 [2k^2 r + m^2 (2r - 3r_0) + 6r_0^3]^2} \left[ 4k^4 r^2 + m^4 (-4r^2 + 6rr_0 - 3r_0^2) \right. \\ 
\left. \qquad + 12m^2 r_0 (r^3 - rr_0^2 + r_0^3) - 12r_0^3 (2r^3 + r_0^3) \right], \quad (A.15) $$
and controls the two point function of $T^{xt}$. In the transverse sector, the field

$$ \psi_{\perp} \equiv r^3 (h_t^{y'} + i\omega h_r^y), \qquad (A.16) $$

obeys the equation of motion

$$ \frac{d}{dr} [r^2 f \psi'_{\perp}] + \frac{\omega^2 - k^2 f - m^2 f + r^3 f f'}{r^2 f} \psi_{\perp} = 0, \qquad (A.17) $$

and controls the two point function of $T^{yt}$. We note that it is also possible to write down decoupled equations of motion for the other two independent degrees of freedom of the system, but for conciseness we will not present them here.

To compute the Green's functions, we must determine the on-shell gravitational action at quadratic order in the fluctuations, after including the Gibbons-Hawking boundary term, and boundary counterterms

$$ S_{GH} + S_{ct} = \int d^3 x \sqrt{-\gamma} \left( -2K - 4 - \mathcal{R}[\gamma] + \sum_{i=1}^{2} \frac{1}{2} \partial_{\mu} \phi_i \partial^{\mu} \phi_i \right), \qquad (A.18) $$

where $\gamma$ is the induced metric and the integral is over the boundary of the spacetime. On-shell, we can replace the fields $h_t^r$, $h_r^r$, $h_x^x$ and $h_y^y$ by solving the constraint equations (A.4), (A.5), (A.6) and (A.12) for these fields. Furthermore, as the on-shell action is a boundary term, we can replace all of the remaining fields by their near-boundary expansions

$$ h_t^x = h_t^{x(0)} + \frac{h_t^{x(1)}}{r} + \frac{h_t^{x(2)}}{r^2} + \frac{h_t^{x(3)}}{r^3} + \frac{h_t^{x(4)}}{r^4} + \dots, \qquad (A.19) $$

and similarly for the other metric and scalar field fluctuations. On-shell, only two of these boundary coefficients are independent: the ‘source’ term $h_t^{x(0)}$ and the ‘expectation value’ term $h_t^{x(3)}$. The result of this is that the on-shell action can be written as a sum of terms which are quadratic in the source and expectation value coefficients. The expectation value terms should be regarded as functions of the source terms, with the precise relationship between them being fixed by demanding ingoing boundary conditions at the horizon. This relationship will be determined via numerical integration.

We have computed the quadratic on-shell action in this form in full generality but, to compute the retarded Green's functions of $T^{tx}$ and $T^{ty}$, we only need to know the on-shell action with all source terms vanishing, except $h_t^{x(0)}$ and $h_t^{y(0)}$. This is

$$ S = \int \frac{d\omega dk}{4\pi^2} \left( \frac{1}{4k^4 - 4m^2\omega^2 + 4\omega^4 + k^2(3m^2 - 8\omega^2)} \left[ \omega h_t^{x(0)} \left\{ -3k(k^2 + m^2)h_t^{(3)} \right. \right. \right. \\ 
+ \frac{3}{2}k(2k^2 + m^2 - 2\omega^2)h_y^{(3)} + \frac{3}{2}k(m^2 + 2\omega^2)h_x^{(3)} + 6\omega(k^2 + m^2)h_t^{(3)} \\ 
- 3im(k^2 - 2\omega^2)\delta\phi_1^{(3)} \left. \left. - \frac{1}{4}r_0(m^2 - 2r_0^2)\left\{ 8\omega^2(m^2 - \omega^2) + k^2(3m^2 + 10\omega^2) \right. \right. \\ 
+ 4k^4\right\}h_t^{x(0)}h_t^{x(0)} \left. - \frac{3}{2(k^2 + m^2 - \omega^2)}h_t^{y(0)}\left\{ (k^2 + m^2)h_t^{(3)} + \omega k h_x^{(3)} + im\omega\delta\phi_2^{(3)} \right. \right. \\ 
+ \frac{1}{2}r_0(m^2 - 2r_0^2)h_t^{y(0)}h_t^{y(0)} \right), \qquad (A.20) $$
and controls the two point function of $T^{xt}$. In the transverse sector, the field

$$ \psi_{\perp} \equiv r^3 (h_t^{y'} + i\omega h_r^y), \qquad (A.16) $$

obeys the equation of motion

$$ \frac{d}{dr} [r^2 f \psi'_{\perp}] + \frac{\omega^2 - k^2 f - m^2 f + r^3 f f'}{r^2 f} \psi_{\perp} = 0, \qquad (A.17) $$

and controls the two point function of $T^{yt}$. We note that it is also possible to write down decoupled equations of motion for the other two independent degrees of freedom of the system, but for conciseness we will not present them here.

To compute the Green's functions, we must determine the on-shell gravitational action at quadratic order in the fluctuations, after including the Gibbons-Hawking boundary term, and boundary counterterms

$$ S_{GH} + S_{ct} = \int d^3 x \sqrt{-\gamma} \left( -2K - 4 - \mathcal{R}[\gamma] + \sum_{i=1}^{2} \frac{1}{2} \partial_{\mu} \phi_i \partial^{\mu} \phi_i \right), \qquad (A.18) $$

where $\gamma$ is the induced metric and the integral is over the boundary of the spacetime. On-shell, we can replace the fields $h_t^r$, $h_r^r$, $h_x^r$ and $h_y^r$ by solving the constraint equations (A.4), (A.5), (A.6) and (A.12) for these fields. Furthermore, as the on-shell action is a boundary term, we can replace all of the remaining fields by their near-boundary expansions

$$ h_t^x = h_t^{x(0)} + \frac{h_t^{x(1)}}{r} + \frac{h_t^{x(2)}}{r^2} + \frac{h_t^{x(3)}}{r^3} + \frac{h_t^{x(4)}}{r^4} + \dots, \qquad (A.19) $$

and similarly for the other metric and scalar field fluctuations. On-shell, only two of these boundary coefficients are independent: the 'source' term $h_t^{x(0)}$ and the 'expectation value' term $h_t^{x(3)}$. The result of this is that the on-shell action can be written as a sum of terms which are quadratic in the source and expectation value coefficients. The expectation value terms should be regarded as functions of the source terms, with the precise relationship between them being fixed by demanding ingoing boundary conditions at the horizon. This relationship will be determined via numerical integration.

We have computed the quadratic on-shell action in this form in full generality but, to compute the retarded Green's functions of $T^{tx}$ and $T^{ty}$, we only need to know the on-shell action with all source terms vanishing, except $h_t^{x(0)}$ and $h_t^{y(0)}$. This is

$$ S = \int \frac{d\omega dk}{4\pi^2} \left( \frac{1}{4k^4 - 4m^2\omega^2 + 4\omega^4 + k^2(3m^2 - 8\omega^2)} \left[ \omega h_t^{x(0)} \left\{ -3k(k^2 + m^2)h_t^{(3)} \right. \right. \right. \\ 
+ \frac{3}{2}k(2k^2 + m^2 - 2\omega^2)h_y^{(3)} + \frac{3}{2}k(m^2 + 2\omega^2)h_x^{(3)} + 6\omega(k^2 + m^2)h_t^{(3)} \\ 
- 3im(k^2 - 2\omega^2)\delta\phi_1^{(3)} \left\} - \frac{1}{4}r_0(m^2 - 2r_0^2) \left\{ 8\omega^2(m^2 - \omega^2) + k^2(3m^2 + 10\omega^2) \right. \right. \\ 
+ 4k^4 \left\} h_t^{x(0)}h_t^{(0)} \left[ -\frac{3}{2(k^2 + m^2 - \omega^2)}h_t^{y(0)} \left\{ (k^2 + m^2)h_t^{(3)} + \omega k h_x^{(3)} + im\omega\delta\phi_2^{(3)} \right\} \right. \\ 
+ \frac{1}{2}r_0(m^2 - 2r_0^2)h_t^{y(0)}h_t^{(0)} \left. \right] \right) \qquad (A.20) $$
where, in each quadratic pair of field perturbations, the first has argument $(-\omega, -k)$ and the second has argument $(\omega, k)$. In general, we will neglect the terms $h_t^{x(0)}(-\omega, -k)h_t^{x(0)}(\omega, k)$ and $h_t^{y(0)}(-\omega, -k)h_t^{y(0)}(\omega, k)$ because these are real contact terms, whose only effect is to shift the real part of the dual Green's function by their prefactor. In particular, these terms do not affect the real (dissipative) parts of the conductivities.

