Joule heating in bad and slow metals

Heat supplied to a metal is absorbed by the electrons and then transferred to the lattice. In conventional metals energy is released to the lattice by phonons emitted from the Lindhard continuum. However in a `bad' metal, with short mean free path, the low energy Lindhard continuum is destroyed. Furthermore in a `slow' metal, with Fermi velocity less than the sound velocity, particle-hole pairs are kinematically unable to emit phonons. To describe energy transfer to the lattice in these cases we obtain a general Kubo formula for the energy relaxation rate in terms of the electronic density spectral weight $\text{Im} \, G^R_{nn}(\omega_k,k)$ evaluated on the phonon dispersion $\omega_k$. We apply our Kubo formula to the high temperature Hubbard model, using recent data from quantum Monte Carlo and experiments in ultracold atoms to characterize $\text{Im} \, G^R_{nn}(\omega_k,k)$. We furthermore use recent data from electron energy-loss spectroscopy to estimate the energy relaxation rate of the cuprate strange metal to a high energy optical phonon. As a second, distinct, application of our formalism we consider `slow' metals. These are defined to have Fermi velocity less than the sound velocity, so that particle-hole pairs are kinematically unable to emit phonons. We obtain an expression for the energy relaxation rate of a slow metal in terms of the optical conductivity.


Joule heating
The dc electrical conductivity σ of a closed quantum system is defined via the linear response of the system to a uniform electric field j = σE.However, beyond linear order the electric field causes Joule heating.Energy density ε is pumped into the system at a rate ε = j • E = σE 2 .If the system is truly closed then this energy has no way to dissipate, the system will heat up and a stationary state is not possible, rendering linear response ill-defined.
Real-world condensed matter systems are not closed and can dissipate energy to the environment.An important energy sink are the neutral lattice vibrations.The transfer of energy from the electronic degrees of freedom to the lattice has been long studied theoretically [1][2][3].More recently, the timescale τ over which the electronic degrees of freedom release energy to the lattice has been probed using ultrafast spectroscopy [4].In these experiments, energy supplied to the material naturally heats up the electrons relative to the lattice, as the electronic degrees of freedom have a lower specific heat.The subsequent cooling of the electronic subsystem, as energy is transferred to the lattice, can be followed in detail using time-and angle-resolved photoemission (tr-ARPES) [5][6][7][8].
In the experiments described above, it is observed that the electronic subsystem thermalizes on a much faster timescale than τ .This means that throughout the cooling process the electrons have a well-defined, time-dependent temperature T (t).Such rapid redistribution of energy among electronic degrees of freedom requires electronic interactions.However, in some tension with this fact, the established theory of energy relaxation to the lattice describes phonons emitted by the Lindhard continuum of non-interacting particlehole pairs [1,3].In conventional metals, as we explain below, this tension is resolved by the fact that most phonons are emitted by low energy fluctuations of particle-hole pairs on inverse momentum scales shorter than the electronic mean free path.Such fluctuations are not strongly affected by interactions.
In non-Fermi liquids, however, strong electronic interactions can lead to the absence of coherent particle-hole excitations at any momenta.While many theoretical and experimental results support this statement, perhaps the most directly relevant for our purposes are recent measurements of the dynamic charge susceptibility using momentum-resolved electron energy-loss spectroscopy (M-EELS) in the non-Fermi liquid regime of a cuprate [9,10].These have revealed the conspicuous absence of a well-defined Lindhard continuum in the charge density spectral function Im G R nn (ω, k).If there is indeed no Lindhard continuum then the established theory of energy loss to the lattice is clearly inapplicable.
In this work we obtain a Kubo formula for the timescale τ that does not start from non-interacting electrons.Instead, our approach will make a virtue of the fact that electronic thermalization is faster than equilibration with the lattice.Energy transfer to the lattice can therefore be treated as a perturbative process that occurs in an electronically thermalized medium.This leads to our formula (14) below for τ in terms of the equilibrium electron density spectral function Im G R nn (ω, k) integrated along the phonon dispersion ω = ω k .This particular spectral function appears because we will consider lattice vibrations that couple to the charge density via a deformation potential. 1It is the same object that can be extracted from M-EELS measurements and, needless to say, is among the most basic quantities characterizing an electronic medium.To illustrate the application of our Kubo formula to an incoherent metal, we will use the recent M-EELS data to estimate the rate of energy dissipation from the cuprate strange metal into a high-frequency optical phonon.
In conventional metals with a long mean free path our framework will be shown to precisely recover the classic results [1,3].More generally, our formula can be used with or without coherent particle-hole excitations to quantitatively obtain the relaxation rate 1/τ once the phonon dispersions and Im G R nn (ω, k) are known from experiment or numerics.This connection can be tested in spectroscopic pump-probe experiments and also in measurements of non-linear current response, as we describe below.
Bad metals [11][12][13] are especially well-suited to our Kubo formula because the very short mean free path means that the entire low energy Lindhard continuum is destroyed, and Im G R nn (ω, k) instead acquires a universal diffusive form.We obtain a simple expression for the relaxation rate 1/τ in this limit, which we proceed to evaluate using recent measurements of diffusion in an ultracold atom simulation of the Hubbard model at high temperatures [14].
These high temperatures are comparable to those arising in pump-probe experiments in which the electrons become very hot.
A second, distinct, application of our formula is to 'slow' metals, where the Fermi velocity is small compared to the sound speed.In this case the phonon dispersion is outside of the Lindhard continuum.Electronic interactions are therefore essential to broaden the charge density spectral weight to low momenta, allowing phonons to be emitted.We obtain the decay rate 1/τ using our Kubo formula together with a low momentum spectral weight consistent with a Drude conductivity.Small Fermi velocities could in principle arise in semimetals, in doped semiconductors, or if an effective mass divergence strongly renormalizes the Fermi velocity down, as occurs close to a quantum critical point.

