A criterion for strange metallicity in the Lorenz ratio

The Wiedemann-Franz (WF) law, stating that the Lorenz ratio L = κ/(Tσ) between the thermal and electrical conductivities in a metal approaches a universal constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}_{0}={\pi }^{2}{k}_{B}^{2}/(3{e}^{2})$$\end{document}L0=π2kB2/(3e2) at low temperatures, is often interpreted as a signature of fermionic Landau quasi-particles. In contrast, we show that various models of weakly disordered non-Fermi liquids also obey the WF law at T → 0. Instead, we propose using the leading low-temperature correction to the WF law, L(T) − L0 (proportional to the inelastic scattering rate), to distinguish different types of strange metals. As an example, we demonstrate that in a solvable model of a marginal Fermi-liquid, L(T) − L0 ∝ − T. Using the quantum Boltzmann equation (QBE) approach, we find analogous behavior in a class of marginal- and non-Fermi liquids with a weakly momentum-dependent inelastic scattering. In contrast, in a Fermi-liquid, L(T) − L0 is proportional to − T2. This holds even when the resistivity grows linearly with T, due to T − linear quasi-elastic scattering (as in the case of electron-phonon scattering at temperatures above the Debye frequency). Finally, by exploiting the QBE approach, we demonstrate that the transverse Lorenz ratio, Lxy = κxy/(Tσxy), exhibits the same behavior.


INTRODUCTION
The properties of the anomalous normal state of high-T c superconductors and other quantum materials, commonly dubbed 'strange metals,' are one of the most elusive mysteries in condensed matter physics 1,2 .In particular, despite myriad works, it is still unclear to what extent the underlying physics of such systems departs from Landau's Fermi-liquid (FL) paradigm and necessitates a non-FL (NFL) description.
One of the hallmark characteristics of strange metals is the T -linear resistivity at extremely low temperatures.This behavior has been empirically linked with the notion of Planckian dissipation [2][3][4][5][6] , showing a degree of universality throughout different experimental setups and hinting towards a strongly-correlated NFL nature for these systems.Albeit at odds with standard FL theory, T -linear resistivity can appear in FLs in the presence of certain scattering mechanisms, at least in some intermediate-to low-T window [7][8][9][10] .It is thus crucial to develop ways to identify the mechanism of T -linear resistivity in strange metals.
Here we present a simple criterion for weakly disordered metals that sharply distinguishes different sources of T -linear resistivity.Our criterion is based on the behavior of the low-T leading correction to the Lorenz ratio, L(T ) = κ T σ , with κ and σ being the thermal and electrical conductivities, respectively.
The Weidemann-Franz (WF) law 11 states that as T → 0. Here, L 0 = π 2 k 2 B /(3e 2 ) is the so-called Lorenz number (we set e = k B = 1 henceforth).Roughly speaking, the deviation of L(T ) from 1 serves as a measure for the relative contribution of inelastic scattering to charge and thermal transport (L(T ) ≈ 1 implies that elastic or quasi-elastic scattering is dominant) 12 .Dominantly inelastic scattering leads to deviations from the WF law in many circumstances [13][14][15][16] .
The validity of the WF law is often used as a test for the existence of FL-like quasi-particle excitations at the lowest temperatures [17][18][19][20][21] .However, the fact that WF is obeyed does not necessarily imply that transport is carried by FL quasi-particles [22][23][24] .Indeed, as we shall show, one can construct solvable models of NFLs where the WF law is obeyed at T → 0. The known mechanisms for T -linear resistivity (not necessarily extending down to T → 0) in FLs involve elastic or quasi-elastic scattering.These include electron-phonon scattering 9 or static charged impurities in 2D 7,8 .In contrast, T -linear resistivity associated with NFLs is typically associated with inelastic scattering [25][26][27][28][29] .In both cases, however, assuming that (T -independent) impurity scattering dominates at sufficiently low T , we expect the WF law to be obeyed at T → 0. Hence, in order to learn about the FL or NFL origin of the T -linear resistivity, one must consider the leading low-temperature deviation from the WF law (see Fig. 1).
Our criterion is applicable to systems that obey the WF law at T → 0, as in the cuprates at sufficiently low temperature 17,20,21 .In this context, it is worth noting that certain weakly disordered 2D systems with Coulomb interactions are expected to violate the WF law at T → 0 [30][31][32] .However, in metals, the deviation from the WF law is significant at an exponentially small temperature in k F ℓ, where k F is the Fermi momentum and ℓ is the elastic mean free path.Our discussion applies above this energy scale.
Figure 1.Schematic plot of the low-T behavior of the normalized Lorenz ratio for systems that obey the Wiedemann-Franz law at T = 0. Here, T is assumed to be smaller than Γ, the elastic scattering rate.The T dependence of the leading deviation from L = L0 serves as a criterion for strange metallicity: Fermi liquids (FL) exhibit L/L0 − 1 ∝ −T 2 ; Fermi liquids with hot spots (FL+HS) are characterized by L/L0 − 1 ∝ −T β , 1 < β < 2; and certain marginal Fermiliquids (MFL) have L/L0 − 1 ∝ −T .