The final step required to determine the Green's functions is to evaluate the specific combinations of expectation value terms, on a solution that is ingoing at the horizon and which has only one source term ($h_t^{x(0)} = 1$ or $h_t^{y(0)} = 1$) non-vanishing at the boundary. This is where the advantage of the gauge-invariant variables (A.13) and (A.16) is manifest. Near the boundary, these have the expansions

$$ \psi_{\perp}(r \to \infty) \to (k^2 + m^2) h_t^{y(0)} + \omega k h_x^{y(0)} + im\omega\delta\phi_2^{(0)} - \frac{3}{(k^2 + m^2 - \omega^2)r} \left[ (k^2 + m^2) h_t^{y(3)} + \omega k h_x^{y(3)} + im\omega\delta\phi_2^{(3)} \right] + O(1/r^2), \quad (A.21) $$

and

$$ \psi_{\parallel}(r \to \infty) \to -\frac{8k}{m(k^2+m^2)} \frac{1}{4k^4 - 4m^2\omega^2 + 4\omega^4 + k^2(3m^2 - 8\omega^2)} \times \left\{ \begin{aligned} & \frac{3}{2}k(k^2+m^2)h_t^{(3)} - \frac{3}{4}k(2k^2+m^2-2\omega^2)h_y^{(3)} - \frac{3}{4}k(m^2+2\omega^2)h_x^{(3)} \\ & - 3\omega(k^2+m^2)h_t^{(3)} + \frac{3}{2}im(k^2-2\omega^2)\delta\phi_1^{(3)} \\ & - \frac{3}{8}r_0(m^2-2r_0^2)(3k^2+2m^2-2\omega^2)(kh_t^{(0)} - 2\omega h_t^{(0)}) \end{aligned} \right\} + O(1/r), \quad (A.22) $$

respectively. Firstly, we will deal with the transverse correlator $G_{T^{ty}T^{ty}}^R$. A solution in which $h_t^{y(0)} = 1$ and all other source terms vanish, is a solution in which $\psi_{\perp}(r \to \infty) \to k^2+m^2$. It is straightforward to find such a solution: one just integrates the equation of motion (A.17) from the horizon, with ingoing boundary conditions, to the boundary. Multiplying this solution by $(k^2+m^2)$ and dividing by its value at the boundary will yield a solution with precisely the boundary conditions required, and thus

$$ (k^2+m^2) \left. \frac{r^2 \psi'_{\perp}(r)}{\psi_{\perp}(r)} \right|_{r \to \infty} = \frac{3}{k^2+m^2-\omega^2} \left[ (k^2+m^2) h_t^{y(3)} + \omega k h_x^{y(3)} + im\omega\delta\phi_2^{(3)} \right], \quad (A.23) $$

where the right hand side is evaluated with the boundary conditions that all fields are ingoing at the horizon and that the only non-vanishing source term at the boundary is $h_t^{y(0)} = 1$. Thus, up to contact terms,

$$ G_{T^{ty}T^{ty}}^{R}(\omega, k) = (k^2 + m^2) \left. \frac{r^2 \psi'_{\perp}(r)}{\psi_{\perp}(r)} \right|_{r \to \infty} . \quad (A.24) $$

A similar procedure works for the longitudinal retarded Green's functions. Near the boundary, the gauge-invariant variable (A.13) has the expansion (A.22). Since the leading order term involves the expectation values ($h_t^{x(3)}$ etc.), there is no simple expression for the
where, in each quadratic pair of field perturbations, the first has argument $(-\omega, -k)$ and the second has argument $(\omega, k)$. In general, we will neglect the terms $h_t^{x(0)}(-\omega, -k)h_t^{x(0)}(\omega, k)$ and $h_t^{y(0)}(-\omega, -k)h_t^{y(0)}(\omega, k)$ because these are real contact terms, whose only effect is to shift the real part of the dual Green's function by their prefactor. In particular, these terms do not affect the real (dissipative) parts of the conductivities.

The final step required to determine the Green's functions is to evaluate the specific combinations of expectation value terms, on a solution that is ingoing at the horizon and which has only one source term ($h_t^{x(0)} = 1$ or $h_t^{y(0)} = 1$) non-vanishing at the boundary. This is where the advantage of the gauge-invariant variables (A.13) and (A.16) is manifest. Near the boundary, these have the expansions

$$ \psi_{\perp}(r \to \infty) \to (k^2 + m^2) h_t^{y(0)} + \omega k h_x^{y(0)} + im\omega\delta\phi_2^{(0)} - \frac{3}{(k^2 + m^2 - \omega^2)r} \left[ (k^2 + m^2) h_t^{y(3)} + \omega k h_x^{y(3)} + im\omega\delta\phi_2^{(3)} \right] + O(1/r^2), \quad (A.21) $$

and

$$ \psi_{\parallel}(r \to \infty) \to -\frac{8k}{m(k^2+m^2)} \frac{1}{4k^4 - 4m^2\omega^2 + 4\omega^4 + k^2(3m^2 - 8\omega^2)} \times \left\{ \begin{aligned} & \frac{3}{2}k(k^2+m^2)h_t^{(3)} - \frac{3}{4}k(2k^2+m^2-2\omega^2)h_y^{(3)} - \frac{3}{4}k(m^2+2\omega^2)h_x^{(3)} \\ & - 3\omega(k^2+m^2)h_t^{(3)} + \frac{3}{2}im(k^2-2\omega^2)\delta\phi_1^{(3)} \\ & - \frac{3}{8}r_0(m^2-2r_0^2)(3k^2+2m^2-2\omega^2)(kh_t^{(0)}-2\omega h_t^{(0)}) \end{aligned} \right\} + O(1/r), \quad (A.22) $$

respectively. Firstly, we will deal with the transverse correlator $G_{T^{ty}T^{ty}}^R$. A solution in which $h_t^{y(0)} = 1$ and all other source terms vanish, is a solution in which $\psi_{\perp}(r \to \infty) \to k^2+m^2$. It is straightforward to find such a solution: one just integrates the equation of motion (A.17) from the horizon, with ingoing boundary conditions, to the boundary. Multiplying this solution by $(k^2+m^2)$ and dividing by its value at the boundary will yield a solution with precisely the boundary conditions required, and thus

$$ (k^2+m^2) \left. \frac{r^2 \psi'_{\perp}(r)}{\psi_{\perp}(r)} \right|_{r \to \infty} = \frac{3}{k^2+m^2-\omega^2} \left[ (k^2+m^2) h_t^{y(3)} + \omega k h_x^{y(3)} + im\omega\delta\phi_2^{(3)} \right], \quad (A.23) $$

where the right hand side is evaluated with the boundary conditions that all fields are ingoing at the horizon and that the only non-vanishing source term at the boundary is $h_t^{y(0)} = 1$. Thus, up to contact terms,

$$ G_{T^{ty}T^{ty}}^{R}(\omega, k) = (k^2 + m^2) \left. \frac{r^2 \psi'_{\perp}(r)}{\psi_{\perp}(r)} \right|_{r \to \infty} . \quad (A.24) $$