Stationary state current with an applied field
Consider a Hamiltonian for electrons and phonons (lattice vibrations) of the form H = H e + H e-ph + H ph . (1) We will obtain the rate at which energy is transferred from the electronic degrees of freedom to the lattice, working perturbatively in the electron-phonon interaction H e-ph .The electronic Hamiltonian H e can be strongly correlated.We consider the case where the electrons and phonons are only slightly perturbed away from equilibrium.
Let δε be the difference of the energy density of the electronic system from the value set by thermal equilibrium with the lattice.The lattice is assumed to be kept at a constant temperature by rapid equilibration with the external environment.The energy density then obeys the conservation equation In this section we will obtain a Kubo formula for the energy relaxation rate 1/τ .Equation (2) is precise if the electronic system locally equilibrates itself on timescales faster than τ , because in that case all non-conserved local quantities will have already decayed and one need only keep track of the energy perturbation δε.More generally, (2) may be a reasonable 'relaxation time approximation' for energy dissipation.The setup is illustrated in Fig. 1 Electrons < l a t e x i t s h a 1 _ b a s e 6 4 = " q P b b 9 v f m B 5 5 f X i f t y 7 r n 1 r 3 H q 1 r j t q i j j E 7 R G b p A H r p G D f S A m q i F C J q h Z / S K 3 q z M e r H e r Y / l a M k q M i f o D 6 z P H w F 5 l Q 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B 0 L f X i f t y 7 r n 1 r 3 H q 1 r j t q i j j E 7 R G b p A H r p G D f S A m q i F C J q h Z / S K 3 q z M e r H e r Y / l a M k q M i f o D 6 z P H w F 5 l Q 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B 0 L f X i f t y 7 r n 1 r 3 H q 1 r j t q i j j E 7 R G b p A H r p G D f S A m q i F C J q h Z / S K 3 q z M e r H e r Y / l a M k q M i f o D 6 z P H w F 5 l Q 0 = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " B 0 L f X i f t y 7 r n 1 r 3 H q 1 r j t q i j j E 7 R G b p A H r p G D f S A m q i F C J q h Z / S K 3 q z M e r H e r Y / l a M k q M i f o D 6 z P H w F 5 l Q 0 = < / l a t e x i t > rapid equilibration to T (0) 6 Y E d z 7 y I q m f l V y n 5 N 6 e F 8 p X s z p y 5 J A c k S J x y Q U p k x t S I T X C y S N 5 J q / k z X q y X q x 3 6 2 M 6 u m T N d g 7 I H 1 i f P 5 + I n F c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 T q u V X v 5 q x S u 5 z F U S S H 5 I i c E I + c k x q 5 J n X S I J w 8 k m f y S t 6 c J + f F e X c + p q 0 F Z z a z T / 7 A + f w B 8 u C W r w = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d f u r a W w z I j g A + H y r m 5 N 8 i 4 a e F f c = "     Firstly, we will illustrate the physics of the timescale τ .Time-independent and spatially uniform solutions to (2) are obtained by balancing the two terms on the right hand side.
Thus we see that the applied field causes a shift in the electronic temperature away from the value T (0) set by the lattice degrees of freedom.That is, T = T (0) + δT with Here c is the electronic specific heat and in the final equality we have worked to leading order in small δT so that j = σ(T (0) )E ≡ σ (0) E. We will assume spatial isotropy for simplicity.
The shift in temperature δT is seen in ( 3) to depend on the applied field, and therefore leads to non-Ohmic conduction.Accounting for this shift Here we expanded σ(T ) = σ (0) + σ (0) δT + • • • , where σ (0) is the temperature derivative of the conductivity.The • • • terms in (4) include other sources of nonlinear current response.
The correction shown in (4) will dominate other effects as τ becomes large.This is when the coupling to the lattice is weak.
Equation (4) shows that while there is no steady state as τ → ∞, as anticipated, for any finite τ the dc conductivity is given by σ (0) for sufficiently small electric fields.More generally, (4) shows that the timescale τ determines the electric field (or, equivalently, the critical current) above which conduction becomes non-Ohmic.From balancing the first two terms on the right hand side of (4), the critical current is Note that in a metal σ (0) < 0, hence the absolute value in the above equation.We will return to this expression once we have a better handle on τ .Once conduction is non-Ohmic, for j > j non-Ohm , higher order corrections to (4) are likely to be important.

