A Proof of the variational principle

# Resistivity bound for hydrodynamic bad metals

## Abstract

We obtain a rigorous upper bound on the resistivity of an electron fluid whose electronic mean free path is short compared to the scale of spatial inhomogeneities. When such a hydrodynamic electron fluid supports a non-thermal diffusion process – such as an imbalance mode between different bands – we show that the resistivity bound becomes . The coefficient is independent of temperature and inhomogeneity lengthscale, and is a microscopic momentum-preserving scattering rate. In this way we obtain a unified and novel mechanism – without umklapp – for in a Fermi liquid and the crossover to in quantum critical regimes. This behavior is widely observed in transition metal oxides, organic metals, pnictides and heavy fermion compounds and has presented a longstanding challenge to transport theory. Our hydrodynamic bound allows phonon contributions to diffusion constants, including thermal diffusion, to directly affect the electrical resistivity.

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Resistivity bound for hydrodynamic bad metals

Andrew Lucas and Sean A. Hartnoll

Department of Physics, Stanford University,

Stanford, CA 94305-4060, USA

ajlucas@stanford.edu
hartnoll@stanford.edu

## 1 Introduction

### 1.1 Motivation: Scattering versus resistivity in strange metals

Electrical resistance arises due to microscopic scattering of charge carriers. The resistivity, however, is a macroscopic observable sensitive to the rate of momentum relaxing scattering events. The challenge of unconventional or ‘strange’ metals is largely associated with relating momentum-conserving to momentum-relaxing scattering rates. Most notably, a -linear resistivity is commonly observed close to quantum critical points in strange metals [1]. And indeed, a scattering rate is both characteristic of quantum criticality [2] and widely seen in strange metals in single particle [3] as well as transport [4, 5] observables. Nonetheless, because the scattering of electrons by quantum critical fluctuations is typically momentum-preserving, this rate does not directly control the resistivity in theoretical models of such systems. The interactions between electrons and collective quantum critical fluctuations are often strong, and hence any momentum transferred from charge carriers to quantum critical modes is rapidly returned [6, 7, 8]. For this reason, among others, a compelling microscopic theory of -linear quantum critical resistivity has remained elusive.

Away from the -linear resistivity characterizing ‘quantum critical fans’, a resistivity is widely observed in strongly correlated metals including transition metal oxides, pnictides, organic metals and heavy fermion compounds [9, 10]. The scaling is usually not considered ‘strange’ because electronic umklapp scattering is a conventional mechanism that can lead to such a resistivity. However, that may be a premature conclusion. In this paper we will describe a new scenario in which a momentum-preserving microscopic scattering rate directly controls the resistivity . This will lead to an alternate path to , without umklapp, that transitions to when the momentum-conserving scattering rate changes from to . In our theory, the mechanism for momentum relaxation – charge density inhomogeneities on length scales longer than the electronic mean free path – does not change across the phase diagram.

### 1.2 Hydrodynamic electron fluids

Denote by the characteristic length scale of spatial inhomogeneities, and by the mean free path of electrons due to momentum-preserving interactions (either with electrons or phonons). When the interactions are strong enough that , then the electron fluid will be locally equilibrated with a well-defined local momentum density that is only relaxed on the length scale . In such cases the conductivity can be calculated entirely within hydrodynamics [11, 12, 13, 14], which is the theory of macroscopic, long-wavelength perturbations from equilibrium. Bad metals in particular have extremely short mean free paths [15, 16, 17]. The same strong interactions that cause the absence of quasiparticles in these materials also lead to efficient local thermalization.

The main technical result in this paper is that the electrical resistivity of a hydrodynamic electron fluid is non-perturbatively (in disorder amplitude) bounded above by a certain spatial average of a matrix of ‘incoherent’ diffusion constants. These incoherent diffusive processes decouple from the conserved momentum density and hence from ‘momentum drag’ effects [18], and are proportional to the microscopic mean free path . We prove our bound using a variational principle for hydrodynamic entropy production developed in [13]. The most well-known incoherent diffusion mode is thermal diffusion [12]. However, at low temperatures thermal diffusion is suppressed. A central observation in this paper is that the presence of an additional, non-thermal diffusive mode has very significant and interesting consequences for low temperature transport in hydrodynamic metals. An example of such a mode is an ‘imbalance’ mode resulting from multiple Fermi bands or pockets, each with an approximately independently conserved number density. See figure 1 below. With such a non-thermal diffusive mode we show that, if our bound is saturated, the temperature dependence of the resistivity is directly determined by a microscopic electronic scattering rate. In this way we obtain a unified description of and -linear resistivity in hydrodynamic electron fluids.