A similar procedure works for the longitudinal retarded Green's functions. Near the boundary, the gauge-invariant variable (A.13) has the expansion (A.22). Since the leading order term involves the expectation values ($h_t^{x(3)}$ etc.), there is no simple expression for the
boundary value of $\psi_\parallel$ in a solution in which all source terms vanish except $h_t^{x(0)}$. However, one can define a new gauge-invariant longitudinal field

$$ \psi_{\parallel}^{\text{new}} \equiv r^2 f \psi_{\parallel}^{\prime}(r) + \frac{3r_0 (m^2 - 2r_0^2)}{2(k^2 + m^2)} \psi_{\parallel}(r), \qquad (A.25) $$

whose near boundary expansion is

$$ \psi_{\parallel}^{\text{new}}(r \to \infty) \to \frac{2\omega k}{m} h_t^{x(0)} + O(1/r), \qquad (A.26) $$

when all other source terms vanish. Using similar arguments to before, one can then deduce that up to contact terms,

$$ G_{T^{tx}T^{tx}}^{R}(\omega, k) = \omega^2 (k^2 + m^2) \frac{\psi_{\parallel}(r)}{r^2 f \psi_{\parallel}^{\prime}(r) + \frac{3r_0(m^2 - 2r_0^2)}{2(k^2 + m^2)} \psi_{\parallel}(r)} \Big|_{r \to \infty} . \qquad (A.27) $$

Using this approach, it is straightforward to show (although we omit the details) that the gauge-invariant field (A.13) also controls the retarded Green's function of the energy density $T^{tt}$, and that it automatically obeys the Ward identity $G_{T^{tt}T^{tt}}^{R} = k^2 G_{T^{tx}T^{tx}}^{R}/\omega^2$, due to this gauge-invariant formulation.

To find the poles of the conductivities as shown in the main text, we simply need to find the poles of the Green's functions i.e. where the denominators of the right hand sides of (A.24) and (A.27) vanish. To compute the real (dissipative) parts of the conductivities as shown in the main text, we simply take the imaginary parts of the Green's functions (A.24) and (A.27) and divide by $\omega$ (as per the definitions (2.11) and (2.23)). In all cases, we have shown, the imaginary part of the subtracted terms in (2.11) and (2.23) vanish, and so we do not need to explicitly do this subtraction. To determine the spatially uniform AC heat conductivity $\kappa(\omega, k=0)$, one can easily take the $k=0$ limits of the transverse equations (A.17) and (A.24).

## B Calculation of Green's functions at self-dual point

When $m = \sqrt{2}r_0$, the gravitational theory exhibits an enhanced symmetry that allows us to explicitly solve the perturbation equations. In the following subsections, we outline how the solutions to these equations at the self-dual point $m = \sqrt{2}r_0$ are related to the two-point retarded Green's functions in the dual field theory.

### B.1 Ward identities

There are 8 independent operators in the sector of the field theory state which is dual to the metric and scalar field perturbations of the gravitational theory. These operators can be divided into transverse and longitudinal sectors ($\{P^y \equiv T^{ty}, T^{xy}, \Phi_2\}$ and $\{P^x \equiv T^{xt}, T^{tt}, T^{xx}, T^{yy}, \Phi_1\}$ respectively), which decouple from one another i.e. cross correlators between an operator in the transverse sector and an operator in the longitudinal sector vanish. Within each sector, there is a further decoupling such that

$$ G_{P^x \Phi_1}^{R}(\omega, k) = G_{\Phi_1 P^x}^{R}(\omega, k) = 0, \quad \text{and} \quad G_{P^y \Phi_2}^{R}(\omega, k) = G_{\Phi_2 P^y}^{R}(\omega, k) = 0. \qquad (B.1) $$
boundary value of $\psi_\parallel$ in a solution in which all source terms vanish except $h_t^{x(0)}$. However, one can define a new gauge-invariant longitudinal field

$$ \psi_{\parallel}^{\text{new}} \equiv r^2 f \psi_{\parallel}^{\prime}(r) + \frac{3r_0(m^2 - 2r_0^2)}{2(k^2 + m^2)} \psi_{\parallel}(r), \qquad (A.25) $$

whose near boundary expansion is

$$ \psi_{\parallel}^{\text{new}}(r \to \infty) \to \frac{2\omega k}{m} h_t^{x(0)} + O(1/r), \qquad (A.26) $$

when all other source terms vanish. Using similar arguments to before, one can then deduce that up to contact terms,

$$ G_{T^{tx}T^{tx}}^{R}(\omega, k) = \omega^2 (k^2 + m^2) \frac{\psi_{\parallel}(r)}{r^2 f \psi_{\parallel}^{\prime}(r) + \frac{3r_0(m^2 - 2r_0^2)}{2(k^2 + m^2)} \psi_{\parallel}(r)} \Big|_{r \to \infty} . \qquad (A.27) $$

Using this approach, it is straightforward to show (although we omit the details) that the gauge-invariant field (A.13) also controls the retarded Green's function of the energy density $T^{tt}$, and that it automatically obeys the Ward identity $G_{T^{tt}T^{tt}}^R = k^2 G_{T^{tx}T^{tx}}^R / \omega^2$, due to this gauge-invariant formulation.

To find the poles of the conductivities as shown in the main text, we simply need to find the poles of the Green's functions i.e. where the denominators of the right hand sides of (A.24) and (A.27) vanish. To compute the real (dissipative) parts of the conductivities as shown in the main text, we simply take the imaginary parts of the Green's functions (A.24) and (A.27) and divide by $\omega$ (as per the definitions (2.11) and (2.23)). In all cases, we have shown, the imaginary part of the subtracted terms in (2.11) and (2.23) vanish, and so we do not need to explicitly do this subtraction. To determine the spatially uniform AC heat conductivity $\kappa(\omega, k=0)$, one can easily take the $k=0$ limits of the transverse equations (A.17) and (A.24).

## B Calculation of Green's functions at self-dual point

When $m = \sqrt{2}r_0$, the gravitational theory exhibits an enhanced symmetry that allows us to explicitly solve the perturbation equations. In the following subsections, we outline how the solutions to these equations at the self-dual point $m = \sqrt{2}r_0$ are related to the two-point retarded Green's functions in the dual field theory.

### B.1 Ward identities

There are 8 independent operators in the sector of the field theory state which is dual to the metric and scalar field perturbations of the gravitational theory. These operators can be divided into transverse and longitudinal sectors ($\{P^y \equiv T^{ty}, T^{xy}, \Phi_2\}$ and $\{P^x \equiv T^{xt}, T^{tt}, T^{xx}, T^{yy}, \Phi_1\}$ respectively), which decouple from one another i.e. cross correlators between an operator in the transverse sector and an operator in the longitudinal sector vanish. Within each sector, there is a further decoupling such that

$$ G_{P^x \Phi_1}^R (\omega, k) = G_{\Phi_1 P^x}^R (\omega, k) = 0, \quad \text{and} \quad G_{P^y \Phi_2}^R (\omega, k) = G_{\Phi_2 P^y}^R (\omega, k) = 0. \qquad (B.1) $$
This is dual to the decoupling of the corresponding perturbations in the gravitational theory. It would be worthwhile to find a good field theoretical reason for this decoupling.