Energy relaxation rate due to phonons
Using ( 6), the energy relaxation rate is seen to be The order of limits is important.We need to take ω small compared with electronic equilibration times in order to be in the low energy regime universally described by (6), while ω must be large compared to 1/τ to pick out the numerator of ( 6).These two conditions are compatible if, as we have already described above, τ is long compared to purely electronic timescales.We noted in the introduction that experiments in strongly correlated materials indicate that the electrons indeed self-thermalize faster than the combined electron-phonon system.This will allow us to treat the electron-phonon interaction as a perturbation of the thermalized electronic dynamics.In the absence of such a hierarchy of timescales, the full coupled electron-phonon problem would need to be solved, which is beyond the methods we develop below.Our approach will also implicitly assume that the backreaction of phonons on the electronic sector, which will be neglected, does not lead to any qualitatively important effects, as we discuss further in section 4 below.
The second line ( 8) is a commonly used step in the memory function approach to relaxation rates [16].It follows from the general relation G R ȦB (ω) = −iωG R AB (ω) + χ ȦB , together with the fact that the susceptibility χ ĖE = 0.The reason to make the step to (8) is that Ė is proportional, as an operator, to the electron-phonon interaction.This is because the total electron energy is conserved without the coupling to the lattice.When 1/τ is small compared to electronic relaxation rates we may formally assume that the electronphonon coupling is 'small' and work perturbative in this coupling.This does not require the dimensionless electron-phonon coupling to be numerically small, it is the comparison to purely electronic timescales that matters.To obtain 1/τ to leading order in this 'small' electron-phonon coupling, then, the Green's function Im G R Ė Ė (ω) itself can be computed in the theory with vanishing electron-phonon interaction.The memory function approach is designed to bring out this simplification, and has been successfully used to describe slow momentum relaxation due to weak translation symmetry breaking [17][18][19] as well as slow phase relaxation due to dilute vortices in superfluid flow [20,21].Here we are extending the method to slow energy relaxation due to weak coupling to the lattice.
To make use of the Kubo formula (8) we need to specify the dynamics of the phonons.
To start with we consider low temperatures and, correspondingly, acoustic phonons.We will take the phonon Hamiltonian to be Here s labels the acoustic phonon modes, with phonon creation operators a † ks .We do not incorporate any anharmonicity of the phonons, which are taken to be in equilibrium with the environment as shown in Fig. 1.For illustrative purposes we will take the electron-phonon interaction to have the textbook form here n k = ´dd xe −ikx n(x), with n(x) the charge density, and the deformation potential is We give a continuum limit derivation of this textbook formula (e.g.[22]) in Appendix A.
The polarization vector appearing in the mode decomposition of the phonon displacement is taken to be odd in k i → −k i , which leads to the relative minus sign between a k and a † −k in (11) -see (41) in the Appendix.Here ρ M is the mass density and C the deformation constant.The longitudinal mode has dispersion The transverse acoustic modes do not appear in this interaction, and we shall suppress the band index in the following.It is clear that (10) is a highly simplified model of electronphonon interactions and ( 12) a simplified model of phonon dispersion.Correlated materials often have exceedingly complex phonon dynamics and this dynamics must be taken as an input to our method.Nonetheless, once (10) and ( 12) are upgraded to realistic expressions for some specified material, the logic of the remaining steps below will go through unchanged.
Because E is conserved within the purely electronic system, so that [H e , E] = 0, and independent of the phonon degrees of freedom, so that [H ph , E] = 0, we have Here we used the fact that V k does not depend on the electronic degrees of freedom and therefore commutes with H e .We now insert (13) into the Kubo formula (8).Using standard manipulations of nonzero temperature Green's functions -given in Appendix B -we obtain This expression is physically transparent: the electronic energy decay rate is obtained by integrating over all processes that can excite a phonon with momentum k and energy ω k .
There is a k 2 weighting due to the derivative coupling of the acoustic phonon to the charge density.Frequencies above ω k ∼ T are not thermally accessible are suppressed.The only electronic quantity that needs to be determined is the spectral weight Im G R nn (ω, k).As we have noted, this is to be evaluated in the purely electronic theory, without electron-phonon coupling.We now discuss this quantity, after first briefly generalizing to optical phonons.

Optical phonons
A non-dispersing optical phonon mode can be modelled with the Hamiltonian Here ω o is a constant, g the electron-phonon coupling and the creation modes obey . An entirely analogous computation to the acoustic phonon case leads to This decay rate will be exponentially small unless the temperature is above the energy of the optical mode, T ω o .
3 Evaluation of the energy relaxation rate