With the inhomogeneity fixed, the microscopic scattering rate can increase arbitrarily at high temperatures. The hydrodynamic fluid description only improves in this limit as the mean free path gets shorter. The framework we have introduced therefore lends itself to the study of bad metals.

## 2 Resistivity bound in hydrodynamics

### 2.1 Hydrodynamic equations

Consider an electron fluid with conserved scalar quantities, with associated densities . Denoting the conserved currents of these scalars by , we find conservation laws: For a minimal linearized hydrodynamics the conserved densities would be the charge density and entropy density , and the corresponding currents would be the electric and entropy currents, and . We are interested in the case where there are additional conserved quantities, such as the imbalance modes discussed in the introduction. There is also a conserved momentum density , and a corresponding conservation law: Here is the stress tensor.

To complete the hydrodynamic description, one must write down the constitutive relations that give the currents in terms of a gradient expansion of the densities (or their conjugate sources). For the currents the constitutive relations are

 jA=nAv−ΣAB0∇μB+⋯. (1)

Here is a ‘chemical potential’ conjugate to the conserved density . For instance, the source conjugate to the charge density is the chemical potential , the entropy density is conjugate to the temperature , and the momentum density is conjugate to the velocity . Within linear response, the densities and sources are related by a matrix of thermodynamic susceptibilities: , and In Galilean-invariant cases, the thermodynamic susceptibility equals the mass density. The first term in the constitutive relations (1) describes how the momentum drags the scalar densities, while are the ‘incoherent’ transport coefficients. The constitutive relation for the stress tensor is

 τij=δijP−ζδij∇⋅v−η(∂ivj+∂jvi−2dδij∇⋅v)+⋯. (2)

Here is the pressure and and are the bulk and shear viscosities, respectively. Note that

Momentum relaxation is incorporated into the above equations by allowing all of the coefficients, thermodynamic susceptibilities and equilibrium expectation values and sources to be space-dependent. This spatial dependence must be sufficiently long wavelength that it is consistent with the hydrodynamic derivative expansion in the constitutive relations.

### 2.2 Variational principle and resistivity bound

We will establish an upper bound on the resistivity using a hydrodynamic variational principle, building upon the results in [13]. The basis for this technique is that the entropy production is smallest on the solutions to the equations of motion. In this regard the variational principle is similar in spirit to that for the kinetic equation [19, 20], Thomson’s principle for resistor networks [21], and holographic conductivity bounds [22, 23]. The rate of entropy density production in linearized hydrodynamics is quadratic in the hydrodynamic variables [24], and can be expressed in terms of the currents and velocities as

 T˙s=η(∂ivj+∂jvi−2d\mbox{%    {1 0 -.18 1}\hbox to 0.0pt{δ}}ij∂kvk)2+ζ(∇⋅v)2+(jA−nAv)(Σ−10)AB(jB−nBv). (4)

Now define the functional

 R[v,jA]≡∫ddxVT˙s(∫ddxVjx)2, (5)

where the entropy production in the numerator is given by (4), viewed as a function of and , and in the denominator is a component of the electric current. is the volume. Following [13], we prove in Appendix A that (i) if the functional (5) is varied over velocities and currents satisfying , then it is minimized on stationary solutions to the hydrodynamic equations of motion in the presence of a uniform applied electric field and (ii) the functional evaluated on these solutions is equal to the electrical resistivity , where in the presence of the applied electric field: . In this last equation, we do not impose any constraints on the average currents ; instead, we demand that the electric field is the only source. It follows that the resistivity is bounded by

 ρxx=min∇⋅jA=0R[v,jA]≤R[v,jA]∣∣∇⋅jA=0. (6)

With translation invariance, none of or in (4) are spatially dependent. Putting , with constant in (6), one finds vanishing resistivity: . This is the dissipationless flow of current in a translationally invariant medium.