Even after imposing these constraints, there are naively many more two-point Green's functions than the number of gravitational degrees of freedom (four). This mismatch can be easily resolved: there are really only four independent Green's functions, and the others are related to them by Ward identities arising from the relations [18, 26]

$$ \partial_{\mu} T^{\mu\nu} = - \sum_{i=1}^{2} \mathcal{O}_i \partial^{\nu} \mathcal{J}_i, \qquad T^{\mu}_{\mu} = 0, \qquad (B.2) $$

where $\mathcal{J}_i$ is the source for the operator $\mathcal{O}_i$. Using these relations, we can express all of the Green's functions in terms of just $G_{P_x P_x}^R$, $G_{P_y P_y}^R$, $G_{\Phi_1 \Phi_1}^R$ and $G_{\Phi_2 \Phi_2}^R$ as follows (up to contact terms)

$$ G_{P_y T^{xy}}^R (\omega, k) = \frac{\omega k}{(k^2 + 2r_0^2)} G_{P_y P_y}^R (\omega, k), \quad G_{T^{xy} \Phi_2}^R (\omega, k) = \frac{\sqrt{2} r_0}{(k^2 + 2r_0^2)} G_{\Phi_2 \Phi_2}^R (\omega, k), \quad (B.3) $$

$$ G_{T^{xy} T^{xy}}^R (\omega, k) = \frac{\omega^2 k^2}{(k^2 + 2r_0^2)^2} G_{P_y P_y}^R (\omega, k) + \frac{2r_0^2}{(k^2 + 2r_0^2)^2} G_{\Phi_2 \Phi_2}^R (\omega, k), $$

$$ G_{P_x T^{xx}}^R (\omega, k) = \frac{k(r_0^2 + \omega^2)}{\omega(k^2 + 2r_0^2)} G_{P_x P_x}^R (\omega, k), \quad G_{P_x T^{yy}}^R (\omega, k) = \frac{k(k^2 + r_0^2 - \omega^2)}{\omega(k^2 + 2r_0^2)} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{xx} T^{xx}}^R (\omega, k) = \frac{r_0^2}{2(k^2 + 2r_0^2)^2} G_{\Phi_1 \Phi_1}^R (\omega, k) + \frac{k^2 (r_0^2 + \omega^2)^2}{\omega^2 (k^2 + 2r_0^2)^2} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{yy} T^{yy}}^R (\omega, k) = \frac{r_0^2}{2(k^2 + 2r_0^2)^2} G_{\Phi_1 \Phi_1}^R (\omega, k) + \frac{k^2 (k^2 + r_0^2 - \omega^2)^2}{\omega^2 (k^2 + 2r_0^2)^2} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{xx} T^{yy}}^R (\omega, k) = -\frac{r_0^2}{2(k^2 + 2r_0^2)^2} G_{\Phi_1 \Phi_1}^R (\omega, k) + \frac{k^2 (r_0^2 + \omega^2) (k^2 + r_0^2 - \omega^2)}{\omega^2 (k^2 + 2r_0^2)^2} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{xx} \Phi_1}^R (\omega, k) = \frac{r_0}{\sqrt{2}(k^2 + 2r_0^2)} G_{\Phi_1 \Phi_1}^R (\omega, k), \quad G_{T^{yy} \Phi_1}^R (\omega, k) = -\frac{r_0}{\sqrt{2}(k^2 + 2r_0^2)} G_{\Phi_1 \Phi_1}^R (\omega, k), $$

and $G_{T^{tt}}^R (\omega, k) = k G_{P_x P_x}^R (\omega, k) / \omega$, due to energy conservation.

## B.2 On-shell action and Green's functions

As outlined in the main text, all of the gravitational equations can be reduced to just two different equations ((4.22) and (4.23)) at the self-dual point. We denote the solutions to these equations as $\varphi$ and $\Psi$ respectively. After solving them, and imposing ingoing boundary conditions at the horizon, their behaviour near the boundary can be used to determine the Green's functions of the dual field theory. This is because their boundary behaviour gives the boundary behaviour of the fluctuations of the metric and scalar fields, and it is these which control the aforementioned Green's functions.

Specifically, near the boundary of the spacetime, the fields $\varphi_{p_x}$ and $\varphi_{p_y}$, defined in equations (4.19) and (4.21), are

$$ \varphi_{p_x} = \frac{3\sqrt{2}k}{r_0 [2k^4 - 4r_0^2\omega^2 + 2\omega^4 + k^2(3r_0^2 - 4\omega^2)]} \left\{ k(\omega^2 + r_0^2)h_x^{(3)} - k(k^2 + 2r_0^2)h_t^{(3)} \right\} \quad (B.4) $$
This is dual to the decoupling of the corresponding perturbations in the gravitational theory. It would be worthwhile to find a good field theoretical reason for this decoupling.

Even after imposing these constraints, there are naively many more two-point Green's functions than the number of gravitational degrees of freedom (four). This mismatch can be easily resolved: there are really only four independent Green's functions, and the others are related to them by Ward identities arising from the relations [18, 26]

$$ \partial_{\mu} T^{\mu \nu} = - \sum_{i=1}^{2} \mathcal{O}_i \partial^{\nu} \mathcal{J}_i, \qquad T_{\mu}^{\mu} = 0, \qquad (B.2) $$

where $\mathcal{J}_i$ is the source for the operator $\mathcal{O}_i$. Using these relations, we can express all of the Green's functions in terms of just $G_{P_x P_x}^R$, $G_{P_y P_y}^R$, $G_{\Phi_1 \Phi_1}^R$ and $G_{\Phi_2 \Phi_2}^R$ as follows (up to contact terms)

$$ G_{P_y T^{xy}}^R (\omega, k) = \frac{\omega k}{(k^2 + 2r_0^2)} G_{P_y P_y}^R (\omega, k), \quad G_{T^{xy} \Phi_2}^R (\omega, k) = \frac{\sqrt{2} r_0}{(k^2 + 2r_0^2)} G_{\Phi_2 \Phi_2}^R (\omega, k), \quad (B.3) $$

$$ G_{T^{xy}T^{xy}}^R (\omega, k) = \frac{\omega^2 k^2}{(k^2 + 2r_0^2)^2} G_{P_y P_y}^R (\omega, k) + \frac{2r_0^2}{(k^2 + 2r_0^2)^2} G_{\Phi_2 \Phi_2}^R (\omega, k), $$

$$ G_{P_x T^{xx}}^R (\omega, k) = \frac{k(r_0^2 + \omega^2)}{\omega(k^2 + 2r_0^2)} G_{P_x P_x}^R (\omega, k), \quad G_{P_x T^{yy}}^R (\omega, k) = \frac{k(k^2 + r_0^2 - \omega^2)}{\omega(k^2 + 2r_0^2)} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{xx}T^{xx}}^R (\omega, k) = \frac{r_0^2}{2(k^2 + 2r_0^2)^2} G_{\Phi_1 \Phi_1}^R (\omega, k) + \frac{k^2 (r_0^2 + \omega^2)^2}{\omega^2 (k^2 + 2r_0^2)^2} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{yy}T^{yy}}^R (\omega, k) = \frac{r_0^2}{2(k^2 + 2r_0^2)^2} G_{\Phi_1 \Phi_1}^R (\omega, k) + \frac{k^2 (k^2 + r_0^2 - \omega^2)^2}{\omega^2 (k^2 + 2r_0^2)^2} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{xx}T^{yy}}^R (\omega, k) = -\frac{r_0^2}{2(k^2 + 2r_0^2)^2} G_{\Phi_1 \Phi_1}^R (\omega, k) + \frac{k^2 (r_0^2 + \omega^2) (k^2 + r_0^2 - \omega^2)}{\omega^2 (k^2 + 2r_0^2)^2} G_{P_x P_x}^R (\omega, k), $$

$$ G_{T^{xx} \Phi_1}^R (\omega, k) = \frac{r_0}{\sqrt{2}(k^2 + 2r_0^2)} G_{\Phi_1 \Phi_1}^R (\omega, k), \quad G_{T^{yy} \Phi_1}^R (\omega, k) = -\frac{r_0}{\sqrt{2}(k^2 + 2r_0^2)} G_{\Phi_1 \Phi_1}^R (\omega, k), $$

and $G_{T^{tt}\mathcal{O}}^R (\omega, k) = k G_{P_x \mathcal{O}}^R (\omega, k) / \omega$, due to energy conservation.