Phonon emission by the Lindhard continuum
Previous computations [1,3] have used non-interacting electronic Green's functions to calculate the energy relaxation rate.We can firstly recover those results from (14).A noninteracting spin-half fermion with dispersion relation ω = k has the Lindhard function 3 Here the Fermi-Dirac distribution n F ( ) = 1/(e /T + 1).We must now insert ( 17) into (14).
To bring out the physics in the simplest possible setting we will take a spherical Fermi surface with parabolic dispersion, so that q = (q 2 − k 2 F )/(2m).The Fermi velocity v F = k F /m. 3 The unscreened charge dynamics must be used to compute the energy relaxation rate into phonons.
The effects of screening by Coulomb interactions have already been incorporated into the electron-phonon interaction -indeed, screening is essential in order to obtain a short range electron-phonon interaction in the first place.The electron-phonon interaction therefore expresses the coupling of the electrons to the total electric field created by lattice vibrations, including screening.The electronic response to the total electric field is then given by the unscreened Green's function, see e.g.§7.4.2 of [22].
The energy conservation delta function in (17) then has argument qk cos θ , with θ the angle between q and k.At temperatures much lower than characteristic electronic energy scales, T E F = k 2 F /(2m), the Fermi-Dirac functions in (17) restrict q to lie close to the Fermi surface so that we may set q → k F in the delta function.The delta function thus picks out the angle cos θ = (2v This is the constraint that the phonon must lie within the Lindhard particle-hole continuum, as illustrated in Fig. 2. When v s < v F there is no constraint from the lower bound in (18).The shaded blue region is the phase space of Lindhard particle-hole pairs.The orange line is the dispersion of an acoustic phonon with v s = 1 20 v F .The constraint ( 18) is obeyed where the phonon is inside the particle-hole continuum.
In Appendix C we perform the standard momentum integrals to evaluate the relaxation rate ( 14) using the Lindhard function (17).In a conventional metal with v s v F the upper cutoff in ( 18) is set by 2k F .It is then natural to express the timescale τ in terms of the Bloch-Grüneisen temperature T BG ≡ 2v s k F and the dimensionless electron-phonon coupling This normalization agrees with e.g.[22].In three dimensions, for example, we obtain where we used the specific heat c = 1 3 k 2 F v F T .In the low temperature limit the integral in (20) gives 480ζ(5) ≈ 498 and ( 20) is then in exact agreement with an expression in [1].This provides a check on our formalism.More generally, the rate ( 20) is plotted as a function of T /T BG in Fig. 3.At low temperatures 1/τ ≈ 237λT 3 /T 2 BG while at high temperatures 1/τ ≈ 0.47λT 2 BG /T .The rate peaks at T ≈ 0.3T BG with the value of 1/τ ≈ 0.9λT BG ≈ 3λT .This maximum rate approaches a 'Planckian' value once λ ∼ 0.3 (cf.[23], page 54).As a crude estimate of the timescale τ , assume the dimensionless electron-phonon coupling λ ∼ 0.2, a typical value for non-superconducting metals [24] and take a typical value T ∼ T BG ∼ 300 K. Restoring factors of and k B we obtain As is clear from Fig. 3, this timescale will get longer at both low and high temperatures.
3.2 Destroying and evading the Lindhard continuum

Bad metals
Interactions lead to a mean free path e and lifetime τ e for the electronic single-particle Bloch states.We will mostly be interested in strongly inelastic collisions that distribute energy as well as momentum among the particles.Such interactions broaden the single-particle states, producing a thermalized many-body medium.At low frequencies ω 2π/τ e and momenta k 2π/ e the charge density spectral weight is universally described by collective many-body dynamics (diffusion, in the simplest case).The Lindblad continuum will only re-emerge at larger frequencies and momenta, at which collisions have not yet occurred.
The collective regime therefore gives a universal contribution to the relaxation rate 1/τ from the low momentum part of the integral in the Kubo formula (14).In a conventional metal the electronic quasiparticles have a long mean free path so that 2π/ e k F .This implies that only a small fraction of the dissipated modes are in the collective regime.Most dissipation instead occurs via modes in the range 2π/ e k 2k F that have low energy but large momentum.These modes can be approximately described by the Lindhard function (17).This fact is illustrated in Fig. 4. We estimate in Appendix D that in conventional metals with a long mean free path the Lindhard contribution strongly dominates the integral for 1/τ , as is suggested by Fig. 4.
This explains why the classic treatment of electronic energy loss in conventional metals [1,3] is justified -the emission of phonons is dominated by density fluctuations with low energy but large momentum.The Fermi surface supports a large number of such fluctuations, the Lindhard continuum, extending up to k = 2k F .These fluctuations occur on inverse momentum scales that are short compared to the mean free path and are therefore not strongly impacted by interactions.
However, if the mean free path becomes sufficiently short the collective regime 'eats up' the entire low energy Lindhard continuum.In particular, the low energy charge dynamics becomes collective over all momenta once 2π/ e ∼ 2k F .The relaxation rate 1/τ is then fully determined by the collective dynamics in this case, rather than particle-hole pairs, as is also illustrated in Fig. 4. Metals with such short mean free paths are called bad metals [11][12][13].
Because the quasiparticle mean free path is likely not a well-defined concept in such regimes, it may be fruitful to define a bad metal as one where the low energy charge dynamics is collective (more specifically, we shall see, diffusive) at all momenta.From this perspective, bad metals show a remarkable universality -collective charge dynamics usually arises as a long wavelength effective theory [25,26], but in a bad metal it is also the microscopic theory!This universality of charge dynamics makes bad metals especially amenable to our Kubo formula for energy dissipation, as we shall explore below.