The velocity appears quadratically in (4) and is unconstrained in the variational principle (6). Therefore we can ‘integrate it out’ exactly. This will give a nonlocal functional of the currents , due to inverting the viscous terms in (4). However, it is clear in (4) that the viscous terms are higher order in derivatives of the velocity field compared to other terms. Therefore, within a strict hydrodynamic expansion in , the viscous terms are subleading and can be dropped. We will therefore set , and minimize over explicitly. Using the fact that is symmetric, we obtain

 ρxx=min∇⋅jA=0∫ddxVjARABjB(∫ddxVjx)2, (7)

where the ‘incoherent resistivity matrix’

 R=Σ−10P≡Σ−10(1−nnTΣ−10nTΣ−10n). (8)

The (skew) projection matrix satisfies .

It is instructive to write in terms of a matrix of incoherent diffusion constants. These diffusion constants are found by plugging the linearized constitutive relations (1) and (2) into their respective conservation laws. The resulting hydrodynamic system will have one sound mode, diffusive scalar modes and a diffusive mode for transverse momentum. In Appendix B we show that the matrix appearing in the formula (7) can be written in terms of the matrix of inverse incoherent diffusion constants, , as

 R=D−1χ−1. (9)

One eigenvalue of is zero, and this is associated with the sound mode. Using (9) in (7), the electrical resistivity of a hydrodynamic electron fluid is bounded from above by the incoherent diffusivity matrix , which is associated with processes that decouple from momentum drag, cf. [25], weighted by a matrix of thermodynamic susceptibilities .

Our expression (7) is exact in the limit, where viscous effects are negligible. A resistivity bound can be obtained by a variational ansatz in which all the are spatially uniform and the . The constant can be taken outside the integrals in (7), leading to

 ρxx≤minJAxJAxRABJBxJ2x,RAB≡1V∫ddxRAB. (10)

Here is the spatial average of the incoherent resistivity matrix . The minimization over in the above equation can now be performed exactly, with a simple argument given in Appendix A. The result is

 ρxx≤1(R−1)nn. (11)

Here is the charge density component of the matrix . This bound is our main result. In one spatial dimension (), where divergence-free currents are necessarily constant, (11) is the exact resistivity. It is possible that with only one incoherent diffusive mode, (11) is far too weak, as currents may flow mostly along a percolating set of contours where the ratio of the two densities is held fixed [12]. With multiple incoherent diffusive modes, we expect that (11) is tight, up to constant prefactors.

The appearance of diffusivities in the bound can be understood as follows. The background inhomogeneities in the densities mean that a purely advective stationary flow is not compatible with simultaneously satisfying all of the conservation laws . The currents are forced to take the more general form of (1). The gradients in the (and in the conjugate densities ) sourced in this way then necessarily trigger diffusive dynamics, with an associated entropy production. In addition, to satisfy the conservation laws one must balance the density gradients with the velocity field, which generically requires . That is to say, the magnitude of the gradients created by the inhomogeneities is independent the wavelength of the inhomogeneities. That is why the inhomogeneity length scale does not appear explicitly in the bound.

While is not invertible – from the definition in (8), – the spatially averaged matrix appearing in (11) is invertible because the null eigenvector of is rotated from point to point in space. However, in the translationally invariant case, is not invertible. Thus the bound (11) is sufficiently powerful to capture the vanishing resistivity in the absence of momentum relaxation.

Furthermore, at leading nontrivial order in weak inhomogeneities, the currents minimizing (7) will be uniform and hence the bound (11) is saturated at leading nontrivial order in this limit. Focusing on the case where the inhomogeneity is due to fluctuations in the local chemical potential (often called charge puddles in the experimental literature), we show in Appendix C that

 ρxx≈¯V2dn2(χRχ)nn, (12)

where all quantities can be evaluated in the clean, homogeneous theory, and with the (weak) averaged disorder strength Here is the Fourier transform of the inhomogeneous chemical potential . The perturbative expression (12) can be obtained by expanding (11) or alternatively from a memory matrix computation. It is straightforward to generalize (12) to alternative types of disorder.

### 2.3 Thermal diffusion, phonons and bad metallic SrTiO3

An example clarifies the formal expressions above. We consider conventional hydrodynamics, where charge, heat and momentum are the only conserved currents (up to disorder). The one incoherent diffusion mode is thermal. Previous work has considered the role of thermal diffusion [12, 14, 25] in transport. For the case of thermal diffusion the matrix takes the form (see Appendix D)

 R=Tκ0(s2/n2−s/n−s/n1), (13)

where is the thermal conductivity. In particular, the perturbative resistivity (12) is given by (see Appendix D for the susceptibilities)

 ρxx=¯V2dTκ0(∂s/n∂μ)2. (14)

This is precisely the expression obtained in [12]. The resistivity (14) is proportional to the inverse of the thermal diffusivity, see again Appendix D. We can now go further and estimate the nonperturbative bound (11). A simple assumption is that the inhomogeneities are such that in (13) have order one variation over space but each retain the same order of magnitude. In that case, the averaged matrix in (10) will have the same form as in (13), but with order one coefficients appearing in each entry. These coefficients generically render invertible and hence from (11)

 ρxx≲s2n2Tκ0. (15)

Here refer to, for example, their average values.