## B.2 On-shell action and Green's functions

As outlined in the main text, all of the gravitational equations can be reduced to just two different equations ((4.22) and (4.23)) at the self-dual point. We denote the solutions to these equations as $\varphi$ and $\Psi$ respectively. After solving them, and imposing ingoing boundary conditions at the horizon, their behaviour near the boundary can be used to determine the Green's functions of the dual field theory. This is because their boundary behaviour gives the boundary behaviour of the fluctuations of the metric and scalar fields, and it is these which control the aforementioned Green's functions.

Specifically, near the boundary of the spacetime, the fields $\varphi_{p_x}$ and $\varphi_{p_y}$, defined in equations (4.19) and (4.21), are

$$ \varphi_{p_x} = \frac{3\sqrt{2}k}{r_0 [2k^4 - 4r_0^2\omega^2 + 2\omega^4 + k^2(3r_0^2 - 4\omega^2)]} \begin{cases} k(\omega^2 + r_0^2)h_x^{(3)} - k(k^2 + 2r_0^2)h_t^{(3)} & (B.4) \\ 0 & (B.5) \end{cases} $$
$$
\begin{equation}
\begin{aligned}
& -i\sqrt{2}k^2 r_0 \delta\phi_1^{(3)} + 2\omega(k^2 + 2r_0^2)h_t^{x(3)} + i2\sqrt{2}r_0\omega^2\delta\phi_1^{(3)} + k(k^2 + r_0^2 - \omega^2)h_y^{y(3)} \\
& \qquad + \frac{k}{2r_0 r} \left\{ \begin{aligned}[t]
& -\sqrt{2}kr_0^2h_x^{x(0)} + \sqrt{2}k(k^2 + 2r_0^2)h_t^{t(0)} + 2ik^2r_0\delta\phi_1^{(0)} - 2\sqrt{2}k^2\omega h_t^{x(0)} \\
& \quad - 4\sqrt{2}r_0^2\omega h_t^{x(0)} - \sqrt{2}k\omega^2h_x^{x(0)} - 4ir_0\omega^2\delta\phi_1^{(0)} - \sqrt{2}k(k^2+r_0^2-\omega^2)h_y^{y(0)} \end{aligned} \right\} + O(1/r^2),
\end{aligned}
\tag{B.5}
\end{equation}
$$

where $h_t^{x(i)}$ etc. are the coefficients in the near boundary expansions of the field perturba-
tions (A.19). These fields both obey the $\varphi$ equation of motion (4.22). Similarly, the fields
$\Psi_x$ and $\Psi_y$, defined in equations (4.18) and (4.20), have near-boundary expansions

$$
\begin{equation}
\begin{aligned}
\Psi_x = \left\{ \sqrt{2} r_0 (h_x^{x(0)} - h_y^{y(0)}) - 2ik\delta\phi_1^{(0)} \right\} \left( 1 - \frac{(k^2 + 2r_0^2 - \omega^2)}{2r^2} \right) & \quad \text{(B.6)} \\
& + \frac{1}{r^3} \left\{ \sqrt{2} r_0 (h_x^{x(3)} - h_y^{y(3)}) - 2ik\delta\phi_1^{(3)} \right\} + O(1/r^4),
\end{aligned}
\end{equation}
$$

$$
\begin{equation}
\begin{aligned}
\Psi_y = \left\{ \sqrt{2} r_0 h_x^{y(0)} - ik\delta\phi_2^{(0)} \right\} \left( 1 - \frac{k^2 + 2r_0^2 - \omega^2}{2r^2} \right) &+ \frac{1}{r^3} \left\{ \sqrt{2} r_0 h_x^{y(3)} - ik\delta\phi_2^{(3)} \right\} \quad \text{(B.7)} \\
& + O(1/r^4).
\end{aligned}
\end{equation}
$$

These fields both obey the $\Psi$ equation of motion (4.23).

To determine the precise relation between the boundary values of the fields and the re-
tarded Green’s functions, we evaluate the total action (given in equations (3.1) and (A.18))
on-shell at the self-dual point $m = \sqrt{2} r_0$. Working to quadratic order in the perturbations,
it is

$$
\begin{equation}
S = \int \frac{d\omega dk}{4\pi^2} \left\{
\begin{aligned}
& \frac{1}{[4k^4 - 8r_0^2\omega^2 + 4\omega^4 + 2k^2(3r_0^2 - 4\omega^2)]} \left[ h_x^{x(0)} \left\{ -\frac{3}{2}k^2(r_0^2 + \omega^2)h_t^{t(3)} \right. \right. \\
& \left. -\frac{3}{4}\left\{ 2k^2(r_0^2 - \omega^2) - 2\omega^2(2r_0^2 - \omega^2) \right\}h_y^{y(3)} + \frac{3}{2}(2r_0^2k^2 - 2r_0^2\omega^2 + \omega^4)h_x^{x(3)} \right. \\
& \left. + 3k\omega(r_0^2 + \omega^2)h_t^{t(3)} - \frac{3}{\sqrt{2}}ikr_0(2k^2 - 3\omega^2)\delta\phi_1^{(3)} \right\} \\
& + h_y^{y(0)} \left\{ -\frac{3}{2}k^2(k^2 + r_0^2 - \omega^2)h_t^{t(3)} + \frac{3}{2}(k^2 - \omega^2)(k^2 + 2r_0^2 - \omega^2)h_y^{y(3)} \right. \\
& \left. - \frac{3}{2}\{k^2(r_0^2 - \omega^2) - \omega^2(2r_0^2 - \omega^2)\}h_x^{x(3)} + 3k\omega(k^2 + r_0^2 - \omega^2)h_t^{t(3)} + \frac{3}{\sqrt{2}}ikr_0(k^2 - \omega^2)\delta\phi_1^{(3)} \right\} \\
& \left. + \left(h_t^{t(0)} - \frac{2\omega}{k}h_t^{x(0)}\right)\left\{ \frac{3}{2}k^2(k^2 + 2r_0^2)h_t^{t(3)} - \frac{3}{2}k^2(k^2 + r_0^2 - \omega^2)h_y^{y(3)} - \frac{3}{2}k^2(\omega^2 + r_0^2)h_x^{x(3)} \right\}
\end{aligned}
\right\} \tag{B.8}
\end{equation}
$$
$$
\begin{align}
& -i\sqrt{2}k^2 r_0 \delta\phi_1^{(3)} + 2\omega(k^2 + 2r_0^2)h_t^{x(3)} + i2\sqrt{2}r_0\omega^2\delta\phi_1^{(3)} + k(k^2 + r_0^2 - \omega^2)h_y^{y(3)} \nonumber \\
& \qquad + \frac{k}{2r_0 r} \left\{ \begin{aligned}[t]
& -\sqrt{2}kr_0^2h_x^{x(0)} + \sqrt{2}k(k^2 + 2r_0^2)h_t^{t(0)} + 2ik^2r_0\delta\phi_1^{(0)} - 2\sqrt{2}k^2\omega h_t^{x(0)} \\
& \quad - 4\sqrt{2}r_0^2\omega h_t^{x(0)} - \sqrt{2}k\omega^2h_x^{x(0)} - 4ir_0\omega^2\delta\phi_1^{(0)} - \sqrt{2}k(k^2 + r_0^2 - \omega^2)h_y^{y(0)}
\end{aligned} \right\} + O(1/r^2), \nonumber \\
& \varphi_{py} = \left\{ (k^2 + 2r_0^2)h_t^{y(0)} + \omega k h_x^{y(0)} + i\sqrt{2}r_0\omega\delta\phi_2^{(0)} \right\} \tag{B.5} \\
& \qquad - \frac{3}{(k^2 + 2r_0^2 - \omega^2)r} \left\{ (k^2 + 2r_0^2)h_t^{y(3)} + \omega k h_x^{y(3)} + i\sqrt{2}r_0\omega\delta\phi_2^{(3)} \right\} + O(1/r^2), \nonumber
\end{align}
$$