A criterion for strangeness
Consider weakly disordered metals (in 2 or 3 spatial dimensions), such that the dc resistivity has the following form as T → 0: ρ (T ) = ρ 0 + AT α , where ρ 0 is the residual resistivity, and A, α > 0. We assume that impurity scattering dominates at sufficiently low T , and the WF law is satisfied at T → 0. In this case, the low-T electronic thermal resistivity takes the form ρ th (T ) ≡ T /κ el = ρ th,0 + BT β with B, β > 0. The normalized Lorenz ratio (1) then takes the following form: (2) We claim that the exponent β is universal and provides information on the nature of the system.In ordinary FLs, β = 2 (logarithmic corrections may arise due to electron-electron interactions in 2D 33,34 ).Systems where a portion of Fermi surface (FS) is 'hot', while the rest is FL-like, have 1 < β < 2. Most interestingly, if β ≤ 1, the system is not described by any existing theory of a FL.
In particular, the case α = β = 1 arises in certain models that realize marginal Fermi-liquids (MFLs) 35 .We therefore argue that α = β = 1 could serve as a criterion for 'strangeness', in the sense that it signals a full departure from FL theory.See Fig. 1 for a schematic illustration of the different cases.

Fermi liquids
We consider a weakly disordered FL with electronelectron (el-el) or electron-phonon (el-ph) interactions.We assume that the WF law is obeyed at T → 0 due to the dominance of elastic scattering [12][13][14][36][37][38][39] . Here, nd in the following section, the disorder corresponds to static impurities, which provide a source of elastic scattering with rate Γ.
At T > 0, el-el and el-ph interactions provide inelastic scattering mechanisms that lead to deviations from the WF law.1][42] ), which translates to where the negative slope is related to the additional contribution of forward scattering that relaxes the thermal current, but not the electrical current [12][13][14] .The elph contribution to the electrical (thermal) resistivity is O T d+2 (O T d ), respectively (where d > 1), as long as T is much smaller than T BG , the Bloch-Gruneisen temperature 12 .That is, the el-ph contribution is subleading in 3D, while in 2D it may modify the non-universal slope, such that the form (3) holds at sufficiently low T in a FL.
In fact, Eq. ( 3) applies even in cases where the resistivity of a FL is T -linear.For example, Coulomb screening of charged impurities, treated within the random phase approximation, leads to a T -linear resistivity in a 2D FL, due to thermal suppression of the FL polarizability 8,43 .(In 3D, this contribution to the resistivity is O T 2 7,8,32 .)However, in this case, the T −linear scattering is still essentially elastic, and the deviations from the WF law still obey Eq. ( 3).
Unlike the case of charged impurities, T -linear resistivity from el-ph interactions emerges only at temperatures T ≳ T BG 12 .Hence, this mechanism is always irrelevant at the limit T → 0. On a more practical note, if T BG sets a particularly small energy scale, the T -linear resistivity due to el-ph scattering might appear to extend down to the lowest experimentally accessible temperatures (as long as T ≳ T BG ).However, in this "equipartition" regime, phonons are essentially classical and the el-ph scattering is quasi-elastic.Hence, the WF law is essentially obeyed in this regime 12 .

Fermi surfaces with hot spots
We now consider systems where a portion of the Fermi surface becomes 'hot', i.e., it experiences enhanced scattering with an anomalous T -scaling.In some situations, such 'hot spots' can lead to an anomalous T dependence of the transport coefficients.This situation arises either when the system is on the verge of a finite wavevector instability [44][45][46][47][48][49] , or when the system is turned to a Van Hove singularity where the topology of the Fermi surface changes 10,15 .
Consider the low-T behavior of L (T ) in a 2D system where a Van Hove singularity (VHS) crosses the FS in the vicinity of a Lifshitz transition 10,15 .In this case, we refer to the Fermi surface regions near the VHS as 'hot'.The transport scattering rates are dominated by processes where a 'cold' electron (away from the VHS) is scattered by a 'hot' one, or two cold electrons are scattered and one of them ends up near the VHS.In clean systems, this leads to ρ ∼ T 2 log(1/T ) 10,45 and ρ th ∼ T 3/215 .This behavior persists in the presence of impurities, namely, ρ = ρ 0 + AT 2 log(1/T ) and ρ th = ρ th,0 + BT 3/215,45 , such that the deviation from WF law satisfy We proceed by considering a weakly disordered FL near an antiferromagnetic (AFM) quantum critical point in 3D, as studied in Refs. 46,47.In this case, the FS contains 'hot lines' connected by the non-zero AFM wavevector, where the scattering off spin fluctuation is most effective.The hot lines then acquire anomalous, NFL-like, scattering rates which may manifest in transport coefficients.In the absence of impurities, these hot lines are short-circuited by the remaining 'cold' parts of the FS such that transport coefficients follow the conventional FL behavior at sufficiently low T 44 .However, introducing impurities enables the hot lines to participate in transport, since, loosely speaking, the scattering rate is averaged over the entire FS.Ref. 46 showed that this leads to an anomalous T -scaling of the resistivity, where ρ = ρ 0 + AT 3/2 at the lowest temperatures.By extending the analysis of 46 to the thermal conductivity, we find that the thermal resistivity follows the same anomalous behavior: ρ th = ρ th,0 + BT 3/2 , see Supplementary Material.Combining the two resistivities, the deviation from WF law follows Eq. ( 4).
Interestingly, a straightforward generalization of the analysis above to 2D yields ρ = ρ 0 + AT 47 .The same reasoning is expected to hold for the thermal resistivity, which would imply that L − 1 ∝ −T in 2D.However, this analysis is based on the Hertz-Millis treatment of the AFM QCP, which breaks down at sufficiently low temperatures in the 2D case 50,51 .