For a degenerate Fermi liquid, the temperature scaling of the various quantities in (15) is and (here is the specific heat), leading to . At low temperatures this small resistivity will be overwhelmed by the viscous contribution that we dropped in the paragraph above equation (7). While viscous effects are suppressed by the long wavelength of the inhomogeneities, thermal transport is inefficient at low temperatures due to the factors of the entropy density in (15); the and limits do not commute [26]. The first signature of hydrodynamic transport at low temperature will be viscous in these cases; such effects have been observed experimentally [27, 28, 29] and have also been emphasized in theoretical discussions such as [30, 31, 11, 32, 33, 34, 35]. At low temperatures, therefore, the transport bound (15) is not accurate, as there will be a further viscous contribution on the right hand side that does not vanish at low temperatures. It is possible that a bound can be found in this regime by balancing viscous and thermal diffusive effects, along the lines suggested in [12] (see also discussion below (11)). The bound (15) is more likely to be relevant at higher temperatures, as we now discuss.

An intriguing possibility allowed by (15) is that non-electronic contributions to the thermal conductivity and entropy directly influence the electrical resistivity. In particular, the thermal conductivity and entropy can have a significant contribution from phonons. Recent transport experiments on the bad metal regimes of underdoped YBCO [36] and niobium doped SrTiO [37, 38] have found evidence that is related to the phonon velocity. The relevance of (15) is especially compelling in the nondegenerate regime of SrTiO. At high temperatures the thermal diffusivity of SrTiO (with or without a small amount of niobium doping) is measured to go roughly like [38]. Here is the sound speed. In Appendix D, we estimate the temperature scaling . If the bound (15) is approximately saturated, we obtain The resistivity and thermal conductivity of (doped and undoped) SrTiO have been reported in [37, 38]: roughly together with . At intermediate temperatures (around K) while at higher temperatures . This scaling is consistent with our hydrodynamic prediction.

## 3 Resistivity from microscopic scattering

### 3.1 Imbalance: Consequences of a non-thermal diffusive mode

The transport bound (11) is especially powerful in the presence of an additional, non-thermal, charge-carrying diffusive mode. The contribution of such modes to the resistivity bound will not be suppressed at low temperature by the temperature dependence of the entropy density, as in (15). It also becomes more difficult to find parametric improvements to the simple bound (11). An example of a non-thermal diffusive mode is an imbalance mode arising due to the approximate independent conservation of the number of fermions in different bands or pockets: see figure 1.

The key feature of non-thermal electronic diffusive modes is that the corresponding eigenvalues of will approach a finite value as . Thus, the temperature dependence of our bound (11) is

 ρxx≲1χeffD, (16)

where is the dominant non-thermal diffusivity in (9) and is a disorder-dependent constant, proportional to , which is temperature-independent at degenerate temperatures. If (16) is approximately saturated, we have achieved the objective of directly relating the resistivity to intrinsic momentum-conserving dynamics. As emphasized previously, the inhomogeneity length scale does not appear in the resistivity bound (11). Instead the resistivity depends on the (temperature dependent) microscopic mean free path . The mean free path determines the diffusivity via . Here is a velocity scale of the microscopic degrees of freedom, and is the time scale of momentum-conserving collisions. We will take to be temperature-independent; this can be justified in weakly interacting microscopic approaches such as kinetic theory [26]. Furthermore, angle-resolved photoemission data on bad metals indeed shows that despite significant broadening in energy, sharp features in momentum space survive, with a dispersion about the Fermi surface [39].1 A further temperature-independent microscopic velocity that can appear is the phonon velocity , when electron-phonon coupling is strong.

We emphasize that (16) is not a standard Einstein relation in which conductivity and diffusivity are related by . In (16), is not the charge diffusion constant, but rather an incoherent diffusion constant describing processes which are decoupled from the dynamics of momentum.