where $h_t^{x(i)}$ etc. are the coefficients in the near boundary expansions of the field perturbations (A.19). These fields both obey the $\varphi$ equation of motion (4.22). Similarly, the fields $\Psi_x$ and $\Psi_y$, defined in equations (4.18) and (4.20), have near-boundary expansions

$$
\begin{equation}
\begin{split}
\Psi_x = \left\{ \sqrt{2} r_0 (h_x^{x(0)} - h_y^{y(0)}) - 2ik\delta\phi_1^{(0)} \right\} \left( 1 - \frac{(k^2 + 2r_0^2 - \omega^2)}{2r^2} \right) &+ \text{O}(1/r^4), \\
& + \frac{1}{r^3} \left\{ \sqrt{2} r_0 (h_x^{x(3)} - h_y^{y(3)}) - 2ik\delta\phi_1^{(3)} \right\} + \text{O}(1/r^4),
\end{split}
\tag{B.6}
\end{equation}
$$

$$
\begin{equation}
\begin{split}
\Psi_y = \left\{ \sqrt{2} r_0 h_x^{y(0)} - ik\delta\phi_2^{(0)} \right\} \left( 1 - \frac{k^2 + 2r_0^2 - \omega^2}{2r^2} \right) &+ \frac{1}{r^3} \left\{ \sqrt{2} r_0 h_x^{y(3)} - ik\delta\phi_2^{(3)} \right\} \quad \text{(B.7)} \\
& + O(1/r^4).
\end{split}
\end{equation}
$$

These fields both obey the $\Psi$ equation of motion (4.23).

To determine the precise relation between the boundary values of the fields and the re-
tarded Green's functions, we evaluate the total action (given in equations (3.1) and (A.18))
on-shell at the self-dual point $m = \sqrt{2} r_0$. Working to quadratic order in the perturbations,
it is

$$
\begin{equation}
\begin{split}
S = \int \frac{d\omega dk}{4\pi^2} & \left\{ \frac{1}{[4k^4 - 8r_0^2\omega^2 + 4\omega^4 + 2k^2(3r_0^2 - 4\omega^2)]} \right\} \\
& \left[ h_x^{x(0)} \left\{ -\frac{3}{2}k^2(r_0^2 + \omega^2)h_t^{t(3)} \right. \right. \tag{B.8} \\
& \left. \left. -\frac{3}{4}\{2k^2(r_0^2 - \omega^2) - 2\omega^2(2r_0^2 - \omega^2)\}h_y^{y(3)} + \frac{3}{2}(2r_0^2k^2 - 2r_0^2\omega^2 + \omega^4)h_x^{x(3)} \right. \right. \\
& \left. \left. + 3k\omega(r_0^2 + \omega^2)h_t^{x(3)} - \frac{3}{\sqrt{2}}ikr_0(2k^2 - 3\omega^2)\delta\phi_1^{(3)} \right\} \right. \\
& \left. + h_y^{y(0)} \left\{ -\frac{3}{2}k^2(k^2 + r_0^2 - \omega^2)h_t^{t(3)} + \frac{3}{2}(k^2 - \omega^2)(k^2 + 2r_0^2 - \omega^2)h_y^{y(3)} \right. \right. \\
& \left. \left. - \frac{3}{2}\{k^2(r_0^2 - \omega^2) - \omega^2(2r_0^2 - \omega^2)\}h_x^{x(3)} + 3k\omega(k^2 + r_0^2 - \omega^2)h_t^{x(3)} + \frac{3}{\sqrt{2}}ikr_0(k^2 - \omega^2)\delta\phi_1^{(3)} \right\} \right] \\
& \left. + \left( h_t^{t(0)} - \frac{2\omega}{k}h_t^{x(0)} \right) \left\{ \frac{3}{2}k^2(k^2 + 2r_0^2)h_t^{t(3)} - \frac{3}{2}k^2(k^2 + r_0^2 - \omega^2)h_y^{y(3)} - \frac{3}{2}k^2(\omega^2 + r_0^2)h_x^{x(3)} \right. \right.
\end{split}
\end{equation}
$$
$$ -3k\omega(k^2 + 2r_0^2)h_t^{x(3)} + \frac{3}{\sqrt{2}}ikr_0(k^2 - 2\omega^2)\delta\phi_1^{(3)} \\ + \delta\phi_1^{(0)}\left\{-\frac{3}{\sqrt{2}}ikr_0(k^2 - 2\omega^2)h_t^{t(3)} - \frac{3}{\sqrt{2}}ikr_0(k^2 - \omega^2)h_y^{y(3)} + \frac{3}{\sqrt{2}}ikr_0(2k^2 - 3\omega^2)h_x^{x(3)}\right. \\ \left. + 3\sqrt{2}ir_0\omega(k^2 - 2\omega^2)h_t^{x(3)} + 6(k^2 - \omega^2)^2\delta\phi_1^{(3)}\right\} \\ + \frac{3}{2(k^2 + 2r_0^2 - \omega^2)}\left[h_x^{y(0)}\left\{(2r_0^2 - \omega^2)h_x^{y(3)} - k\omega h_t^{y(3)} - i\sqrt{2}r_0k\delta\phi_2^{(3)}\right\} - h_t^{y(0)}\left\{(k^2 + 2r_0^2)h_t^{y(3)} + k\omega h_x^{y(3)} + i\sqrt{2}r_0\omega\delta\phi_2^{(3)}\right\} + \delta\phi_2^{(0)}\left\{i\sqrt{2}r_0\omega h_t^{y(3)} + i\sqrt{2}r_0k\delta\phi_2^{(3)} + (k^2 - \omega^2)\delta\phi_2^{(3)}\right\}\right\}, $$

where, in each quadratic pair of field perturbations, the first has argument $(-\omega, -k)$ and the second has argument $(\omega, k)$. With this action, the usual holographic procedure (as described in appendix A) can be used to find expressions for each two-point Green's function in terms of the near-boundary coefficients of the fields ($h_t^{x(3)}$ etc.). These coefficients can be related to the near-boundary coefficients of the fields $\varphi = \sum_i \varphi^{(i)} r^{-i}$ and $\Psi = \sum_i \Psi^{(i)} r^{-i}$ using equations (B.4) and (B.6), yielding (for example) the results (4.24) to (4.27). All of the other remaining Green's functions obey the appropriate Ward identities (B.3), and have the symmetry property $G_{AB}^R = G_{BA}^R$.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

[1] D. Forster, Hydrodynamic fluctuations, broken symmetry, and correlation functions, Advanced book classics, Perseus Books, U.S.A. (1990).

[2] S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, Theory of the Nernst effect near quantum phase transitions in condensed matter and in dyonic black holes, Phys. Rev. B 76 (2007) 144502 [arXiv:0706.3215] [INSPIRE].

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$$ -3k\omega(k^2 + 2r_0^2)h_t^{x(3)} + \frac{3}{\sqrt{2}}ikr_0(k^2 - 2\omega^2)\delta\phi_1^{(3)} \\ + \delta\phi_1^{(0)}\left\{-\frac{3}{\sqrt{2}}ikr_0(k^2 - 2\omega^2)h_t^{t(3)} - \frac{3}{\sqrt{2}}ikr_0(k^2 - \omega^2)h_y^{y(3)} + \frac{3}{\sqrt{2}}ikr_0(2k^2 - 3\omega^2)h_x^{x(3)} \right. \\ \left. + 3\sqrt{2}ir_0\omega(k^2 - 2\omega^2)h_t^{x(3)} + 6(k^2 - \omega^2)^2\delta\phi_1^{(3)}\right\} \\ + \frac{3}{2(k^2 + 2r_0^2 - \omega^2)}\left[ h_x^{y(0)}\left\{(2r_0^2 - \omega^2)h_x^{y(3)} - k\omega h_t^{y(3)} - i\sqrt{2}r_0k\delta\phi_2^{(3)}\right\} \right. \\ \left. - h_t^{y(0)}\left\{(k^2 + 2r_0^2)h_t^{y(3)} + k\omega h_x^{y(3)} + i\sqrt{2}r_0\omega\delta\phi_2^{(3)}\right\} \right. \\ \left. + \delta\phi_2^{(0)}\left\{i\sqrt{2}r_0\omega h_t^{y(3)} + i\sqrt{2}r_0k h_x^{y(3)} + (k^2 - \omega^2)\delta\phi_2^{(3)}\right\}\right] \right\} $$