Slow metals
A second, distinct, circumstance in which the classic formula for energy relaxation will not hold is when the Fermi velocity is small, v F < v s .In this case the acoustic dispersion is outside of the Lindhard continuum at low energies ω E F , as illustrated in Fig. 5 We will refer to such systems as 'slow' metals.It follows that at temperatures T E F the rate of energy loss through the Lindhard continuum is exponentially small at best.Fig. 6 illustrates why a single electron is kinematically unable to emit or absorb a phonon when v F < v s (in the absence of inter-band scattering, cf.[27]).However, multi-electron Figure 6: An electron with momentum q absorbing or emitting a phonon with momentum k acquires final momentum q ± k and energy q ± v s k.These two conditions can be satisfied processes are able to emit phonons.In particular, interactions will spread the charge density spectral weight down to low momenta, outside of the non-interacting Lindhard continuum.
At low energies this spectral weight will have a universal collective form, as is shown in Fig. 5.
More generally, when the sound speed is large compared the Fermi velocity, the relevant charge density spectral weight can be obtained in terms of the k = 0 optical conductivity σ(ω).As we will see below, our Kubo formula again gives a robust and simple expression for the decay rate 1/τ in this limit.

Phonon emission by an incoherent spectrum
As a first explicit example of phonon emission by an incoherent spectrum, we will use the recent M-EELS measurements in optimally doped BSCCO that were mentioned in the introduction.These measurements reported an incoherent charge density spectrum without the dispersive features present in the Lindhard continuum.At T = 295 K and for frequencies ω 0.1 eV [9]: The constant A ≈ 28 × 10 −3 /(eV Å 3 ), the lattice spacing a = 3.81Å, and ω c (k) ≈ 1 eV, with only a weak k dependence.
The conventional formula for energy relaxation -based on the Lindhard function -is manifestly inapplicable in this system.The dissipation of energy into a high energy optical phonon mode with frequency above 0.1 eV can instead be obtained by using the experimentally determined Im G R nn (ω, k) from ( 22) directly in our Kubo formula (16).Because the temperature at which (22) was measured is relatively small compared to these energies, decay into such a mode will be Boltzmann suppressed.Presumably emission into lower energy phonon modes will be dominant in practice, but those processes are not described by (22) which is restricted to ω 0.1 eV.We will use (22) to explicitly compute the decay rate into an ω o ≈ 0.1 eV phonon, to illustrate the use of our formula.From ( 16) we obtain In the first line ∆z ≈ 15Å is the interlayer spacing.To get the (very rough) final estimate we integrated k x and k y from −π/a to π/a, set g 2 = ω 3 o a 2 ∆z to define an order one dimensionless electron-phonon coupling (in three dimensions) and estimated the three dimensional electronic specific heat c ∼ 0.5/(a 2 ∆z) at T = 295 K.This crude estimate is obtained from the fact that several other cuprates have electronic specific heat c/T ∼ 1 mJ/(g-at.K 2 ) [28].
Despite the Boltzmann suppression and the incoherence of the spectrum, the energy dissipation time τ into this mode is seen to be faster than our previous estimate (21) for a conventional metal at the same temperature (and lower phonon frequency).The timescale for the excitation of a high energy optical phonon may be observable in a mode-resolved pump-probe experiment and is a prediction of the spectral function measured by M-EELS.