Marginal Fermi liquids
In this section, we construct a solvable model of a 2D weakly disordered MFL that shows T -linear resistivity down to the lowest temperatures and obeys the WF law at T → 0, with a leading correction of the form In addition, we comment on the expected behavior of other tractable models of MFLs in 2 and 3 dimensions, suggesting that Eq. ( 5) could be a robust signature of a class of weakly disordered MFLs.We further corroborate this expectation using the Quantum Boltzmann Equation (QBE) approach in the following section.
Consider a weakly disordered variant of the model studied in Ref. 28 , based on a 2-band lattice generalization of the Sachdev-Ye-Kitaev (SYK) model [52][53][54] .The model is defined on a D-dimensional lattice, and contains two species of fermions, {c} and {f }, each containing N orbitals per unit cell, governed by the Hamiltonian H = H c + H f + H cf , where The hopping matrix t r,r ′ is diagonal in orbital space and depends only on the distance |r − r ′ |.The last term in H c describes on-site disorder for the c-fermions, where W ijr are site-dependent Gaussian random independent potential, satisfying The couplings in H cf and H f are site-independent Gaussian random independent variables, satisfying V ijkl = 0, V 2 ijkl = U 2 cf and similarly for U ijkl (with variance U 2 f ).The function Υ determines the spatial dependence of the cf -interaction.Note that for W = 0, the model is translationally invariant for every realization of the interactions.We first consider the case of onsite interaction as in 28 : Υ r,r ′ = δ r,r ′ .Spatially extended Υ will be considered later on.
The model is solvable in the N → ∞ limit, where its properties are dictated by replica-diagonal saddle-point of the real-and imaginary-time effective action 28 .The low-energy saddle-point equations describe SYK-like, incoherent f -fermions.These f -fermions constitute a local quantum critical bath for the c-fermions, giving rise to a weakly disordered MFL form for the Green's function of the c-fermions.Importantly, the onsite disorder W for the c-fermions does not alter the low-energy behavior of the f -fermions, rather it only enters as an additional T -independent, elastic scattering term to the c-fermions.For example, at T = 0, the Matsubara frequency Green's function is of the form where Γ = 2πν 0 W 2 is the disorder energy scale.We proceed to consider transport.We compute the electrical and thermal conductivities using the Kubo formula.By virtue of the locality of the f -fermions, both conductivities are given in terms of the bare bubble expressions, similarly to Refs. 28,29.We obtain the thermal  7) with local cf -interaction.Γ is the elastic scattering rate.The inset shows the limit T ≪ Γ which obeys Eq. ( 5). conductivity, and the electrical conductivity, The imaginary part of the retarded self-energy is given by 2π 2 U f and where ψ(z) is the digamma function 29 .Using Eq. ( 10) and Eq. ( 11), we find that the WF law is obeyed at T → 0, despite the fact that the MFL description of the c-fermions persists to the lowest temperatures, and that the leading deviation from the WF law obeys Eq. ( 5).The Lorenz ratio L(T )/L 0 as a function of T is shown in Fig. 2. As can be seen in the figure, L/L 0 decreases linearly with T at small T , and saturates to the value correspodning to the clean case, L/L 0 ≈ 0.71 29,55 , at T ≳ Γ.
In order to examine the robustness of these results to details of the model, we consider the addition of spatially extended cf -interactions: Υ r,r ′ = δ r,r ′ + η δ=±x,±ŷ δ r,r ′ +δ with η being a small control parameter.This modification does not change the MFL form of the self-energy of the c-fermions.In addition, the form of the thermal current operator is unchanged, see Supplementary Material.Hence, to leading order in η, the conductivities are given by α η = α 0 + δα for α = σ, κ, where we have denoted α η=0 ≡ α 0 , and the correction δα is O (η) and corresponds to the current bubble with an insertion of a single cf -interaction rung, see Supplementary Material.These corrections alter the Lorenz ratio, such that for T ≫ Γ, which demonstrates that the saturation value is not universal.Importantly, the spatially-extended cfinteraction do not alter the T → 0 behavior of the Lorenz ratio, which obeys Eq. ( 5).We will demonstrate this and further highlight the conditions for which Eq. ( 5) is valid within the framework of the QBE in the next section.
It is worth commenting that the simplicity of the analysis of ( 7) comes with a price in the form of a residual T → 0 extensive entropy due to the SYK-nature of the ffermions 28,29,53,54 .The residual entropy is relieved upon allowing quadratic terms in the f -fermions, but these also lead to FL behavior at low temperatures 28 .Nevertheless, we expect Eq. ( 5) to be a robust property of weakly disordered MFLs in 2 and 3 dimensions that show T -linear electrical resistivity, as we discuss in the next section.
Let us briefly note that the results presented here and in the next section can be generalized to f -fermions governed by an SYK q (q > 4) Hamiltonian, while the cfinteraction is unchanged.For q > 4, the c-fermions realize an incoherent, NFL description with ρ = ρ 0 + AT 4/q and ρ th = ρ th,0 + BT 4/q , such that L − 1 ∝ −T 4/q 28 .