where, in each quadratic pair of field perturbations, the first has argument $(-\omega, -k)$ and the second has argument $(\omega, k)$. With this action, the usual holographic procedure (as described in appendix A) can be used to find expressions for each two-point Green's function in terms of the near-boundary coefficients of the fields ($h_t^{x(3)}$ etc.). These coefficients can be related to the near-boundary coefficients of the fields $\varphi = \sum_i \varphi^{(i)} r^{-i}$ and $\Psi = \sum_i \Psi^{(i)} r^{-i}$ using equations (B.4) and (B.6), yielding (for example) the results (4.24) to (4.27). All of the other remaining Green's functions obey the appropriate Ward identities (B.3), and have the symmetry property $G_{AB}^R = G_{BA}^R$.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

## References

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[37] C.P. Herzog, P. Kovtun, S. Sachdev and D.T. Son, Quantum critical transport, duality and M-theory, Phys. Rev. D 75 (2007) 085020 [hep-th/0701036] [INSPIRE].

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[40] R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero energy states, JHEP 06 (1999) 036 [hep-th/9906040] [INSPIRE].

[41] C. Charmousis, B. Gouteraux, B.S. Kim, E. Kiritsis and R. Meyer, Effective holographic theories for low-temperature condensed matter systems, JHEP 11 (2010) 151 [arXiv:1005.4690] [INSPIRE].

[42] B. Gouteraux and E. Kiritsis, Generalized holographic quantum criticality at finite density, JHEP 12 (2011) 036 [arXiv:1107.2116] [INSPIRE].

[43] E. Witten, SL(2, Z) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].

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[45] B. Julia, J. Levie and S. Ray, Gravitational duality near de Sitter space, JHEP 11 (2005) 025 [hep-th/0507262] [INSPIRE].

[46] R.G. Leigh and A.C. Petkou, Gravitational duality transformations on (A)dS₄, JHEP 11 (2007) 079 [arXiv:0704.0531] [INSPIRE].

[47] S. de Haro, Dual gravitons in AdS₄/CFT₃ and the holographic Cotton tensor, JHEP 01 (2009) 042 [arXiv:0808.2054] [INSPIRE].
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[49] I. Bakas, Duality in linearized gravity and holography, Class. Quant. Grav. 26 (2009) 065013 [arXiv:0812.0152] [INSPIRE].

[50] S. Deser and A. Waldron, PM = EM: partially massless duality invariance, Phys. Rev. D 87 (2013) 087702 [arXiv:1301.2238] [INSPIRE].

[51] K. Hinterbichler, Manifest duality invariance for the partially massless graviton, arXiv:1409.3565 [INSPIRE].

[52] D. Anninos, S.A. Hartnoll and D.M. Hofman, Static patch solipsism: conformal symmetry of the de Sitter worldline, Class. Quant. Grav. 29 (2012) 075002 [arXiv:1109.4942] [INSPIRE].

[53] R.J. Anantua, S.A. Hartnoll, V.L. Martin and D.M. Ramirez, The Pauli exclusion principle at strong coupling: holographic matter and momentum space, JHEP 03 (2013) 104 [arXiv:1210.1590] [INSPIRE].

[54] P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].

[55] S. Jain, N. Kundu, K. Sen, A. Sinha and S.P. Trivedi, A strongly coupled anisotropic fluid from dilaton driven holography, arXiv:1406.4874 [INSPIRE].

[56] A. Karch and A. O'Bannon, Metallic AdS/CFT, JHEP 09 (2007) 024 [arXiv:0705.3870] [INSPIRE].

[57] A. Karch and A. O'Bannon, Holographic thermodynamics at finite baryon density: some exact results, JHEP 11 (2007) 074 [arXiv:0709.0570] [INSPIRE].

[58] S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [INSPIRE].

[59] C. Hoyos-Badajoz, A. O'Bannon and J.M.S. Wu, Zero sound in strange metallic holography, JHEP 09 (2010) 086 [arXiv:1007.0590] [INSPIRE].

[60] D.-W. Pang, Probing holographic semilocal quantum liquids with D-branes, Phys. Rev. D 88 (2013) 046002 [arXiv:1306.3816] [INSPIRE].

[61] P. Dey and S. Roy, Zero sound in strange metals with hyperscaling violation from holography, Phys. Rev. D 88 (2013) 046010 [arXiv:1307.0195] [INSPIRE].

[62] M. Edalati and J.F. Pedraza, Aspects of current correlators in holographic theories with hyperscaling violation, Phys. Rev. D 88 (2013) 086004 [arXiv:1307.0808] [INSPIRE].

[63] A. Karch, D.T. Son and A.O. Starinets, Zero sound from holography, arXiv:0806.3796 [INSPIRE].

[64] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Striped instability of a holographic Fermi-like liquid, JHEP 10 (2011) 034 [arXiv:1106.3883] [INSPIRE].

[65] R.A. Davison and A.O. Starinets, Holographic zero sound at finite temperature, Phys. Rev. D 85 (2012) 026004 [arXiv:1109.6343] [INSPIRE].

[66] M. Kulaxizi and A. Parnachev, Holographic responses of fermion matter, Nucl. Phys. B 815 (2009) 125 [arXiv:0811.2262] [INSPIRE].

[67] B.S. DiNunno, M. Ihl, N. Jokela and J.F. Pedraza, Holographic zero sound at finite temperature in the Sakai-Sugimoto model, JHEP 04 (2014) 149 [arXiv:1403.1827] [INSPIRE].
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[49] I. Bakas, Duality in linearized gravity and holography, Class. Quant. Grav. 26 (2009) 065013 [arXiv:0812.0152] [INSPIRE].

[50] S. Deser and A. Waldron, PM = EM: partially massless duality invariance, Phys. Rev. D 87 (2013) 087702 [arXiv:1301.2238] [INSPIRE].

[51] K. Hinterbichler, Manifest duality invariance for the partially massless graviton, arXiv:1409.3565 [INSPIRE].

[52] D. Anninos, S.A. Hartnoll and D.M. Hofman, Static patch solipsism: conformal symmetry of the de Sitter worldline, Class. Quant. Grav. 29 (2012) 075002 [arXiv:1109.4942] [INSPIRE].

[53] R.J. Anantua, S.A. Hartnoll, V.L. Martin and D.M. Ramirez, The Pauli exclusion principle at strong coupling: holographic matter and momentum space, JHEP 03 (2013) 104 [arXiv:1210.1590] [INSPIRE].

[54] P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].

[55] S. Jain, N. Kundu, K. Sen, A. Sinha and S.P. Trivedi, A strongly coupled anisotropic fluid from dilaton driven holography, arXiv:1406.4874 [INSPIRE].

[56] A. Karch and A. O'Bannon, Metallic AdS/CFT, JHEP 09 (2007) 024 [arXiv:0705.3870] [INSPIRE].

[57] A. Karch and A. O'Bannon, Holographic thermodynamics at finite baryon density: some exact results, JHEP 11 (2007) 074 [arXiv:0709.0570] [INSPIRE].

[58] S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [INSPIRE].

[59] C. Hoyos-Badajoz, A. O'Bannon and J.M.S. Wu, Zero sound in strange metallic holography, JHEP 09 (2010) 086 [arXiv:1007.0590] [INSPIRE].

[60] D.-W. Pang, Probing holographic semilocal quantum liquids with D-branes, Phys. Rev. D 88 (2013) 046002 [arXiv:1306.3816] [INSPIRE].