Phonon emission and diffusion in the high T Hubbard model
We have explained above that once the mean free path becomes of order 1/(2k F ) the low energy Lindhard continuum is fully dissolved into collective charge dynamics.If the collisions are inelastic and thermalize the system, the charge dynamics at ω 2π/τ e and k 2π/ e is strongly constrained due to conservation of charge and spatial locality of the microscopic interactions.If we furthermore assume that the electronic scattering degrades the total momentum, and that no symmetries are spontaneously broken, then this collective dynamics will be diffusive. 4The simplest model spectral weight that describes the onset of diffusion beyond a timescale τ e is [15] Here χ is the charge susceptibility and D the diffusivity.At low frequencies ωτ e 1 the form (25) becomes purely diffusive.At low momenta Dk 2 /ω 1 it instead acquires a form that is consistent with a Drude optical conductivity: As always, the dc conductivity σ dc = χD.
Remarkably, but consistent with our general expectations outlined above, the form (25) has recently been shown to control the charge dynamics of the high temperature Hubbard model as simulated in an ultracold atomic lattice [14].Furthermore, recent Monte Carlo computations of the spectral weight Im G R nn (ω, k) in [29][30][31] also appear to be compatible with the form (25) over a substantial portion of the Brillouin zone, although this was not remarked upon.Fig. 7 shows Im G R nn (ω, k) according to (25) as a function of ω and k, which can be compared to Fig. 8 in [31] between the Γ and K points.While the two  dimensional Lindhard function is similarly peaked along a linearly dispersing curve, its peak is highly asymmetric in ω, vanishing for ω > v F k F .In contrast the peaks seen in [29][30][31] are Lorentzian as a function of ω, as in (25).While this comparison is promising, the numerical lineshapes in [29][30][31] are obtained using analytic continuation from imaginary time data and therefore may not be accurate.Future high resolution numerical or experimental study of Im G R nn (ω, k) in the Hubbard model strange metal is desirable to corroborate the diffusive model that we develop here.Finally, once τ e ω 1 it is natural for D to become k dependent, so that the peak follows the electronic dispersion, as is seen in [31].The simplification of a constant D and τ e in (25) will not affect the estimates below.
In Appendix C we evaluate the Kubo formula (14) on the diffusive spectral function (25).We consider an incoherent regime where the form (25) holds up to the Brillouin zone cutoff k ∼ π/a.The resulting expression is simplified by estimating the relative importance of various terms.Assuming that v F v s and restricting to temperatures Note that τ e does not appear in this expression.This is because, due to small v s , the ω terms in (25) are subleading along the phonon dispersion.
The electronic quantities χ and D appearing in (27) have been measured in a simulation of the two dimensional Hubbard model in an ultracold atomic system [14] -where precisely the diffusive form (25) was used to fit the data.The model was simulated with U ≈ 7.5t and near-critical density n ≈ 0.82/a 2 (with a the lattice spacing) over the temperature range 0.3 T /t 8.Over this range it was found that These lead to near-linear resistivity ρ = 1/(χD) over this temperature range.These results are furthermore corroborated by quantum Monte Carlo numerics [32].With the cupratelike value t = 4000 K, for example, the lower end of the temperature range considered is T ≈ 1200 K.As we noted in the introduction, such high electronic temperatures are created in the pump-probe experiments that measure electronic energy relaxation to the lattice.
To obtain the relaxation timescale from (27) we furthermore need the specific heat c.It is important to compute the specific heat at constant density, as this differs from the specific heat in the grand canonical ensemble at high temperatures where thermoelectric effects are significant (see e.g.[33], supplementary material).The specific heat was not determined in the experiment but is known from Monte Carlo, e.g.[34,35].The data in those papers is decently fit over the relevant temperature range by Using ( 28) and ( 29) in (27) the decay rate can be written as The function F (T /t) is shown in Fig. 8.As above, the Debye scale is defined as T D ≡ v s π/a This coupling is numerically very close to the previous coupling (19) if we were to use k F ≈ 2.3/a, that follows from the density via n = k 2 F /(2π), and extract the velocity v F by setting D = 1 2 v 2 F τ e .Using D given above and τ e as given in [14] leads to an approximately temperature-independent velocity v F ≈ 3ta.With this velocity the length e = v F τ e decreases from around 2a to a over the temperature range, consistent with the bad metallic nature of this transport.Furthermore, this velocity leads to an estimate of the Fermi energy of E F ≈ 1 2 v F k F ≈ 3t which is consistent with the crossover scale of the susceptibility χ in (28) from degenerate to high temperature Curie behavior.Fig. 8 also shows, for comparison, the corresponding relaxation rate from a two dimensional Lindhard continuum.This rate has been computed as described in Appendix C, with k integrated up π/a.To make the comparison meaningful we proceed to use the values of k F and v F described in the previous paragraph, and we write the answer in terms of the coupling λ .At high temperatures T T D the formula in the Appendix gives c τ ≈ λ T 2 D /t × 0.1/a 2 .To obtain the decay rate 1/τ in a way that allows us to focus on the effects of spectral incoherence, we have used the Hubbard model specific heat (29) in Fig. 8.The increase in 1/τ at high temperatures, visible in both plots in Fig. 8, is due to the fact that the constant density specific heat becomes small at temperatures T t.
Fig. 8 suggests that the absence of the Lindhard continuum in bad metals leads to slower energy dissipation into phonons, for a fixed specific heat.At T ∼ t the difference between the two curves is a factor of about 5. The numerical similarity between the two dissipation mechanisms is likely because the short mean free path e ∼ a ∼ 1/k F means that there is no small parameter in bad metals.In general interactions redistribute spectral weight, but because our formula ( 14) involves an integral over k it is not a given whether this redistribution will cause the energy dissipation rate to increase or decrease.As the temperature is lowered, eventually the mean free path will grow long enough, e 1/(2k F ), that the Lindhard continuum will re-emerge.In Appendix D we verify that once this occurs the Lindhard contribution will rapidly swamp the small diffusive regime, as is suggested by Fig. 4 and Fig. 8, leading to faster energy relaxation. 5

Phonon emission from slow metals
We explained in §3.2.2 above that when v F < v s the acoustic phonon dispersion is outside of the Lindhard continuum.Energy dissipation must therefore go via collective charge dynamics at low momenta.The limit v F v s is especially tractable, and we will focus on this case.The phonon remains at very small momenta relative to electronic momentum scales, so that we may take the k → 0 limit of Im G R nn (ω, k).As we recalled in (26) above the low momentum spectral weight can be expressed in terms of the optical conductivity σ(ω).The Kubo formula (14) becomes Given an experimentally measured σ(ω) this expression predicts the energy relaxation rate.
To estimate the magnitude and temperature dependence of the relaxation rate, we can consider a three dimensional slow metal with a Drude optical conductivity as in (26).We furthermore use the Drude formula σ dc = nτ e /m with n = We also restrict to sub-Planckian metals with τ e T 1, the integral in (32) may be done explicitly.We obtain Here λ is defined in (19) and the unfamiliar relation T F T BG and hence T F /T BG is small in (33).In particular, at low temperatures T T BG the rate ( 33) is suppressed by a factor of (T F /T BG ) 3 relative to the low temperature limit of a conventional metal discussed below (20).Thus we see that, at low temperatures, slow metals have significantly longer energy relaxation times τ than conventional metals with comparable phonon parameters.