Quantum Boltzmann equation approach
Even in the absence of well-defined quasi-particles, we may still derive a QBE for a generalized Fermi distribution function in the model of the previous section.Here we briefly outline the idea behind the QBE approach for MFLs and the conditions for which it is applicable.In addition, we highlight its implications on the validity of the WF law and the criterion for strangeness in a certain class of MFLs, using a generalization of the model (7) as a simple representative.We elaborate on several issues and supply technical details in the Supplementary material.
To derive a QBE in the absence of well-defined quasiparticles, we utilize the MFL form of the self-energy and the fact that the spectral function of the c-fermions is sharply peaked at the FS as a function of ε k (this is in contrast to the QBE approach for FLs which relies on the sharp quasi-particle peak as a function of ω).Within this approximation, known as the Prange-Kadanoff (PK) reduction scheme 56,57 , the momenta of the c-fermions are restricted to the FS.Roughly speaking, the PK reduction is valid when the width of the electronic spectral function A(ω ∼ T, k) as a function of k is smaller than the typical momentum transfer in both elastic and inelastic scattering events, see Supplementary Material and 58 .
Considering the MFL model (7), the QBE approach illustrates that (i) The WF law may hold at T → 0 due to the dominance of elastic scattering, regardless of the existence of well-defined quasi-particles; (ii) The leading deviation from the WF law obeys Eq. ( 5) in weakly disordered MFLs that admit the PK reduction scheme; where (ii) can be understood as a consequence of Matthiessen's rule.We further find that the deviation in Eq. ( 5) hold for a class of generalized models with spatially extended cf -interactions, see e.g., the previous section, which confirms that (i) and (ii) have a much broader regime of validity in weakly disordered MFLs (and NFLs).Specifically, assuming that the momentumdependence of the inelastic scattering rate is sufficiently weak (as defined above), such that PK reduction can be applied, the QBE approach suggests that WF law should hold at T → 0.Moreover, since in these circumstances the transport relaxation rate are proportional to the single particle scattering rate, the leading low-T deviation from the WF law is expected to satisfy Eq. ( 5).

Transverse Lorenz ratio
We employ the QBE approach to generalize our discussion to the transverse Lorenz ratio: where σ xy and κ xy denote the transverse electrical and thermal conductivities, respectively.Specifically, by solving the linearized QBE of the weakly disorder MFLs (7), we find that the leading deviation from the (transverse) WF law for a class of MFLs follows the same scaling as the longitudinal: as in Eq. ( 5); see Supplementary Material.Moreover, while the derivation of the transverse conductivities is slightly more involved due to the presence of a weak magnetic field, the key ingredient remains the validity of the PK reduction scheme.This has the remarkable implication that, as long as the PK reduction scheme is valid, our conclusions for the longitudinal Lorenz ratio (i.e.(i) and (ii) from the previous section) equally apply to the transverse Lorenz ratio of weakly disordered MFLs (or NFLs).In addition, while the transverse conductivities are proportional to the applied magnetic field, this proportionality factor cancels in L xy such that the leading deviation is independent of the magnetic field.Note further that the extension of our criterion to the transverse Lorenz ratio holds also for weakly disordered FLs, where the leading deviation satisfies L xy − L 0 ∝ −T 212 .The same conclusion is expected to hold for Fermi surface with hot spots since, within the conventional Boltzmann transport theory (for sufficiently weak magnetic field that can be treated perturbatively), the dominant inelastic scattering rate that governs longitudinal transport also governs transverse transport.

DISCUSSION
Naively, one may have expected the WF law to hold at T → 0 only in weakly disordered Fermi liquids with welldefined quasiparticles.This is because, within the conventional Landau-Boltzmann description of transport, the universal value L 0 originates from integrating over Fermi functions, implying that the existence of welldefined quasiparticles is necessary.In contrast, as shown in this work, a broad class of weakly disordered non-Fermi liquid metals with no well-defined quasiparticles (in the sense that the electron scattering rate is either comparable to, or larger than, the energy) also satisfy the WF law at T → 0. Intuitively, the fact that this class of systems obey the Wiedemann-Franz law may be understood from the fact that, while there is no welldefined Fermi surface with a sharp jump in the fermion momentum occupation function, the generalized energy distribution function f (ω) = −i ´dε 2π G < (ε, ω), is a Fermi function (see Supplementary Material).A sufficient condition for the WF law to hold is that the QBE approach is applicable; this requires, in particular, that (i) The width of electronic spectral functions at zero energy is smaller than the Fermi momentum, and that (ii) The dependence of the electronic scattering rate on momentum is non-singular.Note that, in particular, condition (i) implies that the resistivity is small compared to the Mott-Ioffe-Regel limit.
Thus, the fact the WF is obeyed at T = 0 is not sufficient to deduce that these systems are conventional Fermi liquids in disguise.Instead, we propose to examine the deviation of the Lorenz ratio L(T ) from L 0 as T → 0. Since this quantity depends on the degree of inelastic scattering, it can distinguish different sources of strange metallicity, such as Fermi liquids with a source of T −linear nearly-elastic scattering (such scattering from an Einstein bosonic mode whose frequency is lower than T ), from "true" non-Fermi liquids where the scattering is inelastic (see Fig. 1).
In practice, our criterion is applicable under experimental conditions where the electronic degrees of freedom dominate heat transport at low T .For the longitudinal case, while these conditions can be met in some scenarios (for example 42,59 ), it could also be the case that other degrees of freedom, e.g., phonons, will dominate the thermal conductivity which would make our criterion inaccessible.To separate the electronic contribution, the transverse Lorenz ratio L xy is often used (since κ xy is often, although not always 60 , dominated by the electronic contribution).Here we showed that our criterion applies to the longitudinal and transverse cases at once, and therefore expect it to be widely applicable.
An intriguing issue concerns the application of our criterion to theories of quantum-critical metals, especially in cases where the electrical resistivity is T −linear 27,61,62 .
In this regard, we point out Ref. 59 , that reported low-T transport measurements in a weakly disordered 3D system at a ferrmomagnetic critical point.It was found that at low T , ρ = ρ 0 +AT 5/3 while ρ th = ρ th,0 +BT , such that L − 1 ∝ −T , consistent with MFL behavior by our criterion 51,63 .This observation is further corroborated by evidence for a T log (1/T ) behavior in the specific heat 64 , as expected for a MFL 35 .