[61] P. Dey and S. Roy, Zero sound in strange metals with hyperscaling violation from holography, Phys. Rev. D 88 (2013) 046010 [arXiv:1307.0195] [INSPIRE].

[62] M. Edalati and J.F. Pedraza, Aspects of current correlators in holographic theories with hyperscaling violation, Phys. Rev. D 88 (2013) 086004 [arXiv:1307.0808] [INSPIRE].

[63] A. Karch, D.T. Son and A.O. Starinets, Zero sound from holography, arXiv:0806.3796 [INSPIRE].

[64] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Striped instability of a holographic Fermi-like liquid, JHEP 10 (2011) 034 [arXiv:1106.3883] [INSPIRE].

[65] R.A. Davison and A.O. Starinets, Holographic zero sound at finite temperature, Phys. Rev. D 85 (2012) 026004 [arXiv:1109.6343] [INSPIRE].

[66] M. Kulaxizi and A. Parnachev, Holographic responses of fermion matter, Nucl. Phys. B 815 (2009) 125 [arXiv:0811.2262] [INSPIRE].

[67] B.S. DiNunno, M. Ihl, N. Jokela and J.F. Pedraza, Holographic zero sound at finite temperature in the Sakai-Sugimoto model, JHEP 04 (2014) 149 [arXiv:1403.1827] [INSPIRE].
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[69] M. Ammon, K. Jensen, K.-Y. Kim, J.N. Laia and A. O'Bannon, Moduli spaces of cold holographic matter, JHEP 11 (2012) 055 [arXiv:1208.3197] [INSPIRE].

[70] J. Tarrio, Transport properties of spacetime-filling branes, JHEP 04 (2014) 042 [arXiv:1312.2902] [INSPIRE].

[71] R.C. Myers, S. Sachdev and A. Singh, Holographic quantum critical transport without self-duality, Phys. Rev. D 83 (2011) 066017 [arXiv:1010.0443] [INSPIRE].

[72] W. Witczak-Krempa and S. Sachdev, The quasi-normal modes of quantum criticality, Phys. Rev. B 86 (2012) 235115 [arXiv:1210.4166] [INSPIRE].

[73] D. Chowdhury, S. Raju, S. Sachdev, A. Singh and P. Strack, Multipoint correlators of conformal field theories: implications for quantum critical transport, Phys. Rev. B 87 (2013) 085138 [arXiv:1210.5247] [INSPIRE].

[74] W. Witczak-Krempa and S. Sachdev, Dispersing quasinormal modes in 2 + 1 dimensional conformal field theories, Phys. Rev. B 87 (2013) 155149 [arXiv:1302.0847] [INSPIRE].

[75] W. Witczak-Krempa, E. Sorensen and S. Sachdev, The dynamics of quantum criticality via quantum Monte Carlo and holography, Nature Phys. 10 (2014) 361 [arXiv:1309.2941] [INSPIRE].

[76] W. Witczak-Krempa, Quantum critical charge response from higher derivatives in holography, Phys. Rev. B 89 (2014) 161114 [arXiv:1312.3334] [INSPIRE].

[77] S. Bai and D.-W. Pang, Holographic charge transport in 2 + 1 dimensions at finite N, Int. J. Mod. Phys. A 29 (2014) 1450061 [arXiv:1312.3351] [INSPIRE].

[78] D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].

[79] C. Charmousis, B. Gouteraux and J. Soda, Einstein-Maxwell-dilaton theories with a Liouville potential, Phys. Rev. D 80 (2009) 024028 [arXiv:0905.3337] [INSPIRE].

[80] E. Katz, S. Sachdev, E.S. Sørensen and W. Witczak-Krempa, Conformal field theories at nonzero temperature: operator product expansions, Monte Carlo and holography, Phys. Rev. B 90 (2014) 245109 [arXiv:1409.3841] [INSPIRE].

[81] P. Kovtun, Fluctuation bounds on charge and heat diffusion, arXiv:1407.0690 [INSPIRE].

[82] S.A. Hartnoll and C.P. Herzog, Ohm's law at strong coupling: S duality and the cyclotron resonance, Phys. Rev. D 76 (2007) 106012 [arXiv:0706.3228] [INSPIRE].

[83] S.A. Hartnoll and C.P. Herzog, Impure AdS/CFT correspondence, Phys. Rev. D 77 (2008) 106009 [arXiv:0801.1693] [INSPIRE].

[84] D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].

[85] M. Kaminski, K. Landsteiner, J. Mas, J.P. Shock and J. Tarrio, Holographic operator mixing and quasinormal modes on the brane, JHEP 02 (2010) 021 [arXiv:0911.3610] [INSPIRE].
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[69] M. Ammon, K. Jensen, K.-Y. Kim, J.N. Laia and A. O'Bannon, Moduli spaces of cold holographic matter, JHEP 11 (2012) 055 [arXiv:1208.3197] [INSPIRE].

[70] J. Tarrio, Transport properties of spacetime-filling branes, JHEP 04 (2014) 042 [arXiv:1312.2902] [INSPIRE].

[71] R.C. Myers, S. Sachdev and A. Singh, Holographic quantum critical transport without self-duality, Phys. Rev. D 83 (2011) 066017 [arXiv:1010.0443] [INSPIRE].

[72] W. Witczak-Krempa and S. Sachdev, The quasi-normal modes of quantum criticality, Phys. Rev. B 86 (2012) 235115 [arXiv:1210.4166] [INSPIRE].

[73] D. Chowdhury, S. Raju, S. Sachdev, A. Singh and P. Strack, Multipoint correlators of conformal field theories: implications for quantum critical transport, Phys. Rev. B 87 (2013) 085138 [arXiv:1210.5247] [INSPIRE].

[74] W. Witczak-Krempa and S. Sachdev, Dispersing quasinormal modes in 2 + 1 dimensional conformal field theories, Phys. Rev. B 87 (2013) 155149 [arXiv:1302.0847] [INSPIRE].

[75] W. Witczak-Krempa, E. Sorensen and S. Sachdev, The dynamics of quantum criticality via quantum Monte Carlo and holography, Nature Phys. 10 (2014) 361 [arXiv:1309.2941] [INSPIRE].

[76] W. Witczak-Krempa, Quantum critical charge response from higher derivatives in holography, Phys. Rev. B 89 (2014) 161114 [arXiv:1312.3334] [INSPIRE].

[77] S. Bai and D.-W. Pang, Holographic charge transport in 2 + 1 dimensions at finite N, Int. J. Mod. Phys. A 29 (2014) 1450061 [arXiv:1312.3351] [INSPIRE].

[78] D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].

[79] C. Charmousis, B. Gouteraux and J. Soda, Einstein-Maxwell-dilaton theories with a Liouville potential, Phys. Rev. D 80 (2009) 024028 [arXiv:0905.3337] [INSPIRE].

[80] E. Katz, S. Sachdev, E.S. Sørensen and W. Witczak-Krempa, Conformal field theories at nonzero temperature: operator product expansions, Monte Carlo and holography, Phys. Rev. B 90 (2014) 245109 [arXiv:1409.3841] [INSPIRE].

[81] P. Kovtun, Fluctuation bounds on charge and heat diffusion, arXiv:1407.0690 [INSPIRE].

[82] S.A. Hartnoll and C.P. Herzog, Ohm's law at strong coupling: S duality and the cyclotron resonance, Phys. Rev. D 76 (2007) 106012 [arXiv:0706.3228] [INSPIRE].

[83] S.A. Hartnoll and C.P. Herzog, Impure AdS/CFT correspondence, Phys. Rev. D 77 (2008) 106009 [arXiv:0801.1693] [INSPIRE].

[84] D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].

[85] M. Kaminski, K. Landsteiner, J. Mas, J.P. Shock and J. Tarrio, Holographic operator mixing and quasinormal modes on the brane, JHEP 02 (2010) 021 [arXiv:0911.3610] [INSPIRE].