In the presence of generic disorder, we expect the bound (16) will be qualitatively saturated in the presence of imbalance modes. Putting the above formulae together then gives

 ρxx(T)∼1χeffv2F1τee(T), (17)

where have made explicit the fact that the temperature dependence of the resistivity is completely inherited from the temperature dependence of the momentum-preserving microscopic scattering rate.

### 3.2 From T2 to T-linear resistivity

In a Fermi liquid . Thus (17) leads to a novel mechanism for , without umklapp scattering. In the case where the extra conserved mode is associated with conservation of fermion number in different Fermi pockets or bands, this mechanism is perhaps reminiscent of Baber scattering [42], where a ‘light’ band which carries the current scatters off a ‘heavy’ band which efficiently relaxes momentum (it is not sufficient to simply have bands with carriers of different mass). In contrast, in our hydrodynamic mechanism, the two bands can even have identical masses. Furthermore, the resistivity is given by (17) due to the large imbalance gradients which arise; unlike in Baber scattering, all microscopic collisions conserve momentum.

Strong electronic correlations are necessary in order for a Fermi liquid to experimentally exhibit a resistivity. In several classes of compounds that do show a resistivity regime – including transition metal oxides, heavy fermions and organic metals – it has been noted that the coefficient of the resistivity can be expressed in terms of electronic properties of the compound [9, 10]. In particular, this means that the coefficient cannot not have a strong dependence of the amount of disorder in the material. This is compatible with the fact that (16) has no singular dependence on the wavelength of the inhomogeneities. Indeed, many of these materials have complicated band structures with multiple pockets and bands which may provide imbalance modes. Many of these materials are also layered: dopant impurities are geometrically separated from the conduction layers. This will tend to increase the wavelength of the Coulomb impurity potential created by the impurities [26].

Furthermore, the phase diagrams of strongly correlated materials showing resistivity are replete with quantum phase transitions and associated ‘critical fans’ exhibiting -linear resistivity. For instance transition metal oxides [43, 44, 45], organic metals [46], heavy fermion compounds [47, 48] and pnictides [49, 50] show this behavior. For an overview see [1]. Just like the resistivity, the coefficient of the -linear resistivity is known to often be determined by purely electronic properties [5], and does not appear sensitive to the purity of the sample.

As noted previously, quantum critical dynamics is characterized by a ‘Planckian’ [51] microscopic scattering rate [2]. Using (17), we therefore predict a transition from to -linear resistivity upon entering quantum critical fans, due to the change in microscopic scattering rate. The mechanism for momentum relaxation (long wavelength inhomogeneities) does not change. This is an attractive scenario because while the resistivity in these materials could be due to umklapp scattering, there is no compelling candidate for a -linear momentum-relaxing microscopic scattering process at low temperature.

Many -linear materials are in fact bad metals at higher temperatures, with a resistivity above the Mott-Ioffe-Regel limit [15, 16, 17]. Our bound (16) is consistent with this behavior, and makes precise the idea that bad metals with -linear resistivity saturate a transport bound [18]. What is novel about our mechanism is that this bound can be saturated without each collision relaxing momentum. The diffusion constant for charge in the inhomogeneous theory can be reliably estimated from the incoherent diffusion constants of the clean, homogeneous theory. Unlike in [18], we do not need to postulate rapid local momentum relaxation. There is some theoretical [52, 53] and experimental [36, 54] evidence that the momentum-conserving scattering rates of clean systems could be bounded . Our formalism shows how a Planckian bound on could be directly responsible for a transport bound .

## 4 Outlook

Without assuming a weakly interacting quasiparticle description, we have described a novel mechanism through which the microscopic scattering rate directly controls the temperature dependence of the resistivity. To determine whether this mechanism is relevant for a specific material, one must go beyond hydrodynamics. In a complementary paper we show how the bound (11) emerges from kinetic theory [26], in the weak coupling limit. Within kinetic theory, we can capture both the hydrodynamic regime, as well as the crossover to a ballistic regime when .

Experimentally, it is crucial to establish whether or not the electron fluids in strange metals are in the hydrodynamic regime. The objective is to establish the presence of a velocity field in the long wavelength dynamics. This can be more difficult in a non-Fermi liquid with a non-Galilean hydrodynamic limit [55]. If imbalance modes are generic in the hydrodynamic regime of strange metals, they will complicate the search for other signatures of hydrodynamic flow, such as viscous “whirlpools” [33, 34], in all but the simplest metals, such as graphene.