METHODS
All analytical calculations are explicitly presented in the Supplementary note.
Thermal conductivity in a FL near AFM criticality in 3D We consider the model studied in Ref. 46 .In the framework of the variational Boltzmann approach 12 , the low-T inverse electronic thermal conductivity is given by Here 1 κimp is the contribution from static impurities that, when combined with the residual electrical resistivity, satisfies the WF law at T → 0. In addition, v k = ∂ k ε k where ε k is the dispersion relation, µ is the chemical potential.We are interested in the leading deviation from the WF law, associated with the second term in Eq. ( 15).We mainly follow the notation of 12 , where the full out-of-equilibrium distribution function is given by The transition rate associated with critical AFM spin fluctuations is given by To proceed we use the ansatz Φ k = η (ε k − µ) u • k, with a small η and where u is a unit-vector in the direction heat current, which is appropriate at the low-T limit, where impurity scattering dominates κ.From this point the analysis is analogous to the one in Refs. 44,46, namely, the low-T scaling of 1/κ is determined by the T -dependent phase space of the hot-lines that scales as √ T .In total, we obtain that the thermal conductivity is given by where A and B are related to the impurity and spin contributions.Finally the thermal resistivity obeys ρ th = T /κ = ρ th,0 + BT 3/2 .

Spatially extended cf -interaction
We discuss the modifications to the self-energy and resistivities in the presence of a spatially extended cf -interaction.
Supplementary Figure 3.Leading diagrams in 1/N for the electrical and thermal current-current correlation function.Solid line correspond to c-fermions, Dashed lines correspond to f -fermions and blue dotted lines denote averaging over realizations of V ijkl .

Self energy
As in the main text, we follow the notation of 28 .We first show that the MFL form of the c-fermions self-energy is unchanged by extending the range of the cf -interactions, up to a non-universal numerical coefficient.Let us set Υ r,r ′ = δ r,r ′ + η δ=±x,±ŷ δ r,r ′ +δ as in the main text.To show that the self-energy is unchanged, we note that to leading order in η, the cf -contribution to the self-energy of the c-fermions is given by where Υ (q) = 1 + 4η (cos q x + cos q y ) + O η 2 .Considering k on the FS, we approximate ε k+q ≈ v F q cos θ where θ is the angle between k and q.In addition, to obtain the most singular contribution we further approximate Υ (q) ≈ 1 + 8η + O q 2 .Carrying the momentum and frequency integrals in the usual way (see 28 for details), it follows that, in the low-energy limit, where α Υ = 8η, namely, the MFL form of the self-energy is unchanged.

Thermal current operator
Note that the thermal current is given by j th (t kl ċkl as long as the cf -interaction is local in the c-fermions.For brevity, let us show this in a simplified 1D continuum model given by where We use the continuity equation to identify the thermal current operator: ḣr = ∂ r j th,r .Here the local Hamiltonian density is given by and ḣr = i [h r , H]. Ignoring the time derivatives of the f -fermions, which do not participate in transport, we have that To proceed we rewrite the first term as Then, we insert the equation of motion for c r , c † r (i.e.∂ 2 r c r = −i ċr − ´r′ V r,r ′ c r ), which is local as long as the cf -interaction is local in the c-fermions, and arrive at with j th,r = 1 2m ċ † r ∂ r c r + ∂ r c † r ċr .Going back to the lattice, the total thermal current is indeed given by Generalizing this to higher dimensions is straightforward.

Vertex corrections
Consider the effect of the nearest-neighbors (n.n.) interaction on the conductivities.Here we demonstrate the effect on the electrical conductivity.The thermal conductivity follows exactly the same considerations.First recall that in the absence of n.n.cf -interaction, all vertex corrections vanish due to the locality of the f -fermions.The conductivities are then associated with the current-current bubble diagram shown in Fig. 3a.Adding n.n.interactions lead to an additional contribution, which is associated with the bubble diagram with a single rung inserted, see Fig. 3b.Importantly, while the diagram shown in Fig. 3c is the same order in 1/N , it does not contribute to the conductivities because it is non-zero only at second order in η.
For our purposes, it is sufficient to show that the contribution of Fig. 3b does not vanish.It then follows that, to leading order in η, there are vertex corrections to σ and κ that generically alter the high-T (T ≫ Γ) saturation value.The contribution due to the insertion of a single rung is given by We can now see that in the absence of n.n.interaction, i.e., when Υ = 1, this contribution vanishes since the integrand is odd in k and in k ′ .However upon introducing the n.n.interaction, there is an additional odd part in k and in k ′ : The product of the odd part of Υ with the velocities is even in k and k ′ and therefore does not vanish under the integral.Note that only taking the leading diagrams in η is equivalent to a perturbative treatment of the current vertex, which justified since we are perturbing about a non-singular point, i.e., since the conductivities are finite at η = 0.