Discussion
The most basic point we wish to make is that the Kubo formula ( 14) is a powerful and general formula for the rate of energy dissipation from an electronic system into a lattice.
While we have evaluated the formula in a few situations, more generally it can be a precision tool if the phonon dispersions and dynamical charge susceptibility Im G R nn (ω, k) are known.A second important notion that we have developed is that when the electronic mean free path becomes small, one expects Im G R nn (ω, k) to acquire a universal diffusive form over a significant portion of the Brillouin zone.There has been significant recent progress in constructing a systematic theory of diffusion, including nonlinearities and fluctuations [25,26], and this machinery can now be applied to the problem of bad metals in general and their coupling to phonons in particular.It also highlights that the diffusivity is a fundamental quantity in bad metals [37].
We have furthermore introduced the notion of a 'slow' metal, with v F < v s .The most obvious candidates for slow metals are those with a very small Fermi surface.In practice, however, it is not straightforward to find examples of this kind: semi-metals typically develop linear electronic dispersion at low density so that the Fermi velocity remains large, while dilute semiconductors typically become insulating.A further basic question is whether quantum critical metals can become slow.In many theories of quantum criticality, the Fermi surface survives at the critical point while the electronic mass diverges and v F → 0.
If this indeed occurs with v s remaining constant, then quantum critical metals will be in a very different kinematic regime of electron-phonon interactions than conventional metals with v s v F .It may be interesting to reconsider theories of quantum criticality coupled to acoustic phonons, including the intermediate regime where v s ≈ v F .
A key motivation behind our work was to clarify the role of electron-phonon interactions in strange metals.In particular, it is crucial to understand, quantitatively, the contribution of electron-phonon scattering to unconventional transport [38,39].The transfer of energy to the lattice gives an unambiguous measure of the presence and strength of electron-phonon interactions, but making use of this fact requires a formula for energy relaxation that does not assume a conventional Lindhard continuum.In this work we have obtained such a formula.The rough estimates we have made using this formula suggest energy relaxation timescales broadly compatible with tr-ARPES data in cuprates such as [5,40] with an order one electron-phonon coupling.However, more precise knowledge of Im G R nn (ω, k) over relevant temperature and energy scales is highly desirable.
There is a technical point related to the previous paragraph.We have treated the electron-phonon coupling perturbatively, assuming that the dominant charge dynamics was due to purely electronic interactions.This is a widely considered scenario, as many models of strange metals omit phonons entirely.However, especially at temperatures above T BG , it is also plausible that quasi-elastic phonon scattering is an important part of the charge dynamics.For example, in a conventional metal this scattering determines the resistivity.Furthermore, the backreaction of phonons on the electrons is necessary to capture spectroscopic phenomena such as kinks in the electronic dispersion and phonon peaks in the M-EELS spectra [41,42].To incorporate such scattering requires a slight extension of the formalism we have developed, because ImG R Ė Ė in (8) must now be evaluated in the theory that includes nonzero electron-phonon interactions.However, because the scattering is quasi-elastic, the energy relaxation rate remains slow compared to the timescale of the interaction and therefore it is likely that our memory-function approach continues to work. 6hile we have focused on timescales that could be measured in pump-probe experiments, let us end by returning to our discussion in §2.1 of non-Ohmic conduction due to Joule heating.To estimate the critical current (5) in a conventional three dimensional material we can use the Drude formula σ (0) = ne 2 τ tr /m, the density n = This is a large current.An even larger similar estimate is obtained from the incoherent cuprate spectrum discussed in (3.3).With a normal state resistivity of ρ ≈ (T /300K) µΩcm for optimally doped BSCCO (e.g.[43]), we obtain j non-Ohm ∼ 100 A/µm 2 .
The critical current is generally reduced at lower temperatures.For example, in a conventional three dimensional metal at T = 3 K, say, with other parameters as in the previous paragraph we obtain j non-Ohm ∼ 5 × 10 6 A/cm 2 .From the discussion around (33), the critical current is predicted to become significantly lower that this in slow metals.This may be an interesting probe of quantum critical systems.
The corresponding phonon Hamiltonian is The electron-phonon interaction is where n(x) = σ c † σ (x)c σ (x) is the electron density, and V ep is the deformation potential ) d e iq(x−y) iq i V (q)u i (y) , where R a i labels the equilibrium position of ion c, and a is the lattice step.Use ˆdd ye −iqy u i (y) = where we used s i (−q) = − s i (q).We have Then, assuming that in the long-wavelength the potential becomes a constant V (q)/a d → C, we find

C.1 Lindhard integral
The integrals that arise from using ( 17) in ( 14) can be performed in two steps.Firstly, the q integrals can be split into the radial direction, where q = k F + q ⊥ , and the remaining angular directions along the Fermi surface.The angular integrals simply give a factor of the area of the Fermi surface.At T E F the dispersion can be linearized in the Fermi-Dirac functions so that q ≈ v F q ⊥ .The q ⊥ integral is then ´∞ −∞ dq ⊥ [n F ( q )−n F ( q +ω)]/ω = 1/v F .
Secondly, the k integration measure can be written as (59) Here, as in the main text, T D ≡ v s π/a and T BG ≡ 2v s k F .We are considering the case where where T D < T BG .If we furthermore impose v s v F and T T D (as this will be the limit considered in Fig. 8) then the expression is simplified to