Our proposal motivates searching for evidence of imbalance modes in strange metal transport. This could potentially be achieved by modifying the band structure of the materials to eliminate Fermi pockets, or by direct observation of imbalance diffusion. Previous measurements of diffusive modes in unconventional metals include [56, 57, 36].

It will also be important to identify other non-thermal diffusive modes that can play the required role, including spin imbalance or phase-fluctuating order parameters such as superconductivity [58] and density waves [59]. For instance, the low temperature resistivity of SrTiO shows a scaling even when the Fermi surface is too small for umklapp scattering to be effective [60]. This system may be an excellent candidate for our non-umklapp mechanism of resistivity, but it remains to identify a candidate non-thermal diffusive mode in this single-band material.

## Acknowledgements

It is a pleasure to thank Kamran Behnia and Steven Kivelson for helpful discussions. AL was supported by the Gordon and Betty Moore Foundation. SAH is partially supported by a DOE Early Career Award.

## Appendix A Proof of the variational principle

There are four statements to prove.

1. The functional (5) in the main text is extremized on stationary solutions to the hydrodynamic equations of motion in the presence of an applied electric field.

Proof: We must extremize (5) with respect to and and subject to the constraints that . The constraints are implemented as usual by Lagrange multipliers . Furthermore, in the presence of ‘electric fields’ , the constitutive relation (1) is modified to

 jA=nvA−ΣAB0(∇μB−EB), (18)

and momentum conservation is modified to

 ˙πi+∂jτji=nAEA. (19)

We will be interested in only having an electric field sourcing the actual electric current , but it is convenient to work with the more general case to start with.

Variation with respect to directly gives the condition

 ∂j(−ζδij∇⋅v−η(∂ivj+∂jvi−2dδij∇⋅v))=(jA−nAv)(Σ−10)ABnB. (20)

We can now see that this relation is identically true on solutions to the hydrodynamic equations of motion. Using (2) in the main text, the term in brackets on the left hand side of (20) is equal to . Similarly the term in brackets on the right hand side of (20) is equal to using (18). Using (19) on stationary solutions to set , along with the thermodynamic identity (3), we see that (20) is simply the generalized Navier-Stokes equation.

Now we vary with respect to . First we consider the currents excluding the electric current:

 (jA−nAv)(Σ−10)AB=(∫ddxjx)2∇ϕB. (21)

At this point we set sourcing all of these currents (which do not include ) to zero. Therefore using the constitutive relation (1) in the main text, the left hand side of (21) is simply . The equation is therefore solved by setting the Lagrange multipliers

 ϕA=−μA(∫ddxjx)2. (22)

 (jA−nAv)(Σ−10)An=(∫ddxjx)2∇ϕn+K, (23)

where the constant . Using the modified constitutive relation (18) for , with an electric field , and setting the Lagrange multiplier , the equation is seen to be solved, with the electric field set to be . ∎

2. The solution to the hydrodynamic equations minimizes the functional.

Proof: The functional (5) in the main text has the abstract form

 R[X]=X⋅A⋅X(a⋅X)2, (24)

for some positive matrix , some vector and with a vector. The positivity of is required in hydrodynamics for positivity of entropy production. The second order variation of about a stationary point is found to be

 ρ(2)xx=δX⋅A⋅δX(a⋅X0)2+(a⋅δX)2(a⋅X0)3, (25)

which is manifestly positive for any variation.∎

3. The variational functional (5) in the main text evaluated on the minimum is equal to the electrical resistivity.

Proof: The variational functional becomes the entropy production evaluated on the solution, divided by the total electric current squared. In the proof of the first statement above we have seen than a uniform electric field is turned on, but there are no other external sources. The entropy production is therefore given by Joule heating: . Furthermore, in the absence of any external sources except an electric field, the electric current is given by . Evaluating the functional on the solution thus leads to

 ρ(0)xx=σxxE2xσ2xxE2x=1σxx≡ρxx, (26)

as required. ∎

4. The solution to the optimization problem in (10) in the main text is equation (11) in the main text.

Proof: We need to find the minimum value of a functional of the abstract form (24). A straightforward derivative gives an extremum when

 ∂R∂X=0=2A⋅X(a⋅X)2−2aX⋅A⋅X(a⋅X)3, (27)

which implies that

 X=cA−1a. (28)

with an arbitrary non-zero constant (since ). Hence we find that

 minR=1a⋅A−1⋅a, (29)

which directly leads to (11), upon replacing with and with a unit vector in the (charge) direction. ∎