High-T saturation value
We recall the Kubo formula for the dc electrical conductivity, expressed in terms of the analytic continuation of the Matsubara frequency current-current correlator: For any T > 0, we can thus define dimensionless frequencies Ω l = Ω l /T and Ω = Ω/T such that Therefore, obtaining the T -scaling of ImΠ x J iΩ l → Ω + iη yields the T -scaling of σ.We will now show that this simple power counting argument can be used to obtain the T -scaling of the conductivity correction.
To begin, consider the expression for the current correlator related to the conductivity correction: We take the odd part of Υ, as above, and approximate sin This allows us to change the momentum integration to energy integration in the usual way.Then, by expressing the Green's functions with the spectral representation and integrating over ε, ε ′ , we obtain ×T 2 n,m∈Z where and F = ImΠ f is obtained from a convolution of two real-time Green's functions of the f -fermions.Using the scaling (SYK) form of the f -fermions, one can write where F is a dimensionless function of the dimensionless variable ϵ = ϵ T .Then, by rescaling ϵ 1 , ..., ϵ 5 , ω n , ω m and Ω l by T , we see that ×T 2 n,m∈Z Here the overline denotes the rescaled variables.
To obtain the scaling behavior of δσ in the limit T ≫ Γ, we set Γ = 0, such that the self-energy has the scaling MFL form, Σ R (ϵ i ) ≈ T Σ cf,R (ϵ i ), which means that As before, the overline denotes dimensionless functions and variables.Inserting this back into the current correlator, we obtain that × n,m∈Z We can now observe that δΠ x J iΩ l is independent of T .Using Eq. ( 26), it follows that Hence the correction δσ scales as 1/T , similarly to σ.For the thermal conductivity, the derivation is similar, and gives δκ ∼ const when T ≫ Γ.The saturation value of the Lorenz ratio is thus altered by vertex corrections.

Quantum Boltzmann equation
Here we outline the idea behind the QBE for the generalized Fermi distribution function, following similar steps to Refs. 57,65.

Derivation of QBE
We begin by introducing the non-equilibrium Green's functions G and Σ that satisfy the Dyson's equation: where and similarly for Σ.G 0 denotes the non-interacting Green's function.As in 65 , the multiplication in Eq. ( 41) denotes integration over a shared space-time variable.In addition, where 1 = (r 1 , t 1 ) and similarly for 2. Note that where G R(A) is the familiar retarded (advanced) Green's functions.The derivation proceeds by changing to centerof-mass and relative coordinates: To derive a self-consistent Dyson's equation for G < , we consider the Fourier transform with respect to the relative coordinates: In the following we slightly abuse the notation above by relabelling the center-of-mass coordinates by r and t.In addition, we will occasionally ignore these arguments for brevity.Note that in thermal equilibrium, where f 0 is the equilibrium Fermi distribution at some temperature, and the spectral function Following 57,65 , the Dyson equation for G < is given by where the multiplication here is standard, e.g., Σ > G < = Σ > (k, ω, r, t) G < (k, ω, r, t), and we have introduced the generalized Poisson brackets, To derive the QBE, we use the fact that Σ R is momentum independent in the low-energy limit and note that the spectral function of c-fermions, is sharply peaked as a function of ε k at the FS for sufficiently small ω.In addition, we note that ´dε 2π A c = 1.Assuming that these features persist if the system is sufficiently close to local equilibrium (limiting ourselves to linear response), it follows that by changing the integration variables from k to ε ≡ ε k and k we can define a generalized distribution function, that describes the distribution of c-fermions with energy ω at position r and time t.
By defining the generalized distribution function (56) together with the above assumptions we effectively restrict the momentum of the c-fermions to the FS, an approximation known as the Prange-Kadanoff (PK) reduction scheme.The PK reduction can be consistently applied provided that the spectral function is sharply peaked at the FS as a function of ε.More precisely, assuming that the typical momentum transfer due to different scattering mechanisms can be characterized by a ball of radius q * (T ), the PK reduction is consistent if |ImΣ R (ω ≲ T ) | ≪ v F q * (T ) for all scattering mechanisms.In our case, the consistency of the PK reduction at low T follows from the locality of the f -fermions, namely, q * = k F and indeed in the T -window of interest we have that Γ, T ≪ v F k F ∼ E F .More generally, we can see that in the presence of disorder scattering, the PK reduction is consistent if the momentum dependence of the single-particle scattering rate is sufficiently weak.
To obtain the QBE for f c , we consider the equation of motion for G < .Restricting ourselves to slowly varying force fields, the QBE is obtained from a gradient expansion 65 followed by an integration over ε of Eq. ( 53) 57 , from which we arrive at where we have introduced the differential operator and ∇ k F g ≡ ∇ k g| k∈FS for some function g.The collision integral is given by , where I dis and I cf denotes the contributions due to disorder and the cf -interactions, respectively.Explicitly, the self-energies are given by Σ Here, n 0 is the equilibrium Bose distribution function.We allow spatially correlated disorder.In addition, we allow for a spatially extended cf -interaction which introduces a momentum dependence to the scattering amplitude from the f -fermions, Υ (q) (as in the Vertex correction section above).We discuss the validity of the PK reduction in more general terms below.For our current discussion, Υ = 1 and W 2 k−q = W 2 which is independent of momentum.These general forms of the different parts of the self-energies will be useful later on.The collision integrals are obtained from the above via an integration over ε k , for example, At this point we can explicitly see that the momentum of the c-fermions is restricted to the Fermi surface: I cf = I cf k, ω, r, t (and similarly for I dis ).
It is worthwhile to comment on two subtle points in the above derivation: (i) The elimination of the second term on the LHS of Eq. (53) (i.e.Σ < , ReG R ) is due to the fact that we have assumed a particle-hole symmetric form for the density of states (i.e., a constant DOS ν(ε) ≈ ν 0 , extending from −W c /2 to W c /2, where W c is the itinerant electron bandwidth).For a more generic DOS, the corresponding correction to this approximation is of the order of |Σ < |/W c .The QBE is valid when this correction is small, namely, when the scattering rate is small compared to the Fermi energy, similarly to the standard Boltzmann equation.(ii) Note that in order to integrate over ε k in the collision integral, we have used the fact that the scattering rate depends weakly on ε k at low energies.This is consistent with the fact that the internal frequency, ν, is always restricted to be of the order of T , given that the external frequency ω ∼ T and the system is close to thermal equilibrium.