C.2 Diffusive integral
Substituting the diffusive form (25) into the Kubo formula (14) gives As previously we set x = v s k/T .The integral runs over the Brillouin zone, so that the upper cutoff is k ∼ π/a.
A few simplifications can be made in (61) with physical assumptions about the values of the parameters.At the upper cutoff x ∼ πv s /(aT ) ∼ T D /T , where the Debye temperature is defined to be T D ≡ πv s /a.If we restrict to bad metals with T T D , as is typically the case, the csch in (61) is well approximated by expanding at small x: csch 2 x 2 → 4/x 2 .Furthermore, the second term in the denominator in (61) dominates over the first term, as we can see proceeding in two steps.Firstly, if we estimate D ≈ 1 d v 2 F τ e then the first term in the brackets dominates when v F v s .In essence this is just the assumption that the characteristic velocity controlling diffusion is electronic.We can therefore neglect the τ e T term.Secondly, at the upper endpoint of the integral the remaining term In the final step we again used the fact that v F v s , as well as the fact that e /a is order one but not small in the bad metals we will consider.We also estimated D ≈ 1 d v F e .It can be checked that the dominance of the second term in the denominator at the endpoint of the integral is a sufficient condition to neglect the first term throughout the integral.We thereby obtain The diffusivity D is still general in (62), the form mentioned in the paragraph above was used only to estimate the relative importance of the two terms in the denominator of (61).
H H e z P M z A s S K Q y 6 7 r d T W F p e W V 0 r r p c 2 N r e 2 d 8 q 7 e 0 0 T p 5 p D g 8 c y 1 n H H e z P M z A s S K Q y 6 7 r d T W F p e W V 0 r r p c 2 N r e 2 d 8 q 7 e 0 0 T p 5 p D g 8 c y 1 n 1 S y O L D t k R + y E e e y C l d g N K 7 M K E + y R P b N X 9 u Y 8 O S / O u / M x b c 0 4 s 5 l 9 9 g f O 5 w 8 q l p W H < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 x c K T X g z E 8 Z 1 a P 1 S y O L D t k R + y E e e y C l d g N K 7 M K E + y R P b N X 9 u Y 8 O S / O u / M x b c 0 4 s 5 l 9 9 g f O 5 w 8 q l p W H < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 x c K T X g z E 8 Z 1 a P 1 S y O L D t k R + y E e e y C l d g N K 7 M K E + y R P b N X 9 u Y 8 O S / O u / M x b c 0 4 s 5 l 9 9 g f O 5 w 8 q l p W H < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 x c K T X g z E 8 Z 1 a P

Figure 1 :
Figure 1: Power is supplied to the electrons by driving a current.The electrons rapidly thermalize to a temperature δT above the temperature of the thermalized lattice and environment.Heat is consequently transferred to the lattice, at a rate 1/τ .

Figure 2 :
Figure 2:The shaded blue region is the phase space of Lindhard particle-hole pairs.The

Figure 3 :
Figure3: Rate of energy loss to phonons from the Lindhard continuum as a function of temperature in three dimensions from(20).The rate becomes small at low temperatures because there are few phonons around.The rate becomes small at high temperatures because high temperature electron-phonon scattering is elastic from the perspective of the electrons and does not efficiently transfer energy out of the electronic subsystem.

Figure 4 :
Figure 4: Collective (shaded red) and Lindhard (shaded blue) regimes in a conventional metal (left) and a bad metal (right).The collective regime extends over k 2π/ e .The good metal has 2π/ e k F while the bad metal has 2π/ e ∼ 2k F .The dispersion of the acoustic phonon is the orange line.The dashed red line shows ω = v F k, with e = v F τ e .

Figure 5 :
Figure 5: The orange line shows the dispersion of an acoustic phonon with v s = 3v F , outside of the Lindhard continuum (shaded blue) but inside the collective regime (shaded red).
as in the left figure, where the four red dots represent the possible final momenta.The blue circles are constant energy contours for the electron.If v s > v F the contours are spaced further apart relative to the phonon momentum and no final momenta satisfy momentum and energy conservation, as in the right figure.

Figure 7 :
Figure 7: Contour plot of 1 χ Im G R nn (ω, k) from (25) as a function of dimensionless frequency and momentum.

Figure 8 :
Figure 8: Solid curve: Energy relaxation rate (30) of the bad metal observed in the ultracold atom experiments of [14].Dashed line: High temperature energy relaxation rate of a Lindhard continuum with the same parameters in the degenerate range T < E F .

k 3 F
3π 2 , the mass m = k F v F and, for example, consider Planckian transport 1 τtr = k B T .Taking a typical value for a conventional metal of k F = 1.5×10 8 cm −1 and at temperature T = 300 K the estimate (21) for the energy relaxation time gives j non-Ohm ∼ 24 A µm 2 .(34) qs − a † −qs ) ,

1 k d− 1 F v 2 F
d d k = k d−1 dk(sin θ) d−2 dθS d−2 , where S d−2 is the area of a d − 2 dimensional sphere.The angle θ is determined by the energy conservation delta function, as described above (18), so that only the k integral now remains, leading toc τ Lind.= C 2 ρ M S d−1 S d−2 4(2π) 2d−T v s d+1 ˆdx[sin θ(x)] (d−3) x d+2 csch 2 x 2 .(58)Here x = v s k/T is subject to the constraint(18) as well as any cutoff at the Brillouin zone boundary k ∼ π/a.In particular, in d = 2 we obtain, integrating up to k = π/a,