## Appendix B Diffusive modes

As noted in the main text, the hydrodynamic equations of motion lead to diffusive scalar modes. These are of particular interest, so let us exhibit them explicitly. To isolate the diffusive modes, we look for linearized solutions of (1) and (2) with . From (3), this means that

 nT\mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}μ=0. (30)

Left-multiplying (1) by the skew projection matrix defined in (8), and using , we arrive at

 Pχ\mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}˙μ=PΣ0∇2\mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}μ=(Σ0−nnTnTΣ−10n)∇2\mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}μ=Σ0PT∇2\mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}μ. (31)

Left-multiplying by we obtain (recall that was defined in (8))

 R\mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}˙n=Rχ% \mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}˙μ≡D−1% \mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}˙μ=PT∇2\mbox{   {1 0 -.18 1}\hbox to 0.0pt{δ}}μ. (32)

The matrix obeys . Hence, one can see that the non-zero eigenvalues of correspond to the incoherent inverse diffusion constants, with (30) obeyed.

## Appendix C Resistivity at weak disorder

When momentum relaxation is weak, then the resistivity may be calculated using the memory matrix formalism, by considering the inhomogeneity as a perturbation of the translationally invariant hydrodynamic state [24]. The resistivity is given by the Drude-like formula

 ρxx=Mn2Γ, (33)

where was defined in §2.1 in the main text and generalizes the role played by the quasiparticle mass density in the usual Drude formula. In the presence of a weak inhomogeneous chemical potential, the momentum relaxation rate is given to leading order by

 Γ=1dM∫ddk(2π)d|V(k)|2k2limω→0ImGRnn(ω,k)ω. (34)

Here the spectral weight is to be calculated in the homogeneous theory. The inhomogeneous source has strength at wavevector . In deriving (34), we have assumed that is isotropic.

When the inhomogeneities are long wavelength (in addition to being weak), so that , then only small wavevectors contribute to the integral in (34). In that case, the Green’s function that appears can itself be evaluated using (translationally invariant) hydrodynamics [61]. The result can be summarized as follows [24]. We write the hydrodynamic equations of motion in the form

 ˙nA+M(k)ABμB=0. (35)

The matrix of retarded Green’s functions for the conserved densities is then

 GR(ω,k)=M1M−iωχχ. (36)

Recall that was defined in §2.1. These Green’s functions can be explicitly evaluated. We know that

 limω→0ImGRAB(ω,k)ω=(χM−1χ)AB, (37)

and from (32) that . Hence,

 k2limω→0ImGR(ω,k)ω=χRχ+O(k2). (38)

This result can be substituted into (34) and the integral over performed in terms of the averaged strength of inhomogeneities

 ¯V2≡∫ddk(2π)d|V(k)|2. (39)

Plugging the result for into (33) we obtain (12).

## Appendix D Formulae for thermal diffusion

Here we give the derivation of (13) in the main text, and exhibit the connection to thermal diffusion. In minimal linearized hydrodynamics the two conservation laws for the charge and entropy density are

 ˙n+∇⋅j=0,˙s+∇⋅(jQ/T)=0. (40)

Within linearized hydrodynamics, it is convenient to take entropy rather than energy to be the conserved quantity. The matrix of transport coefficients appearing in the constitutive relations and the matrix of susceptibilities can be written (recall )

 Unknown environment '% (41)

One then finds from the general formula (9) that the nonzero inverse diffusivity eigenvalue

 1Dth.=(n∂s/n∂T−s∂s/n∂μ)Tκ0, (42)

where

 κ0≡¯κ0−2Tsnα0+Ts2n2σ0, (43)

is the incoherent thermal conductivity with open circuit boundary conditions [24]. This diffusive mode can therefore be thought of as thermal diffusion. In a Galilean invariant system , so that .

From (8) in the main text, the matrix

 R=Tκ0(s2/n2−s/n−s/n1). (44)

It is clear that this matrix is proportional to the inverse diffusivity (42).

### Footnotes

1. In holographic models of compressible matter, characteristic velocities are instead found to scale as , with a dynamic critical exponent [40, 41]. From the perspective of more conventional electronic systems, such behavior would be indicative of a non-degenerate regime where a single-particle dispersion relation leads to a temperature-dependent velocity . This is consistent with the fact that the charge dynamics in holographic models does not show conventional signatures of a degenerate Fermi surface [24].

### References

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