Variational formulation
A direct solution of this QBE is clearly a non-trivial task.Instead, we will compute the resitivities via a variational formulation of the QBE.The validity of the WF law at T → 0 essentially follows from the fact that the elastic scattering term, I dis , dominates the inelastic term I cf , similarly to the conventional QBE description of FLs.To demonstrate this, we linearize the QBE in a manner that allows us to utilize a variational formulation of the QBE along the lines of 12,57 .We parameterize the deviation from equilibrium with the function ϕ k, ω, r, t , such that the full distribution function is approximated as follows, The local equilibrium distribution f 0 nullifies the collision integrals by definition.Also note that f 0 could depend on space and time via local temperature 1/β (r, t) or chemical potential µ (r, t) 57 .Using the form (62), and following the steps in Ref. 57 , one can show that the linearized QBE in the presence of a uniform electric field E, which we will consider for the computation of the electrical resistivity, is given by And similarly for an applied uniform thermal gradient, the linearized QBE we will consider for the computation of the thermal resistivity is given by Remarkably, the form (62) enables us to relate the thermal and electrical resistivities to a variational problem in the function ϕ.Specifically, the physical values of ρ and ρ th correspond to ϕ that minimizes the functionals (a = el, th) where the disorder and cf -interaction contributions to the resistivities are denoted by F dis and F cf , respectively.The derivation of the variational formulation is analogous to the one in Refs. 12,57.We therefore state the final expressions for the resistivities.To do so, we must introduce an inner product defined as for some functions g and h.In addition we define the operators P a as with P a being the equilibrium transitional rates related to the different scattering mechanisms.Using these definitions, the electrical resistivities are conveniently given by Here, where the factor of 2 comes from the equal contribution of an emission and absorption of a 'Π f -boson' 57 , and Anticipating the dominance of elastic scattering as T → 0, we may use the variational ansatzes: where u (v) denotes a unit vector in the direction of the electrical (heat) current and η is a small parameter.By inserting the ansatzes (72) into Eq.( 67) with the above definitions, we can explicitly see that the validity of the WF law at T → 0 follows from the dominance of the T -independent elastic scattering contribution, F dis , over the inelastic term, F cf .In addition, a simple power counting (as in Eq. 34) shows that the scaling form of Π f leads to T -linear resitivities.In total, we confirm that and similarly ρ th = ρ th,0 + BT .Importantly, Eq. ( 73) captures the physical leading low-T behavior, rather than serving as an upper bound.Indeed, consider leading correction to ϕ el dis , δϕ el , such that the minimizer of the resistivity functional is ϕ el = ϕ el dis + δϕ el .Then, by expanding the right-hand-side of Eq. ( 73) in δϕ el and using the fact that ϕ el dis minimizes F el dis [ϕ el ], we obtain that the contribution due to δϕ el is subleading.In practice, ϕ dis is expected to minimizes the full functional F in the case of a local cf -interaction.
So far, the QBE provided a simple perspective for the validity of the WF law in the case of local cf -interactions, i.e., Υ r,r ′ = δ r,r ′ (such that Υ = 1), for which Π f is completely uniform in momentum space.But in fact, since the key ingredient in our derivation was the PK reduction scheme, the above discussion can be generalized to a class of deformed models with spatially extended cf -interactions.See the main text (MFL section) for an example.Indeed, under such deformations, the scattering off of f -fermions obtains a momentum dependence, Π f (ν) → Π f (q, ν) ≡ Υ (q) Π f (ν).However, as long as this momentum dependence is not singular, namely, it can be written as Υ (q) ∼ 1 + ηh(q) with a sufficiently small η and smooth h, the low-energy MFL form of the self-energy of the c-fermions does not change (as demonstrated above).And most importantly, since the PK reduction scheme is valid, we can repeat the analysis above.It thus follows that this class of deformations, and in particular the example presented in the main text (MFL section), also obeys Eq. ( 5) in the main text.
For completeness, let us note that the main effect of spatially extended cf -interactions is to change the nonuniversal prefactor of the leading −T term in L(T ) − L 0 .That is, it modifies the coefficients A and B of the resistivities above.For the electrical resistivity, this is due to the suppressed contribution of small angle scattering events.This can be seen explicitly by noticing that, in this case, ϕ is frequency independent [see Eq. (72)] and the momentum and frequency integrations in Eq. (69) factorize such that For Υ = 1 we retrieve the familiar 1 − cos θ k,k ′ weighting factor, while any momentum dependence in Υ will change the weighting and hence the overall prefactor.The T -scaling is determined by the frequency intergrals and is hence unaffected.Similar, yet more involved, consideration can be applied to the thermal resistivity 12 .The validity of PK reduction was the key ingredient to the derivation above, namely, the same analysis can be applied to generic weakly disordered MFLs (and NFLs), provided that the momentum-dependence of the inelastic scattering rate is sufficiently weak (which enables the PK reduction).Furthermore, this means that the low-T deviation from the WF law (Eq.( 5) in the main text) is not a fine-tuned feature of our model, and could serve as a generic criterion for strangeness, as claimed in the main text.
Lastly, notice that by spatially extending that range of Υ we reduce the characteristic momentum transfer to a narrower region in the Brillouin zone.Denoting the radius of this region by q * , the validity condition for the PK reduction can be roughly estimated as v F q * ≫ Γ.It is interesting that by further extending the range of Υ, such that v F q * ∼ Γ, we effectively invalidate the PK reduction scheme.The PK reduction scheme cannot be consistently used in this case.This scenario is reminiscent of the MFL model studied in 58 .

Generalization to the transverse Lorenz ratio
We obtain the transverse conductivities by a solving the QBE in the presence of a magnetic field for a local cf −interaction (introducing extended cf −interaction does not change the qualitative physical picture).The reason for this alternative derivation is that the variational principal is not directly valid in the presence of a magnetic field since the operators P a are not self-adjoint 12 .Physically, this is related to the fact that time-reversal symmetry is broken such that the probability of a scattering of an excitation ( k, ω) to an excitation ( k′ , ω ′ ) is not equal to the probability in the reverse direction.Nevertheless, the direct solution of the QBE is shows that the transverse Lorenz ratio follows the same T −scaling is the longitudinal one provided that the PK reduction holds.
The introduction of a magnetic field follows the same steps as in the case of an electric field.Specifically, we introduce an electromagnetic vector potential via minimal coupling, ε k → ε k+A where A = A 1 + A 2 such that −dA 1 /dt = E and ∇ × A 2 = B.Then, the QBE is obtained in a similar fashion to the derivation above, where the introduction of electromangetic fields is done via a change of the COM coordinates k → k + A; see e.g. 66.The QBE then takes the form Df c k, ω, r, t = I coll f c k, ω, r, t where Similarly to the conventional Boltzmann equation, the magnetic part of the Lorentz force nullifies the equilibrium distribution and acts nontrivially only on the non-equilibrium piece.
To proceed, we insert the parametrization of the full distribution, f c ≡ f 0 + δf , into I coll .Then, we use (a) the fact that by definition I coll [f 0 ] = 0 67 , and (b) the fact that δf (− k) = −δf ( k), which implies that the terms proportional to δf ( k′ ) vanish in the integration over k′ .It is then straightforward to express the collision integral in terms of the self-energy of the c−fermions: Finally the QBE for in the presence of an electric and magnetic field is given by The QBE in the presence of a small thermal gradient is identical to the above with the replacement E → βω∇ r T .The solution of the QBE is obtained by inserting the ansatz 12 δf k, ω = k F k • δf (ω), which yields (in agreement with 29 ) with F j = E j , βω∂ j T for an applied electric field and thermal gradient, respectively (the magnetic field is assumed to point along ẑ, i.e. out of the plane), and ϵ ij is the antisymmetric tensor in two dimensions.We recall that, in our model, by definition the (expectation values of the) electrical and thermal currents (per flavor) are given by and We can therefore obtain the transverse electrical and thermal conductivities, and where the cyclotron frequency is given by ω c = (v F /k F ) B.
We can now explicitly see that L xy − L 0 ∝ −T in agreement with Eq. ( 5) in the main text.For example, in the simplest case where the magnetic field is sufficiently small (ω c ≪ Γ), the conductivities obey the relation α xy (T ) = (ω c /2Γ)α xx (T ) (α = σ, κ) to leading order in ω c , which automatically guarantees the desired behavior.
Note also that even when Γ → 0 the WF law is obeyed (ω c takes the role of the disorder term) but with a deviation that scales as T 2 .
Analogously to the longitudinal case, the leading T -scaling of the transport rates is governed by the form of ImΠ R f (ν) in I cf , which is unaffected by the introduction of extended interactions.Hence, as in the longitudinal case, spatially extended cf -interactions changes the prefactor of the −T term in L xy (T ) − L 0 , but not its scaling form.More generally, the T -scaling remains unchanged as long as the inelastic scattering mechanism has sufficiently weak momentum dependence.This is exactly the validity condition of the PK reduction.We thus observe that, similarly to the discussion on the longitudinal Lorenz ratio, our conclusion holds for models of weakly disordered MFLs (of NFLs) where the PK reduction scheme can be applied.In other words, the leading deviation of the transverse Lorenz ratio satisfies the same generic behavior as the longitudinal Lorenz ratio, provided that the PK reduction is valid.

Figure 2 .
Figure2.The Lorenz ratio as a function of temperature for the MFL model (7) with local cf -interaction.Γ is the elastic scattering rate.The inset shows the limit T ≪ Γ which obeys Eq. (5).